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Tunneling effect at semiconductor/oxide interfaces

Florius — Sun, 05 May 2024 19:52:49 +0000

1. Step potential: E > V

Electron tunneling is a phenomenon commonly observed at semiconductor/insulator or metal/insulator interfaces, particularly when the insulating layer is thin, typically a few nanometers thick. To grasp its intricacies, let’s simplify the scenario. [latexpage]

We begin by considering a 1D barrier problem for a stream of particles. The current density ($J$), relating to the particle density, can be expressed as:

$\frac{\partial\rho}{\partial t}+\nabla\cdot\textbf{J} = 0\ ,$

where $\rho$ denotes the particle density. If we conceive $\psi$ as the particle beam, then $\mid\psi\mid^2$ also represents the particle density, thus linked to the current $J$.

When examining a semiconductor/insulator/semiconductor interface, we can model it with a rectangular potential comprising three regions, as is shown in Figure 1. Regions 1 and 3 exhibit zero potential, while region 2, ranging from $(-a; a)$, features a finite potential $V$. Here, $V$ remains constant but nonzero.

In classical terms, imagine kicking a football with sufficient kinetic energy—it flies over the barrier. Similarly, in quantum physics, when the energy $E$ vastly exceeds $V$ as is shown in Figure 1, the transmission coefficient nears 1, indicating seamless passage across the boundary. However, as $E$ approaches $V$, this smooth transition is no longer true. When they align, the transmission coefficient oscillates between 0 and 1. Notably, only when the barrier width ($2a$) precisely matches an integral number of half wavelengths, does the barrier become transparent to incoming electrons. In the following section, we delve deeper, exploring the scenario when energy falls below the barrier, necessitating tunneling through it.

2. Step potential: E < V

Continuing from the preceding section, we now examine scenarios where the energy level dips below the potential threshold, illustrated in Figure 2. In classical terms, this situation is similar to tossing a ball, only to have it fall short of clearing the obstacle and rebounding back. But this classical analogy no longer holds at the quantum level.

Similar to earlier, regions 1 and 3 maintain their oscillating nature, governed by a combination of oscillating wavefunctions of an electron. However, in region 2 the particle beam encounters the barrier, it undergoes absorption, characterized by a mix of exponential functions. When an incoming wave with a certain energy enters region 2, its signal diminishes exponentially as it approaches zero upon reaching the boundary condition $x=a$. Yet, under specific conditions -such as sufficient energy or a thin barrier – the wave function doesn’t fully extinguish at $x=a$. Instead, there exists a probability, albeit small, for the beam to traverse the barrier, persisting with an oscillatory behavior. Figure 3 shows an illustration of how this looks like.

Within the potential barrier, a reflecting wave emerges at the boundary $x=a$ with an amplitude $D$. This phenomenon, too, follows an exponential decay pattern as it succumbs to absorption. It’s worth noting that while my description may imply a temporal aspect, the solution to the problem remains time-independent.

The electron wave function is obtained by solving the one-dimensional (time-independent) Schrödinger equation

$\left[ \frac{\hbar^2}{2m^*}\frac{\partial^2}{\partial x^2} + V(x)\right]\psi (x) = E\psi (x)$

where $V(x)$ has two conditions, it is either

$V(x)=0$ when $x<-a$ and $x>a$ or,
$V(x)=V_0$ when it falls between $-a

The solution of $\psi$ in the different regions of space is given by:

$A\textrm{exp}(ikx) + B\textrm{exp}(-ikx)$
$C\textrm{exp}(\gamma x) + D\textrm{exp}(-\gamma x)$
$E\textrm{exp}(ikx)$

Where $A$ is the amplitude of the incoming wave on the left side, and $E$ is the amplitude of the outgoing wave on the right side, furthermore

$k=\frac{\sqrt{2m^* E}}{\hbar}\ ,$

$\gamma = \frac{\sqrt{2m^*(V_0-E)}}{\hbar}$

2.1 Transmission coefficient

The transmission coefficient is then defined as the ratio between the transmitted and incident probability flux.

$T = \frac{\textbf{j}_{trans}}{\textbf{j}_{inc}}=\frac{\mid E\mid^2}{\mid A\mid^2}$

To solve it, we have to impose the continuit of the solution and their derivatives at the boundaries.

$\psi_1(0) = \psi_2(0)$ and $\psi_1^{‘} (0) = \psi_2^{‘}(0)$

$\psi_2(0) = \psi_3(0)$ and $\psi_2^{‘}(0) = \psi_3^{‘}(0)$

In the limit $\gamma a >> 1$, corresponding to the weak transmission limit (electron wave length is small compared to the barrier width W):

$T(E)\approx \left(\frac{4k\gamma}{k^2+\gamma^2}\right)^{2}e^{(-4\gamma a)}\propto e^{\left(-2W\sqrt{2m^* (V_0-E)}/\hbar\right)}$

This relationship highlights the exponential decline of the transmission coefficient concerning both the barrier width, $W$, and its height, $$V_0$$ . Consequently, as barriers become thicker, the likelihood of tunneling diminishes exponentially, approaching zero. This behavior is illustrated in Figure 4, which charts the transmission coefficient $T(E)$ $)$ against energy $E$ relative to $$V_0$$ .

For substantial barrier thicknesses (e.g., 10nm), tunneling activity below $E/V_0 = 1$ is extremely limited, with only a faint tail observed below this threshold. Also note that oscillations emerge around $_{$E \approx V_0$}$ , indicating energy levels at which electrons cannot transmit through. Conversely, smaller barriers (e.g., 2nm) exhibit quite a large transmission coefficient below $E/V_0 = 1$, making a substantial tunneling current possible.

3. MOSFET tunneling

in MOS (Metal-Oxide-Semiconductor) devices such as MOSFETs (Metal-Oxide-Semiconductor Field-Effect Transistors), the voltage drop across the SiO₂ layer is controlled by the gate terminal voltage. By applying a specific voltage to the gate terminal with respect to the source terminal, the electric field generated across the SiO₂ layer can be manipulated, affecting the tunneling behavior of electrons through the oxide layer. We can consider two different regimes for tunneling of electrons through an Si/SiO₂interface, depending on the value of the voltage drop $V_{ox}$ across the insulating layer.

Direct tunneling

When the energy potential, $qV_{ox}$, is less than the potential barrier, $\phi_b$, at the Si/SiO₂ interface (approximately 3.2 eV), electrons undergo tunneling through a trapezoidal barrier, known as direct tunneling. Using the Wentzel–Kramers–Brillouin (WKB) approximation, one can derive:

$T_{DT}(E,V_{ox})\approx\textrm{exp}\left[-\frac{4\sqrt{2m^*}}{3e\hbar E_{ox}}\left((\phi_b-(E-E_F))^{3/2}-(\phi_b-(E-E_F)-eV_{ox})^{3/2}\right)\right]$

Fowler-Nordheim tunneling

When $_{$qV_{ox}$}$ exceeds $\phi_b$ , electrons tunnel through a triangular potential barrier, a phenomenon known as Fowler-Nordheim tunneling. Within the WKB approximation, the tunneling probability is described as:

$T_{FN}(E,V_{ox})\approx\textrm{exp}\left[-\frac{4\sqrt{2m^*}}{3e\hbar E_{ox}}(\phi_b-(E-E_F))^{3/2}\right]$

The current density due to the tunneling of electrons at a semiconductor/oxide interface is given by:

$J=\frac{em^*(k_BT)}{(2\pi)^2\hbar^3}\int^{\infty}_{0}T(E_x)ln\left(\frac{1+\textrm{exp}\left((E_F-E_x)/k_BT\right)}{1+\textrm{exp}\left((E_F-eV-E_x)/k_BT\right)}\right)dE_x$

The tunneling current can be calculated (numerically) by using the expressions of the transmission coefficient ($T$) corresponding to the Direct or Fowler-Nordheim tunneling regimes.

A development of the exponential term of the Fowler-Nordheim tunneling probability for $E\approx E_F$ fits with measured data as can be seen in Figure 7 and leads to an approximate expression for the Fowler-Nordheim tunneling current:

$J_{FN}(V)=A\left(\frac{V_{ox}}{t_{ox}}\right)^{2}\textrm{exp}\left(\frac{-Bt_{ox}}{V_{ox}}\right)\ ,$

where:

$A=\frac{e^3}{16\pi^{2}\hbare\phi_b}$
$B=\frac{4\sqrt{2m^*}}{3e\hbar}\phi^{3/2}_b$

3.1 Advancements in nanoelectronics

Scaling down dimensions proved effective during the microelectronics era, from 1μm to 100nm. However, this required the adoption of new materials and modifications to the MOSFET structure itself to remain viable. Silicon, known for its remarkable properties, served as the cornerstone material, offering semiconductor attributes like n-type, p-type, poly-silicon, and silicides, alongside a stable SiO₂ insulator. Scaling the thickness of SiO₂ reached a limit due to escalating gate leakage caused by electron tunneling. The probability of electron tunneling through the SiO₂ layer is represented by $e^{t_{ox}}$, thus, as $t_{ox}$ decreases, undesired leakage current rises. As previously demonstrated, with $W$ representing the insulator’s thickness.

However, the objective isn’t merely to minimize $t_{ox}$ but to maximize $C_{ox}$. To achieve this, novel materials were introduced with a relative permittivity ($\varepsilon_{HK}$) significantly greater than that of SiO2, denoted as “high-$\kappa$ materials”. The premise is to utilize $t_{HK} >> t_{ox}$. This necessitates the introduction of Equivalent Oxide Thickness (EOT) as a distinct concept from $t_{ox}$, enabling the creation of gates with larger oxide thicknesses while maintaining the same oxide capacitance as SiO₂.

$EOT = t_{HK}\cdot (\varepsilon_{ox}/\varepsilon_{HK})$,

where $\varepsilon_{ox} = 3.9$. The advantage of this approach lies in its scalability, allowing for further reduction of EOT.

However, the persistent challenge remained with Si oxidation, which is inherently difficult to prevent at the interface. One of the advantages of SiO₂ lies in its ability to form a robust interface with thermally grown Si, providing effective surface passivation. Any deposition process results in a compromised interface with increased surface states. Hence, even with the adoption of high-$\kappa$ materials, a thin layer of SiO₂ is retained between the high-$\kappa$ material and Si substrate. It’s essential to note that this adjustment slightly alters the capacitance ($C_i=\varepsilon_i/t_i$).

The total capacitance is given by the formula:

$(C_{tot})^{-1}=(C_{HK})^{-1}+(C_{SiO2})^{-1}$,

and the effective permittivity ($\varepsilon_{eff}$) is calculated as:

$\varepsilon_{eff}=t_{tot}\cdot\frac{\varepsilon_{hk}\varepsilon_{SiO2}}{t_{HK}\varepsilon_{SiO2}+t_{SiO2}\varepsilon_{HK}}$.

4. References

[1] S. Vaziri, M. Belete, E. Dentoni Litta, A. D. Smith, G. Lupina, M. C. Lemmea and M. Östling, “Bilayer insulator tunnel barriers for graphene-based vertical hot-electron transistors”, nanoscale, vol. 30, 2015.

[2] Ralph Smeets, “Intel kondigt transistor met high-k metalen gate aan,” Tweakers, Nov. 4, 2003, [Online]. Available: https://tweakers.net/nieuws/29510/intel-kondigt-transistor-met-high-k-metalen-gate-aan.html. [Accessed: May. 11, 2024]

[3] Contact me if you know the source of this figure.

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Giant (GMR) and Tunnel (TMR) magnetoresistance

Florius — Sun, 28 Apr 2024 18:52:04 +0000

Giant magnetoresistance (and later Tunnel magnetoresistance) is one of the biggest discoveries in thin-film magnetism. Within 10 years after its discovery, it was already used in commercial devices, such as hard disk drive read heads, changing the world. Just like other magnetoresistive effects, GMR is about the change in resistivity as a result of an applied magnetic field on it. For this reason, we will talk in this post about $\Delta R/R$ as a measure of how much the resistance changes under such effect. In the first section, I will mainly talk about spin polarized transport, and how metallic freromagnets can conduct electricity. From there we take a jump to GMR, explain how it works, and one of its applications. Lastly we also take a look at another important structure in spintronics, the magnetic tunnel junction, where the layer between the ferromagnets is an insulator, and the only way an electron can cross this is by quantum tunneling.

1. Spin polarized transport

[latexpage] Magnetoresistance is the property of a material to alter its electrical resistance when subjected to an external magnetic field. This phenomenon was initially observed by William Thomson (Lord Kelvin) in 1857 [12]. When he introduced a metal with a current flowing through it to a magnetic field, either perpendicular or parallel to the direction of the current, he noted a marginal change in its bulk resistance, typically less than $\Delta R/R < 5\%$.

The use of spin-polarized currents opens up significantly more intriguing applications in magnetoresistance. Spin-polarized transport naturally occurs in materials where there exists an imbalance in the spin population at the Fermi level. This imbalance is commonly found in ferromagnetic metals, where the density of states (DOS) available to spin-up and spin-down electrons is nearly identical, albeit with a disparity in energy levels.

1.1 The Mott model

In the Mott model it is proposed that the electrical conductivity in metals can be described in terms of two largely independent conducting channels, corresponding to the spin-up ($\uparrow$) and spin-down ($\downarrow$) electrons. Electrical conduction occurs in parallel across these two channels [13].

In non-magnetic conductors, the scattering remains invariant with respect to the electron spin. However, in ferromagnetic metals, the scattering rates of the spin-up and spin-down electrons are different. Typically, it is presumed that the scattering is strong for electrons with spins anti-parallel to the magnetization direction and weak for electrons with spin parallel to the magnetization direction. This translates into different resistivity for spin-up and spin-down electrons, dependent on the magnetization of the material.

Ferromagnetic materials have a majority of electrons in one spin state (typically the up-state by convention). Figure 1 shows the density of states $Z(E)$ as a function of energy $(E)$ for normal and ferromagnetic materials. This schematic diagram illustrates that in ferromagnetic material (on the right), the available majority spin states (up-state) are fully occupied at the Fermi level, whereas this is not the case for the minority spin states (down-state). This phenomenon directly arises from the spin imbalance induced by Hund’s rule.

1.2 Band structure of magnetic materials

In general, in ferromagnetic materials, there is an asymmetry near the Fermi level, with the density of states being greater for spin-down compared to spin-up. Notably, for Ni, Co and Fe, these differences can be observed in the $3d$ orbitals [2]. Among these materials, distinctions exist; for instance, examining the actual density of states for Fe reveals that although both spin-up and spin-down states are present, one vastly outweighs the other. Conversely, for Co, the scenario presents a different variation, with no more spin-up states available, as is illustrated in Figure 2.

Figure 3 (a) shows the band structure of non-magnetic Cu on the left panel, and the density of states on the right panel, the latter is the same for spin-up and spin-down electrons. It is characterized by the fully occupied $d$ bands and the presence of a dispersive $sp$ band at the Fermi energy level, which result in high conductivity of Cu. The electronic structure of ferromagnetic Co is different for the two spin orientations and is characterized by the exchange-split $d$ bands. The Fermi level lies within the $sp$ band for the majority-spin electrons (b), which leads to high conductivity of majority-spin channel. The Fermi level lies, however, within the $d$ band for the minority-spin electrons resulting in low conductivity of the minority-spin channel (c). In the latter case the $sp$ electrons are strongly hybridized with the $d$ electrons, which diminishes their contribution to conduction [3].

1.3 Spin dependent transport in the diffusion limit

The following section goes further into the conceptual idea, rather then rigorous scientific principles. Fermi’s “golden rule” basically states that the scattering probability of an electron depends on the availability of final states for scattering [14]. Consequently, electrons with spin-up and spin-down orientations exhibit different resistivities, because of the difference in available states to scatter to.

The idea of this model is based on the presumption that spin-flip scattering events are much less likely than regular scattering events (see Figure 4), the spin diffusion length is normally much larger than the mean free path of the electrons [4]. With this in mind, we can use the “two channel” conduction model (see Figure 5) to explain the resistance in metallic ferromagnets. Thus, in ferromagnetic metals the scattering, and as a consequence also the resistance, depends on the spin of the electron.

The ability to manipulate spin within metallic structures presents an intriguing way for controlling resistivity. Figure 6 basic configuration involves two magnetic materials separated by a metallic layer, with a current flowing perpendicular to the plane (CPP). When both ferromagnets (left and right) exhibit identical spin orientations, electrons encounter many available states to traverse from the first ferromagnet through the metal and onto the second ferromagnet, resulting in low resistance. Conversely, for the opposite scenario, while electrons can easily transition from the first ferromagnet to the metal; the absence of available states in the second ferromagnet with the correct spin spin orientation lowers the chance of electron transmission, thereby giving higher resistance [16]. However, CPP devices are generally not fabricated in this way.

2. Giant Magnetoresistance (GMR)

A much more interesting structure makes use of the giant magnetoresistance (GMR) effect in thin magnetic multilayers. This has less to do with the density of states, but more with the scattering mechanism. It was first discovered by Fert et al.[5] that applying a magnetic field to a Fe/Cr multilayer resulted in a reduction of the resistance of the multilayer, with $\Delta R/R \approx 50\%$. The effect was much larger then the ordinary magnetoresistance, and for that reason it was called the “giant magnetoresistance”. Figure 7 shows the drop in resistance for different sets of layer thicknesses as a function of applied magnetic field. Around the same time, a similar discovery was made in Fe/Cr/Fe trilayers by another group under supervision of P. Grunberg [15], albeit with smaller change differences. Due to the importance of these discoveries, both A. Fert and P. Grünberg received the Nobel prize in physics in 2007 (see Figure 8) [6].

By making very thin layers, you ensure that the mean free path of your electron is much larger than the thickness of the layer. If you inject the current, it will scatter into various layers. When the magnetic moment of the two magnetic layers are parallel, the conducting electrons that carry the majority of the current in both layers have the same spin orientation. Since the thickness of the GMR stack must be smaller than the electron mean free path, the conducting electrons move through both layers.

When the magnetic moments are anti-parallel, a majority spin electron in one layer becomes a minority spin electron in the other layer. Every time a majority spin electron in one magnetic layer crosses the non-magnetic material interlayer and enters the other magnetic layer, it is scattered near the interface. This extra scattering near the interface is the reason for the increase in resistance of the anti-parallel state.

A very simple resistor model is useful as a starting point for understanding the origin of the GMR. In this model, each metallic layer (and each interface) is treated as an independent resistor.

If the mean free path is short compared to the layer thickness, then each layer conducts the electric current independently, and the resistors should be added in parallel. The resistance of the parallel and antiparallel configurations are the same, and consequently, the GMR is zero.
If the mean free path is sufficiently long for the electrons to propagate across the spacer layer freely, GMR can be observed. When the mean free path is long compared to the layer thickness, the probability of scattering within the multilayer is the sum of scattering probabilities within each layer and interface. Therefore, within a given spin channel the total resistance is the sum of resistances of each layer and each interface, i.e. the resistors are connected in series.

Within each ferromagnetic layer the electron spin can be either parallel or antiparallel to the magnetization direction. In the former case, the electron is locally a majority spin electron and in the latter case a minority spin electron. The majority- and minority-spin resistivity of the ferromagnetic layer are different and are equal to $\rho_\uparrow$ and $\rho_\downarrow$ respectively.

2.1 GMR read head / Spin valve

One of the primary applications of GMR technology is in the read heads of hard disk drives, which utilize a spin valve design (more common now when talking about TMR). This device comprises thin layers of ferromagnetic/non-magnetic/ferromagnetic metals, illustrated in Figure 9 as a stack of two blue layers sandwiching a yellow layer. Typically, the top layer’s magnetization is fixed, or “latched,” by a fourth layer, often an antiferromagnet. The conducting layer could be a metal like Fe, as described in the previous section, while the ferromagnetic “sensing” layer is highly responsive to magnetic fringe fields.

In the magnetic recording strip, GMR read heads detect changes in magnetic orientation, typically indicated by transitions between areas of opposing magnetization. These transition zones, known as domain walls, mark where magnetization flips direction. Depending on the orientation of the fringe field encountered, the sensing layer aligns either parallel or antiparallel to the fixed layer. This change in alignment induces a change in resistance, as depicted in the preceding figure, which can then be measured by running current through it.

3. Tunneling Magnetoresistance (TMR)

Magnetic tunnel junctions (TMJ) are quite similar to what we discussed in GMR, but these devices consist of two ferromagnetic layers separated by an insulating layer, instead of a metal. The insulating barrier (e.g. aluminum oxide, typically 1-2 nanometers thick) allows an electric current whose magnitude depends on the orientation of both magnetic layers to tunnel through the barrier when subjected to an electric bias. For more in-depth details on tunneling, see my post on tunneling effects. Some initial results with $Al_2O_3$ as insulating barrier gave $\Delta R/R = 10\%$, while larger differences have been measured with MgO, up to 250% and higher.

The tunneling current injected from $M_1$ to $M_2$ is give by the following:

$J_{tunnel} \propto T(E,V)f(E_{F, M1})[1-f(E_{F,M2})]\ ,$

where $T(E,V)$ is the tunnel probability, which depends on the barrier height and thickness of the layer, $f(E_{F, M1})$ are the states in $M_1$ and lastly, $[1-f(E_{F,M2})]$ are the empty states in $M_2$.

Figure 10 shows two configurations. On the left it shows when the magnetic moments of the two ferromagnetic layers are parallel; and spin-down states can be injected from the cathode ($M_1$), and occupy empty spin-down states in the anode ($M_2$). On the right you see the configuration, when the magnetic moments of the two ferromagnetic layers are anti-parallel; there are no empty states with spin-down available at the anode ($M_2$), and the tunneling current is much reduced.

The magnetic tunnel junctions thus allow the injection of electrons with a specific spin orientation, and the transport across these junctions is called spin transport. These structures are also called spin valves.

The orientation and strength of the applied magnetic field has a large effect on the functioning of TMR. Figure 11 illustrates the resistance outcomes that depend on whether the field surpasses or falls below $H_{max}$—a crucial threshold denoting where even the hard magnet succumbs to the external field’s influence.

Within this structure, CoFe ($M_1$) is a hard magnetic material, such that it keeps the orientation of its magnetization under the applied magnetic field, for $HH_{max}$, both the magnetization of $M_1$ and $M_2$ follows the orientation of $H$, and they are always parallel, explaining the low resistance observed in this case.

As a side note, in a more general tunneling model, both majority and minority spins have to be taken into account, and more information can be found in the references [10] and [11].

3.1 Improved tunnel barriers

The efficiency of a magnetic tunnel junction can be increased $\Delta R/R = 400\%$, by using epitaxial (oriented) barriers, such as MgO.

Wang et al. [17] researched the following. The transmission probability of the Bloch electrons (electrons in a crystal lattice) in an epitaxial ferromagnet / insulator / ferromagnet structure depends on the lateral symmetry of the wavefunctions. This symmetry filtering effect is converted into a large spin filtering effect if the wavefunction with the preferred symmetry only exists in one of the two spin channels in the ferromagnetic electrodes.

Such a spin filtering effect was first theoreticall predicted to give rise to a very large tunneling magnetoresistance in magnetic tunnel junctions, followed by the experimental realization in Fe/MgO/Fe, CoFe/MgO/CoFe and CoFeB/MgO/CoFeB MTJs.

The TMR in CoFeB/MgO/CoFeB junctions is developed through the so-called solid state epitaxy process during annearling. The MgO barrier fabricated by RF sputtering has a very strong (001) texture when deposited on the amorphous CoFeB bottom electrode. During the post-growth thermal annealing, the top and bottom interfaces of the highly (001)-oriented MgO layer serve as templates for the crystallization of amorphous CoFeB ayers in the (001) orientation, thus forming the out-of-plane epitaxial CoFeB(001) / MgO(001) / CoFeB(001) sandwich structure.

Coherent tunneling through single crystal MgO tunnel junctions results in a large increase in the magnetoresistance effect. Different attenuations for spin-up and spin-down electrons due to symmetry matching between metal and MgO states predicted up to $\Delta R/R =6000\%$!!

4. References

[1] X. Yao, Q. Duan, J. Tong, Y. Chang, L. Zhou, G. Qin, and X. Zhang, “Magnetoresistance Effect and the Applications for Organic Spin Valves Using Molecular Spacers.” Materials, vol. 11, 721, 2018.

[2] M. Häfner, J. K. Viljas, D. Frustaglia, F. Pauly, M. Dreher, P. Nielaba, and J. C. Cuevas, “Theoretical study of the conductance of ferromagnetic atomic-sized contacts” Phys. Rev. B, vol. 77, 104409, 2008.

[3] E. Tsymbal, and D. Pettifor, “Perspectives of Giant Magnetoresistance”, Solid State Physics, vol. 56. pp. 113-237, 2001.

[4] T. Valet, and A. Fert, “Theory of the perpendicular magnetoresistance in magnetic multilayers”, Phys. Rev. B, vol. 48, 10, 1993.

[5] M. N. Baibich, et al., “Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattices, Phys. Rev. Lett., vol. 61, 21, 1988.

[6] The Nobel Prize in Physics 2007. NobelPrize.org. Nobel Prize Outreach AB 2024. Thu. 2 May 2024.

[7] D. Andelman, and R. E. Rosensweig, “The Phenomenology of Modulated Phases: From Magnetic Solids and Fluids to Organic Films and Polymers”, Series in Soft Condensed Matter, pp. 1-56, 2009.

[8] A. Schuhl, and D. Lacour, “Spin dependent transport: GMR & TMR”, Comptes Rendus. Physique, vol. 6, pp. 945-955, 2005.

[9] Contact me if you know the source of this figure.

[10] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, “Large Magnetoresistance at Room Temperature in Ferromagnetic Thin Film Tunnel Junctions”, Phys. Rev. Lett., vol. 74, 16, 1995.

[11] M. Julliere, “Tunneling between ferromagnetic films” Physic Letters, vol. 54A, 1975.

[12] W. Thomson, “XIX. On the electro-dynamic qualities of metals:—Effects of magnetization on the electric conductivity of nickel and of iron” Proceedings of the Royal Society of London, 1857.

[13] H. Adachi, “Back to basics”. nature Physics, vol. 11, pp. 707-708, 2015.

[14] “Scattering and Decays from Fermi’s Golden Rule including all the $\hbar$’s and $c$’s, (originally by Direc & Fermi), obtained from: https://web2.ph.utexas.edu/~schwitte/PHY362L/QMnote.pdf.

[15] G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, “Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange”, Phys. Rev. B, vol. 39, 4828, 1989.

[16] S. Zhang, “Electric-Field Control of Magnetization and Electronic Transport in Ferromagnetic/Ferroelectric Heterostructures”, Springer, Berlin, https://doi.org/10.1007/978-3-642-54839-0_1

[17] Wang, W.G. et al. “Understanding tunneling magnetoresistance during thermal annealing in MgO-based junctions with CoFeB electrodes.” Phys. Rev. B, vol. 81, 144406, 2010.

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Spin-Transfer Torque: An Introductory Overview

Florius — Thu, 25 Apr 2024 18:35:06 +0000

1. Introduction

Next to the charge, electrons also posses spins. In normal electronic circuits it is of (hardly) no use, as they are orientated randomly in non-ferromagnetic materials. However, when we integrate ferromagnetic components into these devices, the itinerant electrons can become partially spin polarized, with their spins taking on a more ordered configuration. The interaction between spins of electrons and ferromagnetic materials lead to spin-based effects, influencing the flow of current based on their magnetization orientation. Moreover, these interactions enable electron spins to exert influence on the orientation of magnetizations themselves. This phenomenon is termed Spin-Transfer Torque (STT).

[latexpage]Berger and Slonczewski [1][2] demonstrated the theoretical feasibility of utilizing a spin-polarized current to influence local magnetization of a ferromagnet. This idea revealed the potential for inducing steady-state precession or even complete reveral of the magnetization solely through spin transfer. The method of magnetization control through electrical currents, rather than magnetic fields presented a lot of possibilities for practical applications.

The concept of spin-transfer torque has gotten a lot of interest and has been used in many laboratories and actual spintronic devices. Such devices include metallic/ferromagnetic multilayers (GMR effect) and magnetic tunnel junctions (TMR effect) when the spacer between the ferromagnetic material is an insulator (or semiconductor). Despite their differences, they have in common that their cross-sectional area is in the sub-micron range. This small size is crucial, as spin-transfer torque only works under very high current densities, in the order of $10^8$ A/cm$^2$ [3].

2. Spin-Transfer Torque

2.1 Macroscopic spin mode

Take the model in Figure 1 as example. When a spin current is (spin-)filtered by one ferromagnetic layer, is passing through another magnetic layer, whos magnetization is not aligned with the first layer; the second magnet absorbs a portion of the spin angular momentum that is carried by the electron spins. Thus, the magnetization of the ferromagnet changes the flow of the angular momentum of the spin by exerting a torque on the spins to reorientate them. But due to conservation of momentum, the flowing electrons must also exert an equal and opposite torque on the ferromagnet. This torque from the polarized electrons onto the ferromagenet is called a spin-transfer torque.

To explain the principles of STT, we start with a macrospin model, containing two ferromagnetic layers, labeled $F_1$ and $F_2$, separated by a spacer (this can be metallic, insulator or semiconductor) [4]. The magnetizations of the two ferromagnets are misaligned by an angle $\theta$, as shown in Figure 2. An electric current is passing through it, so that the electrons flow from $F_1$ to $F_2$. The electrons that flow into $F_1$ are spin polarized along the direction of magnetization ($M_1$), of this ferromagnetic layer. Similarly, electrons flowing through $F_2$ are polarized along $M_2$.

Now let’s consider an electron in the ferromagnetic layer $F_1$ with a magnetic moment $\vec{\mu}_1$ parallel to $M_1$, and another electron in $F_2$ with a magnetic moment $\vec{\mu}_2$ parallel to $M_2$. As they are not the same, when it is being moved through the total structure, some of this magnetic moment has to be transfered to the system per time unit. As the magnetization vector has a fixed magnitude (length), any variation in time has to be perpendicular to $\textbf{M}$. That means that only the component perpendicular to $\textbf{M}$ of the magnetic moment can be transfered to the total bulk magnetization. This total transfered moment ($T$), is composed into two parts, $T_1$ and $T_2$. The part $T_1$ is the magnetic moment $\vec{\mu}_1$ that is transfered to the magnetization $M_1$.

The $\vec{\mu}_1$ that is transfered to $M_1$ is a vector orthogonal to $M_1$, and lies in the plane created by $M_1$ and $M_2$. The vector that is orthogonal to both $M_1$ and the plane, is $\vec{M}_1 \times \vec{M}_2$, as shown in Figure X, by coming into the plane. Hence, $T_1$ and $T_2$ can be written as

$\vec{T}_{1,2} = T_{1,2}\vec{M}_{1,2}\times(\vec{M}_1\times\vec{M}_2) \ ,$

where $T_{1,2}$ is the length of the torque, and the vector product $\vec{M}_{1,2}\times(\vec{M}_1\times\vec{M}_2)$ creates a vector perpendicular to $\vec{M}_{1,2}$ to the right. This torque was dubbed by Slonczewski as “pseudo-torque”, but now goes by the name of spin-transfer torque [4].

2.2 At the microscoping level

To explain it, we use a 1-dimensional model [5] of an interface between a normal metal and a ferromagnet, as illustrated in Figure 3. We assume a single spin-polarized electron moves in the $x$-direction, incident on a magnetic layer whose magnetization is in the $z$-direction. This electron has the wavefunction [6]:

$\psi_{in} = \frac{e^{ikx}}{\sqrt{\Omega}}\left(cos(\theta /2)\mid\uparrow\rangle + sin(\theta /2)\mid\downarrow\rangle\right)\ ,$

where $\Omega$ is a normalization volume.

We will use the Stoner-model approach to describe the ferromagnetic layer [6]. In this model, we assume that the electrons in the ferromagnet experience an exchange splitting ($\Delta$), which moves the states of the minory-spin carrier band ($\mid\downarrow\rangle$) to a higher energy, compared to the majority-spin carrier band ($\mid\uparrow\rangle$). It is further assumed that electrons move independently of one another and are not significantly influenced by the atomic cores or lattice structure of the material, thus electrons have a free-electron dispersion. Electrons scatter from the energy barrier at position $x=0$, which has different heights for spin-up and spin-down electrons. To make it easier, we shift everything so that for spin-up the height of the potential energy is 0, while for spin-down it is $\Delta$, as is shown in Figure 4. They calculated, by matching wavefunctions and their derivatives at the interface ($x=$ 0), the spin current density of the transmitted and the reflected spin currents [6]:

$\textbf{Q}_{trans} = \frac{\hbar^2}{2m\Omega} \textrm{sin}(\theta)k \textrm{cos}[(k_\uparrow -k_\downarrow) x]\vec{\textbf{x}} -\frac{\hbar^2}{2m\Omega} \textrm{sin}(\theta)k \textrm{sin}[(k_\uparrow – k_\downarrow) x]\vec{\textbf{y}}+\frac{\hbar^2}{2m\Omega} \left[k \textrm{cos}^2(\theta /2)-k_\downarrow\left(\frac{2k}{k+k_\downarrow}\right)^2 \textrm{sin}^2(\theta /2)\right]\vec{\textbf{z}}$

$\textbf{Q}_{refl} = \frac{\hbar^2}{2m\Omega}k\left(\frac{k-k_\downarrow}{k+k_\downarrow}\right)^2 \textrm{sin}^2(\theta /2)\vec{\textbf{z}}$

where $\Omega$ is a normalization volume, $\hbar$ is the reduced planck constant, $\theta$ is the angle of the spin orientated in the $\vec{\testbf{x}}-\vec{\testbf{z}}$ plane with respect to the $\vec{\testbf{z}}$-direction, $k$ is the wavevector, and describes the spatial frequency and direction of propagation of the wave. For these calculations $k_\uparrow = k$ and $k_\downarrow < k$.

With these calculations we can observe a few interesting facts. Firstly, it shows that in the reflected spin current density, there is no transverse component (meaning no $\vec{\testbf{x}}$- and $\vec{\testbf{y}}$-direction). That means that all of the transverse components of the incident spin current density is being transmitted into the ferromagnet. This result follows from the fact that in the Stoner-model, the amplitude of the potential energy barrier for spin-up is zero, while for spin-down it has a finite amplitude, meaning that the reflected part is purely spin-down. Even with this simplistic model, there are many combinations of metal/ferromagnet materials where this is a reasonable approach. As an example Cu/Co, Cu/Ni, or Cr/Fe have a reflection magnitude of almost zero [7][8][6].

The second interesting fact is what happens to the transverse components ($\vec{x}, \vec{y}$) of the spin current density ($\textbf{Q}_{trans}$) once it is inside the ferromagnet. The oscillatory term in those components represent the precession of the spin around its $\vec{z}$-axis as a function of its position $x$ (how far it penetrates the ferromagnet).Because spin-up and spin-down have a difference between $k_\uparrow$ and $k_\downarrow$, they don’t travel at the same speed. This induces a phaseshift between these them. In other words, the spin precesses around the local magnetization with very small spatial prpecession period, in the order of a few atomic spacings. That means that electrons crossing the interface from different directions, follow different paths into the ferromagnet, and the precession of all of them combined, rapidly averages to zero due to classical dephasing. Thus, within 1-2 nm, the spins that crossed the interface will have transfered their momentum to the local magnetization of the ferromagnet[6][8][9].

2.3 Reflected spin current

What would happen if you had a F/M/F stack as shown in Figure 5, and you have a reflecting spin current from our original model in Figure 4. What we used to call reflected spin current, is now incoming spin current for the left ferromagnet. When these electrons enter the ferromagnet, they will start to precess around the local field of that ferromagnet, and transfer their transverse momentum to it. So keep in mind that reflecting electrons are responsible for torque on the left ferromagnet as well [10].

The authors do note that spin currents can flow within a device even if there is no net charge current [6]. This means that STT can also be applied to magnetic materials that do not carry a charge current. One way to look at this, is that the incoming spin-polarized electrons penetrate by diffusive motion into the ferromagnetic material, and transfering their spin angular momentum as discussed before. But for this to happen, an equal amount of electrons leave the material that have (on average) a spin component collinear to the ferromagnet due to conservation of angular momentum.

3. References

[1] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B. vol. 54(13), pp. 9353-9358, 1996

[2] J. C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater, vol. 159, pp. 1-7, 1996

[3] M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, M. Sec, V. Tsoi and P. Wyder, Excitation of a magnetic multilayer by an electric current, Phys. Rev. Lett. vol. 80(90), pp. 4281-4284, 1998.

[4] J. C. Slonczewski, Currents, torques, and polarization factors in magnetic tunnel junctions, Phys. Rev. B. vol. 71(2), 024411, 2005.

[5] M. D. Stiles, and A. Zangwill, Anatomy of spin-transfer torque, Phys Rev. B. vol. 66, 014407, 2002.

[6] D. C. Ralph, and M. D. Stiles, Spin Transfer Torque, J. Magn. Magn. Mater., vol. 320(7), pp. 1190-1216, 2008.

[7] M. D. Stiles, Spin-Dependent Interface Transmission and Reflection in Magnetic Multilayers, J. Appl. Phys., vol. 79, 5805, 1996.

[8] K. Xia, P. J. Kelly, G. E. W. bauer, A. Brataas, and I. Turek, Spin torques in ferromagnetic/normal-metal structures, Phys. Rev. B, vol. 65, 2020401, 2002.

[9] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Calculating scattering matrices
by wave function matching, Phys. Rev. B, vol. 71, 054420, 2005

[10] C. Baraduc, and M. Chshiev, Introduction to spin transfer torque, Nanomagnetism and Spintronics, pp. 173-192, 2010.

The post Spin-Transfer Torque: An Introductory Overview appeared first on Your studyhack.

Micromagnetics – An overview

Florius — Fri, 19 Apr 2024 22:26:35 +0000

1. Micromagnetics

[latexpage]Micromagnetics is a field in physics that deals with the behaviour of magnetics at a sub-micrometer dimension. This theory is based on the assumption that the length of the magnetization vector is constant, and that the energy varies slowly at the atomic scale. This will break down when approaching the atomic size, but it is suitable for resolving magnetic structures, such as domain walls, vortices and more. Due to this assumption of constant magnetization length, it does not work for temperatures far from the Curie temperature. Its principles were first outlined in 1949 by William Fuller Brown.

It all resolves around minimizing he energy by, for example, solving the dynamic equation of motion. The dynamics of the magnetization vector in an external field are governed by the Landau-Lifshitz-Gilbert equation [1]:

$\frac{d\textbf{m}}{dt} = -\gamma_0(\textbf{m}\times\textbf{H}_\textrm{eff}) + \alpha\textbf{m}\times\frac{d\textbf{m}}{dt}\ ,
$

where $\textbf{m} = \frac{\textbf{M}}{M_s}$ is the normalized magnetization vector, $\gamma_0$ is the gyromagnetic ratio, $\alpha$ is the Gilbert damping parameter, and $\textbf{H}_\textrm{eff}$ is the effective field. The latter one consists of external magnetic field, and other contributions that will be described in later sections down here. $\textbf{H}_\textrm{eff}$ can be represented as the first derivation of the total energy density with respect to the magnetization.

$\textbf{H}_\textrm{eff} = -\frac{1}{\mu_0 M_s}\frac{\partial\textbf{E}}{\partial\textbf{m}} \,$

where $M_s$ is the saturation magnetization, and $\textbf{E}$ is the total energy of the ferromagnetic element, which includes exchange, magnetic anisotropy, magnetostatic, and Zeeman energy contributions.

Most often, for numerical calculations, the Gilbert equation [2] is converted into the following form:

$
\frac{d\textbf{m}}{dt} = \frac{1}{1+\alpha^2}\left[-\textbf{m}\times\gamma_0\textbf{H}_\textrm{eff}- \alpha (\textbf{m}\times\textbf{m}\times\gamma_0\textbf{H}_\textrm{eff}) \right]
$

2. Magnetic interactions

The total energy of a ferromagnet with respect to the magnetization depends on many physical effects. Some of them have classical descriptions, such as the Zeeman energy. Other effects have a quantum mechanical origin, and should therefore be converted to the continuous limit of micromagnetism, such as the exchange energy and the anisotropy energy. In this article I will not derive the micromagnetic expressions from the Hamiltonians for all the terms, but will only show the effective magnetic field terms. For full derivations, I will refer you to the PhD dissertation of [3] or on the website of micromagnetics.

2.1 Total effective magnetic field

The dynamics of the system are defined by the effective magnetic fields on each point of the ferromagnet. In the previous section a description on the equations was given, in this equation an effective magnetic field is required that consists of a summation of each one of the fields corresponding with a magnetic interaction,

$
\textbf{H}_\textrm{eff} = \textbf{H}_\textrm{exc} + \textbf{H}_\textrm{ani} + \textbf{H}_{\textrm{DMI}{\bot}} + \textbf{H}_\textrm{zee}\ ,$

where $\textbf{H}_\textrm{exc}$ is the exchange interaction effective field, $\textbf{H}_\textrm{ani}$ is the anisotropy effective field, $\textbf{H}_{\textrm{DMI}{\bot}}$ is the perpendicular DMI effective field, $\textbf{H}_\textrm{zee}$ is the Zeeman term.

These terms were proposed in [4][5], where they derived it by using semi classical approximations. Each of these interactions has an energy associated with it. Through damping, the micromagnetic approach will try to minimize the total energy of these terms. Each of these interactions will be briefly described in the following sections, except the torque. This term should also be added as well into the effective magnetic field, but it will be described in it’s own section.

2.2 Exchange interaction

Ferromagnetism is caused by the exchange interaction between the electrons. The elementary magnetic moments in ferromagnets are subjected to an exchange interaction. This is a quantum mechanical effect that aligns neighboring spins for a positive exchange constant and it creates a uniform magnetization on a macroscopic scale as illustrated in Figure 2. A negative constant would lead to an anti-ferromagnetic state as shown in . In the continuous limit, the exchange interaction is described as an effective magnetic field term with the following equation}:
$
\textbf{H}_{\textrm{exc}} = \frac{2}{\mu_0}\frac{J}{M_s}\nabla^2 \textbf{m}\ ,
$
with $\mu_0$ denotes the permeability of free space, $M_s$ the saturation magnetization, the corresponding exchange interaction constant $J$ and $\textbf{m}$ the normalized magnetic moment.

2.3 Anisotropy

A crystal lattice favours or impedes the magnetization in a certain preferential direction. This effect is called anisotropy and it has its origin in the material spin-orbit interactions [7]. The magnetization will prefer to align itself along the easy axis to minimize the energy. The anisotropy is described in the following term of the effective magnetic field.
$
\textbf{H}_{\textrm{ani}} = \frac{2}{\mu_0}\frac{K}{M_s}\textbf{m}z\ ,
$
where K is the anisotropy constant and the rest of the parameters are the same as above.
Material anisotropy is responsible for permanent magnetization. Soft and hard magnets are characterized by small and large magnetic anisotropy respectively.

2.4 Dzyaloshinskii-Moriya interaction

Dzyaloshinskii-Moriya interaction is an anti-symmetric exchange, as shown in Figure 3, that originates from a combination of large spin-orbit coupling (SOC) and broken inversion symmetry. This chiral interaction is a key aspect of the creation of skyrmions. I. Dzyaloshinskii and T. Moriya [8][9] showed that the DMI between two spins can be expressed with the Hamiltonian:
$
\mathcal{H}_{\textrm{DM}}=-\textbf{D}_{12}\cdot(\textbf{S}_1\times\textbf{S}_2)\ ,
$
where $\textbf{S}_1$ and $\textbf{S}_2$ represent the two spin atoms and $\textbf{D}_{12}$ is the DMI vector. There are two types of DMI, one that can exist in the bulk of certain materials, and the second type is at the interface between a ferromagnet with a heavy metal layer. Both types will be explained below.

2.4.1 Bulk DMI

In a bulk material, the DMI vector is shown in as $\textbf{D}_{12} = \textbf{D}_\textrm{bulk} = D_\parallel\textbf{r}_{12}$, where $D_\parallel$ is the strength of the bulk DMI and $\textbf{r}_{12}$ is the vector between both spin atoms [10]. The bulk DMI originates from the breaking of the inversion symmetry in the non-centrosymmetric lattice of a bulk material itself [11]. It can be seen from Figure 4(a), that this is also called the parallel ($\parallel$) DMI, as it is parallel to the vector between $\textbf{S}_1$ and $\textbf{S}_2$.
The corresponding effective magnetic field term is derived to be [13]:
$
\textbf{H}_{\textrm{DMI}\parallel} = -\frac{2}{\mu_0}\frac{D_{\parallel}}{M_s}(\nabla\times\textbf{m})\ ,
$
where the parameters are the same as described above.

2.4.2 Interfacial DMI

In an interface, the mechanics are somewhat different. The interfacial DMI vector is shown in as $\textbf{D}_{12} = \textbf{D}_\textrm{int} = D_\bot(\textbf{z}\times\textbf{r}_{12})$, where $D_\bot$ is the strength of the interface DMI, $\textbf{r}_{12}$ is the vector between both spin atoms and $\textbf{z}$ is the interface normal direction [10]. In this case, the inversion symmetry is broken at the interface between the ferromagnetic film with a heavy metal layer with a strong SOC [12]. This type of DMI, is also called the perpendicular ($\bot$) DMI, as the vector is along ($z\times r$).
The corresponding effective magnetic field term is derived to be [13]:
$
\textbf{H}_{\textrm{DMI}\bot} = -\frac{2}{\mu_0}\frac{D_{\bot}}{M_s}((\nabla\cdot\textbf{m})z – \nabla \textbf{m}_z)\ ,$

where the parameters are the same as described above.

2.5 External field

In the field of magnetism, in addition to the magnetic induction (\textbf{B}), there is another quantity called the magnetic field (\textbf{H}), that differ on the material response, that produces its own magnetization (\textbf{M}). This gives the following equation:
$
\textbf{B} =\mu_0(\textbf{H} + \textbf{M})\ ,
$
where $\mu_0$ is the permeability of free space and the rest is the same as described above. By substituting \textbf{B} into the LLG equation, the following partial cross product is obtained
$
\textbf{M}\times\textbf{B} = \mu_0\textbf{M}\times\textbf{H} + \mu_0\textbf{M}\times\textbf{M} = \mu_0\textbf{M}\times\textbf{H}\ .
$
Because ($\textbf{M}\times\textbf{M}) = 0$, there is no need to account for the material response $\textbf{M}$. Hence, \textbf{B} is only proportional to $\textbf{H}$ with a constant $\mu_0$. This is called the Zeeman term of the magnetic field ($\textbf{H}_\textrm{zee}$).

2.6 Spin-transfer torque

2.6.1 Spin-polarized current

In a ferromagnet, the spin occupation is imbalanced as shown in Figure 5. For a given energy level, there is more of one spin compared to the other. In this example, the d-band with spin $\uparrow$ is nearly completely filled, while the band with spin $\downarrow$ is nearly empty. Any itinerant electrons with a spin $\downarrow$ will get trapped by the empty states, while the electrons with a spin $\uparrow$ are free to move. This creates a current that has a majority of a certain spin, also known as a spin-polarized current. The polarization is not perfect as there are still free states available in the d-band, but the effect is significantly in favour of the spin $\uparrow$ polarized current.

2.6.2 Spin-transfer torque

In a ferromagnet, the spins of electrons are bound to the atoms of the lattice, whereas in a metal, the spin current of the electrons flows across the material. Spin transfer torque refers to the interaction between the spins of the ferromagnet and those of the metal. Due to the conservation of momentum, this interaction results in torques of equal magnitude but opposite signs acting on these spins.

Adding the torque to Eq. (\ref{eqn:newLL}) results in the following:
\begin{equation}\label{eqn:LLG+torque}
\frac{d\textbf{M}}{dt} = -\frac{1}{1+\alpha^2}\textbf{M}\times\gamma_0\textbf{H}_\textrm{eff}- \frac{1}{1+\alpha^2}\frac{\alpha}{M_s}(\textbf{M}\times\textbf{M}\times\gamma_0\textbf{H}_\textrm{eff}) + \textbf{T}\ ,
\end{equation}
where the torque ($\textbf{T}$) is expressed as [15]:
\begin{multline}\label{eqn:zhang}
\textbf{T}= \frac{1}{1+\xi^2}\left[-\frac{n_0}{M_s}\frac{\delta\textbf{M}}{\delta t} + \frac{\xi n_0}{M_s^2}\textbf{M}\times\frac{\delta\textbf{M}}{\delta t}-\frac{\mu_BP}{eM_s^3}\textbf{M}\right.\\ \left.\times[\textbf{M}\times(\textbf{j}_e\cdot\nabla)\textbf{M}] -\frac{\mu_BP\xi}{eM_s^2}\textbf{M}\times (\textbf{j}_e \cdot\nabla)\textbf{M}\right] \ ,
\end{multline}
where $n_0$ is the local equilibrium spin density, $M_s$ is the magnetization saturation, $\mu_B$ is the Bohr magneton, $P$ is polarization of the spin-polarized current, $e$ is the electron charge, $\textbf{j}_e$ is the current density and $\textbf{M}$ is the magnetization. $\xi$ models the strength of the electronic diffusion in the metal. $\xi$ is zero if the current is transported ballistically and different from zero if electron scattering is present.

2.6.3 Conversion into effective field

According to Zhang and Li [15], the torque consists of four terms, as shown in Eq. (\ref{eqn:zhang}). The first two are from the magnetization variation in time and the other two describe the magnetization variation in space. However, the temporal torques can be absorbed by the redefinition of the gyromagnetic ratio and damping constant and can therefore be ignored. Hence, only the spatial magnetization part of the torque will be used. This term consists of the current density, $\textbf{j}_e$, which is further changed to model the proposed device as will be clear in the next section. The final expression for the spatial torque is the following:
\begin{equation}\label{torquey}
\textbf{T}=\frac{b_j}{M_s^2}\textbf{M}\times\textbf{M}\times(\textbf{j}_e\cdot\nabla)\textbf{M} + \frac{c_j}{M_s}\textbf{M}\times(\textbf{j}_e\cdot\nabla)\textbf{M}\ ,
\end{equation}
where
\begin{equation}
b_j = -\frac{\mu_B P}{eM_s(1+\xi^2)}\ ,\ c_j = \xi b_j .
\end{equation}

Eq. (\ref{eqn:LLG+torque}) can not easily be solved by the numerical integration programs. Instead, the torque can be rewritten as an effective magnetic field through several steps. To do that, the LL equation can be converted back into an LLG expression, following the steps in this post on the LLG equation, but in the reverse order. Then the torque can be rewritten as an effective field and added to the rest. Finally this LLG equation can be transformed back into the LL form to numerically solve it.
\begin{multline}\label{solvingnumer}
\frac{d\textbf{M}}{dt} = -\gamma_0(\textbf{M}\times\textbf{H}_\textrm{eff}) + \frac{\alpha}{M_s}\textbf{M}\times\frac{d\textbf{M}}{dt} + \\ \textbf{M}\times\left[\textbf{M}\times\frac{b_j}{M_s^2}(\textbf{j}_e\cdot\nabla)\textbf{M}+ \frac{c_j}{M_s}(\textbf{j}_e\cdot\nabla)\textbf{M}\right]
\end{multline}
Next, Eq. (\ref{solvingnumer}) is made dimensionless in order to solve it numerically. Through several mathematical steps that are explained in more detail in Appendix \ref{app:torqueintegration}, and with the aid of relationship $\textbf{M} = M_s\textbf{m}$ the following result is obtained:
\begin{multline}
\frac{d\textbf{m}}{dt} = -\gamma_0(\textbf{m}\times\textbf{H}_\textrm{eff}) + \alpha\textbf{m}\times\frac{d\textbf{m}}{dt} +\\ \textbf{m}\times \left[\textbf{m}\times b_j(\textbf{j}_e\cdot\nabla)\textbf{m} + c_j (\textbf{j}_e \cdot\nabla)\textbf{m}\right]\ ,
\end{multline}
where $\textbf{m}$ is the dimensionless magnetization and the other parameters are the same as above. This adimensionalization avoids the usual loss of precision associated with numerical calculations, because it avoids to use large and/or small numbers.
By taking certain parameters together and rewriting the equation further, it gives the following intermediate result:
\begin{equation}\label{eqn:intermediatetorque}
\frac{d\textbf{m}}{dt} = -\gamma_0\textbf{m}\times\left[\textbf{H}_\textrm{eff} + \textbf{m}\times\frac{\textbf{B}}{-\gamma_0} + \frac{\textbf{C}}{-\gamma_0}\right] +\alpha\textbf{m}\times\frac{d\textbf{m}}{dt}\ ,
\end{equation}
where
\begin{equation}
\textbf{B} = b_j(\textbf{j}_e\cdot\nabla)\textbf{m} \textrm{ and }\textbf{C} = c_j(\textbf{j}_e\cdot\nabla)\textbf{m}\ .
\end{equation}
From Eq. (\ref{eqn:intermediatetorque}), the part inside the brackets with $\textbf{B}$ and $\textbf{C}$ can be taken into a single term $\textbf{H}_\textrm{torq}$ that describes the torque as an effective magnetic field. The final equation will be a dimensionless version in the LLG form, as given by:
\begin{equation}\label{eqn:H”}
\frac{d\textbf{m}}{dt} = -\textbf{m}\times(\gamma_0\textbf{H}_\textrm{eff} + \textbf{H}_\textrm{torq}) +\alpha\textbf{m}\times\frac{d\textbf{m}}{dt}\ ,
\end{equation}
where the torque appears as an effective magnetic field is described as:
\begin{equation}\label{eq:integratedtorque}
\textbf{H}_\textrm{torq} = \textbf{m}\times \left(\textbf{u}\cdot\nabla)\textbf{m}\right) + (\xi\textbf{u}\cdot\nabla)\textbf{m}\ ,
\end{equation}
where $\textbf{m} = \frac{\textbf{M}}{M_s}$ the dimensionless magnetization and $\textbf{u} = -b_j\textbf{j}_e$.

It can be shown that the term $\textbf{H}_\textrm{torq}$ has the same units has the magnetic field, thus it can be absorbed in the effective magnetic field equation of section 2.1. To solve the remaining standard LLG expression, it can be further transformed into the LL equation that has no time dependency on the right hand side of equation by using the transformation results obtained in the article on the LLG equation.

3. Micromagnetic modelling

A lot has changed since 1949, when W. Brown developed the mathematical model for micromagnetism. Large-scale computers are the main reason why micromagnetic modelling has been developed so rapidly. The numerical equations in the previous chapters can be integrated into a software program, that can simulate the magnetization dynamics in a 2D (and in some cases 3D) interface. I will start by giving a general overview of how I tackled this problem, and then show you more professional software tools that are worth looking into.

3.1 Self-made software

As I will not publish my own software, I will give some tips on how to tackle it. First off, to create your own program, it’s good to have a basic in coding languages, such as Python, or Fortran. Secondly, almost all the information is already available in the form of numerical equations in the previous sections. The task of the software is to solve these equations with initial parameters and specified boundary conditions.

The surface can be discretized into a grid of points and the LLG equation needs to be solved at each point on this grid. For that we need to set initial conditions for each point on that grid, and specify the boundary conditions, so we do not create artifacts at the edges. The initial conditions in Figure 6 were most likely that the magnetization at all grid points, besides the center, were pointing down, indicated by the color blue (red stands for pointing up). Note that in my experiments I was simulating the interface between a ferromagnetic layer with a metallic layer, hence we have interfacial DMI, that allows skyrmions.

From here, we can calculate the effective magnetic field ($\textbf{H}_\textrm{eff}$) of each point separtely, with the initial conditions given. From here the right hand side of the numerical LLG equation can be calculated. A different module performs the numerical integration of the LLG equation, using methods such as Euler, Heun and Dormand-Prince. Depending on the complexity of the system, you will find that certain methods are faster than others. Lastly, you will have to find something to display the information, this can be done through OriginLab, GNUplot, or other options available to you.

When dealing with modelling magnetic structures, you will have to create models that initialize them on your interface. The main issue is to get it to become stable. Take a skyrmion for example, which only has a small range of useable parameters, see Figure 7. When initializing the skyrmion with a number of requirements, such as the initial radius, the type (Nëel or Bloch), the magnetization direction at its center and its initial coordinates, you will have to allow it to evolve to its minimum energy configuration

3.2 Professional software tools

There are several micromagnetic simulation tools avaible, each with their own strength and capabilities. Here are some of the popular ones.

OOMMF (Object Oriented MicroMagnetic Framework): It is a widely used open-source micromagnetic simulation package developed by the National Institute of Standards and Technology (NIST). It supports various simulation techniques and models, including LLG dynamics, energy minimization, and more.

MuMax3: This is a GPU-accelerated micromagntic simulation tool developned by the DyNaMat group at the Gent University. it supports features such as parallel computing, non-uniform grids, and advanced material properties. However, you’ll need an NVIDIA graphics card for it (I got an AMD…).

MAGPAR: This is a finite element micromagnetic package that offers a range of simulation capabilities, including LLG dynamics and thermal effects. It is highly efficient and scalable.

Vampire: This shouldn’t be in the list, as it is an atomistic simulation tool for magnetic materials. However, since it is a free and open-source package i will include it in the list. It bridges the gap between micromagnetics approaches and electronic structure by treating a magnetic material at the natural atomic length scale. I’ve had the pleasure to meet the creator R. Evans and it left a great impression on me.

4. References

[1] L. Landau and E. Lifshitz, “On the theory of magnetic permeability in ferromagnetic bodies.” Phys. Z. Sowjetunio, vol. 8, pp. 153-169, 1935.

[2] T. Gilbert, “A lagrandian formulation of the gyromagnetic equation of the magnetization field.” Phys Rev, vol. 100, pp. 1243, 1955.

[3] C. Abert “Discrete Mathematical Concepts in Micromagnetic Computations” PhD. Thesis. Universitat Hamburg ,Germany, 2013.

[4] Y. Liu, D.J. Sellmyer and D. Shindo, “Handbook of advanced magnetic materials. Volume 1: nanostructural effects”, Springer, 2005.

[5] C. Abert, “Micromagnetics and spintronics: models and numerical methods.”, Eur. Phys. J. B., vol. 92, 120, 2019.

[6] NanoScience, “Noncollinear spins.”, Feb. 13, 2022. [Online] Available: http://www.nanoscience.de/HTML/research/noncollinear_spins.html [Accessed: April. 24, 2024].

[7] N. A. Spaldin, “Magnetic materials: fundamentals and applications.“, Cambridge : Cambridge University Press, 2013.

[8] I. Dzyaloshinsky, “A thermodynamic theory of weak ferromagnetism of antiferromagnetics.”, J. Phys. Chem. Solids, vol. 4, pp. 241-255, 1958.

[9] T. Moriya, “New mechanism of anysotropic superexchange interaction”, Physical review letters, vol. 4, 5, 1960.

[10] K.-W. Kim, K.-W. Moon, N. Kerber, J. Nothhelfer and K. Everschor-Sitte, “symmetric skyrmion Hall effect in systems with a hybrid Dzyaloshinskii-Moriya interaction.”, Phys. Rev. B, vol. 97, 224427, 2018.

[11] S. Muhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch and A. Neubauer, et al., “Skyrmion Lattice in a Chiral Magnet.”, Science, vol. 14, pp. 915-919, 2009.

[12] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, et al., “Chiral magnetic order at surfaces driven by inversion asymmetry.”, Nature, vol. 447, pp. 190-193, 2007.

[13] S. Rohart and A. Thiaville, “Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction.”, Phys. Rev. B, vol. 88, 184422, pp. 1-8, 2013.

[14] A13ean, “Stoner model of ferromagnetism. [Online] Available: en.wikipedia.org/wiki/File:Stoner model of ferromagnetism.svg [Accessed: April 24, 2024].

[15] S. Zhang and Z. Li, “Roles of Nonequilibrium Conduction Electrons on the Magnetization Dynamics of Ferromagnets.”, Physical Review Letters, vol. 93, 12, pp. 12704, 2004.

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The Landau-Lifshitz-Gilbert equation

Florius — Fri, 19 Apr 2024 17:48:20 +0000

1. Landau-Lifshitz equation

[latexpage] To model the motion of the magnetization in the time domain, the Landau-Lifshitz (LL) equation is used. It describes the evolution in time of the magnetization in a solid. The equation is as follows [1]:

$\frac{d\textbf{M}}{dt} = -\gamma_0\textbf{M}\times\textbf{H}_\textrm{eff} -\frac{\lambda}{M_s}\textbf{M}\times(\textbf{M}\times\textbf{H}_\textrm{eff})\ ,$

where $\lambda$ is a constant characteristic of the material itself, $\gamma_0 = \mu_0\gamma$, with $\mu_0$ the permeability of free space and $\gamma$ the gyromagnetic ratio, $M_s$ is the magnetization saturation, $\textbf{H}_\textrm{eff}$ is the effective magnetic field and $\textbf{M}$ the magnetization.

In an ideal case, with only an applied magnetic field and no damping, the magnetic moment of a single spin would precess indefinitely around the effective field, this is the so-called Larmor precession, given by the term $-\gamma_0\textbf{M}\times\textbf{H}_\textrm{eff}$ as illustrated in Figure 1. This works for a single sin, but for a large ensemble of spins, the precession is influenced by the interaction with the neighbors as well. For actual materials, you have to account for the energy losses due to dissipation, Landau and Lifshitz added the second term at at the right hand side of Eq (1). that models the damping of the the system to the nearest local minimum, which is represented by the effective magnetic field, $\textbf{H}_\textrm{eff}$. That is how the complete Landau-Lifshitz (LL) equation came to be.

2. Landau-Lifshitz-Gilbert equation

Later on, T. Gilbert [3] suggested that the term $\textbf{M}\times\frac{d\textbf{M}}{dt}$ could be used to model the damping. Figure 2 illustrates the damping of the magnetization, indicated by the inward spiraling red line. In the same figure, the torque ($\textbf{T}$) is also illustrated. If this term is large enough to overcome the damping, it can flip the magnetization to $-\textbf{H}_\textrm{eff}$. The exact mechanics of the torque will be discussed in a later section.

There is a substantial difference between the term coined by Gilbert and the one described in Eq. (1), but mathematically it can be proven to be equivalent with a normalization factor. This substitution leads to the Landau-Lifshitz-Gilbert (LLG) equation:

$\frac{d\textbf{M}}{dt} = -\gamma_0(\textbf{M}\times\textbf{H}_\textrm{eff}) + \frac{\alpha}{M_s}\textbf{M}\times\frac{d\textbf{M}}{dt} + \textbf{T}\ ,
$
with $\alpha$ the Gilbert damping coefficient and the rest of the parameters are the same as above.

2.1 Proof of equavalence

Both the LL = and LLG equations are used to obtain the normalization factors and to integrate the torque and as such, the derivation to go from one to the other is shown. M. d’Aquino [4] demonstrated the equivalence between these equations. Starting with the original LLG and multiplying both sides with $\textbf{M}$ to obtain:
$\textbf{M}\times\frac{d\textbf{M}}{dt} = -\gamma_0\textbf{M}\times(\textbf{M}\times\textbf{H}_\textrm{eff}) + \textbf{M}\times\left(\frac{\alpha}{M_s}\textbf{M}\times\frac{d\textbf{M}}{dt}\right)\ .$

Making the substitution $a\times(b\times c) = b(a\cdot c) – c(a\cdot b)$ to the previous equation and by applying the following relationship $\textbf{M}\cdot\frac{d\textbf{M}}{dt} = 0$. The last relationship can be deduced assuming that $M_s$ is onstant in time, therefore any variation in time must be a perpendicular vector to the magnetization.

$\textbf{M}\times\frac{d\textbf{M}}{dt} = -\gamma_0\textbf{M}\times(\textbf{M}\times\textbf{H}_\textrm{eff}) – \alpha M_s\frac{d\textbf{M}}{dt}\ .
$
This result can be substituted back in the LLG equation, giving the following intermediate result:
$
\frac{d\textbf{M}}{dt} = -\gamma_0(\textbf{M}\times\textbf{H}_\textrm{eff})-\frac{\gamma_0\alpha}{M_s}\textbf{M}\times(\textbf{M}\times\textbf{H}_\textrm{eff}) -\alpha\frac{d\textbf{M}}{dt}\ .
$
By taking out the time dependency on the right hand side and moving it to the left side and grouping certain parameters together, the LLG equation can be described in the form of the Landau-Lifshitz equation in the following way:
$
\frac{d\textbf{M}}{dt} = -\frac{\gamma_0}{1+\alpha^2} \textbf{M}\times\textbf{H}_\textrm{eff} – \frac{\gamma_0\alpha}{(1+\alpha^2)M_s} \textbf{M}\times(\textbf{M}\times\textbf{H}_\textrm{eff})\ .
$
This form of the LLG equation is mathematically the same as the original LL equation, provided that the two parameters $\gamma_0$ and $\lambda$ in those equations are changed to the following:
$
\gamma_{0}(LL) = \frac{\gamma_0}{1+\alpha^2}\ ,\ \lambda = \frac{\gamma_0\alpha}{1+\alpha^2}\ .
$
Expressions are the same if the damping coefficient vanishes, Furthermore, Mallinson [5] proved that when $\lambda$ and $\alpha$ both go to infinite, then the LL equation and LLG equation give respectively:
$
\frac{d\textbf{M}}{dt} \rightarrow \infty\ , \ \frac{d\textbf{M}}{dt} \rightarrow 0 \ .
$
This shows that the LLG equation is more appropriate to describe the damping of the magnetization, because a large damping coefficient should have a slow motion statement, which is true for the final LLG equation and for the expression of the LL equation that considers the normalization factors. The reason to use the latter expression, is that the format of the LL equation is numerically easier to solve as there is no time dependency in the right hand side.

In conclusion, the equation that I often used, is obtained by replacing $\gamma_0$ and $\lambda$ from the original LL equation with the obtained normalization factors. This can be further rewritten to the mathematical form as programmed in the code,
$
\frac{d\textbf{M}}{dt} = \frac{1}{1+\alpha^2}\left[-\textbf{M}\times\gamma_0\textbf{H}_\textrm{eff}- \frac{\alpha}{M_s}(\textbf{M}\times\textbf{M}\times\gamma_0\textbf{H}_\textrm{eff}) \right]\ ,
$
where $\textbf{M}$ is the magnetization, $\gamma_0 = \mu_0\gamma$, with $\mu_0$ the permeability of free space and $\gamma$ the gyromagnetic ratio, $M_s$ is the magnetization saturation, $\textbf{H}_\textrm{eff}$ is the effective magnetic field and $\alpha$ is the damping constant.

3. Temperature dependancy

The system described up to this point, is all purely for fictional zero degree temperatures. Non-zero temperatures would give a jiggle on the motion of the precession, effecting it ever so lightly. With high enough temperature, it can even flip the magnetization. This is one of the reasons why most of the lab work on these type of materials is done at low temperature, to keep this fluctuation to a minimum. There are many different methods to include the temperature fluctuation, such as Monte Carlo simulations, or just by adding an additional term to the LLG equation that takes this into account. For this I’d like to refer to the Master thesis of D. Schürhoff, Atomistic Spin Dynamics with Real Time Control of Simulation Parameters and Visualization (2016). It is definitely a pleasure to read such well written and informational thesis. With this, it completes the discussion of the LLG equation, its terms and their effect on a single spin.

4. References

[1] L. Landau and E. Lifshitz, “On the theory of magnetic permeability in ferromagnetic bodies.” Phys. Z. Sowjetunio, vol. 8, pp. 153-169, 1935.

[2] G.P. Muller, “Exploration of skyrmion energy landscapes,” M.S. Thesis, RWTH Aachen and FZ Julich, Germany, 2015.

[3] T. Gilbert, “A lagrandian formulation of the gyromagnetic equation of the magnetization field.” Phys Rev, vol. 100, pp. 1243, 1955.

[4] M. d’Aquino, “NONLINEAR MAGNETIZATION DYNAMICS IN THIN-FILMS AND NANOPARTICLES.” PhD Thesis, Universita degli studi di Napoli “Federico II”, Italy, 2004.

[5] J.C. Mallinson, “On damped gyromagnetic precession.” IEEE Transactions on Magnetics, vol 23, pp. 2003, 1987.

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Reference & Citations overview

Florius — Thu, 18 Apr 2024 20:50:50 +0000

When writing a manuscript or an article, you will have to deal with citations, references and creating a reference list. That can be cumbersome and complicated. That is why every topic will be explained in great detail, including examples to help you start. If you rather use a citation manager app, there are several citation generators, such as Mendeley, websites such as CiteThisForMe or MyBib, and it can even be done in MS Word. But it is always helpful to know what you are doing!

Where can I find these sources?

Web of Science (formerly known as Web of Knowledge), is a database of online resources, such as journals, books, etc.
Your university most likely has a paid subscription to a variety of journals that you access online.
Certain large journals, such as Nature Communication, have their articles accessable online for free.
Your local (university) library.

Traditionally, sources were mostly books, journals, encyclopedias, and other published materials. However, with the advent of the internet, the definition of a source has been expanded to include online resources such as e-books, online libraries, videos, and audio material. These resources can be classified as physical (printed) or online (digital) or text-based or visual/audio-based. It ultimately comes down to a resource from which you can obtain an intellectual idea that is not your own^[1]Suny Empire State College “Types of information sources,” subjectguides.esc.edu, 24 October, 2022. [online] Available: https://subjectguides.esc.edu/researchskillstutorial/sourcetypes. [Accessed: … Continue reading.

It is quite common nowadays to listen to podcasts. They have topics from science to sport and can be a great source of information.

Tip

When using an online article or book, their complete reference is normally listed on the journal’s webpage!

Both in-text citations and references serve as pointers to sources of information in academic texts. They are closely related, with the source being within the text, either through a number or author name pointing to a reference list, which is typically found at the end of the main text.

When using an intellectual idea from someone else, it is important to reference the source. This is typically done using a shorthand notation within your own written text, referred to as an in-text citation. Different referencing styles, such as the IEEE style, have different ways of using in-text citations. These citations can also serve as signals to the reader, indicating that a source is being used ^[2]“Signal and Lead-in Phrases,” [online] Available: https://owl.purdue.edu/owl/research_and_citation/using_research/quoting_paraphrasing_and_summarizing/signal_and_lead_in_phrases.html.[Accessed: 5 … Continue reading. For example, including the author’s name in the phrase that introduces the topic, such as “Fink et al. argued that…”.

When using a direct quotation from a source, it is important to enclose the text in quotation marks. When paraphrasing or summarizing a source, it is still necessary to cite it, but quotation marks are not needed. However, there is an important difference between paraphrasing and citation, which should be carefully considered.

It is not necessary to cite your own intellectual ideas or work unless they have been previously published ^[3]Suny Empire State College “Types of information sources,” subjectguides.esc.edu, 24 October, 2022. [online] Available: https://subjectguides.esc.edu/researchskillstutorial/citationparts. … Continue reading. Similarly, common knowledge, or information that the reader does not need to look up, does not need to be cited ^[4]“What is Common Knowledge,” [online] Available: https://academicintegrity.unimelb.edu.au/plagiarism-investigation-and-penalties. [Accessed: 10 November, 2022]..

At the end of your work, it is essential to include a reference list or bibliography that provides detailed information on how to locate the original sources.

Why citations?

References and citations are important to use in academic writing for several reasons:

To give credit to the original authors of the ideas and information used in the work. This is a basic principle of academic integrity and helps to avoid plagiarism.
To allow readers to locate the original sources and conduct further research. This enhances the credibility of the work and allows readers to fact-check and verify the information presented.
To demonstrate the breadth and depth of the research conducted. A well-referenced work shows that the author has done their due diligence and has a good understanding of the literature on the topic.
To provide context and background information. Citations and references can provide additional context and background information to help readers understand the significance of the ideas presented in the work.
To support the arguments and claims made in the work. Citing sources can provide evidence and support for the arguments and claims made in the work, making it more convincing and credible.
To comply with academic conventions and guidelines. In most academic fields, it is expected that work will be properly referenced and cited, failure to do so is considered as a violation of academic integrity. Faculty members of your university have access to “turnitin“, which checks your work against other articles, millions of websites, papers, etc.

Since there is no universal referencing style, it is important to check your university’s guidelines when choosing which one to use. The best approach is to ask your professor or teaching assistant which style they prefer. If there is no specific guidance, it is recommended to use the APA style and maintain consistency throughout your work.

In this section, we will provide a brief overview of different referencing styles. By following the links provided, you can access more detailed guides with plenty of examples. Each guide includes information on how to cite in-text, how to correctly cite your sources, and the format for the bibliography list. Additionally, a full-page example is provided to help you get started.

APA style:

The American Psychological Association (APA) style is primarily used for studies in the fields of psychology, history, economics, and political science. The APA updates their style every few years, and currently, the 7th edition is in use.

IEEE style:

The Institute of Electrical and Electronics Engineers (IEEE) style is mainly used for studies in the fields of electronics, electrical engineering, software, and IT. The IEEE style guide provides all the information and examples you need to properly cite any work. The guidelines are based on the latest manual, published in 2019.

MLA style:

The Modern Language Association (MLA) style is primarily used for studies in the fields of arts and humanities. The MLA style guide provides all the information and examples you need to properly cite any work. This guide is based on the 9th edition of the MLA handbook.

Other styles

More styles still upcoming, please send me a request if your preferred style is missing.

Reference list and bibliography

A reference list (or Work Cited list) is where you provide all the details of your sources in an organized fashion. The information of each source includes:

Author
Title of the work
Date
Container
Location (where you found it).

The entries are either labelled with a number in the order of when it first appeared in the context, or in an alphabetical order according to the surname of the author. Each style has its own format, such as APA, IEEE and MLA formatting styles.

Presenting someone else’s work or intellectual ideas as your own without proper citation constitutes plagiarism, whether it is done with or without the original author’s permission. This applies to all forms of published and non-published work, such as reports. Plagiarism is a serious disciplinary offense and can result in penalties, including suspension from studies. There are several forms of plagiarism, and even if it is unintentional, there may still be an academic penalty for poor citation practices.

It is important to note that the use of references and citations is not limited to texts, but applies to all types of media, such as computer code, websites, manuscripts, or a fellow student’s report.

To avoid plagiarism and ensure academic integrity, it is essential to start learning proper citation techniques early on in your academic career. When first starting as a student, it may feel overwhelming to have to cite everything, and reports may become cluttered with citations. This is normal, and it shows that you are actively engaging with the topic. With time and experience, you will learn how to think critically and assess which arguments are most relevant, allowing you to draw your own conclusions.

Key takeaways

A source contains an intellectual idea that is not your own. When you use the concept in your own work, you need to use citations to refer to it.
There are different referencing styles, depending on your study and university, such as IEEE/APA/MLA/etc.
Each reference style has a bibliography that contains information on all the used sources.
Failing to cite, can lead to plagiarism, which can result in academic penalties.

References & Links

 [+] [−]

References
↑1	Suny Empire State College “Types of information sources,” subjectguides.esc.edu, 24 October, 2022. [online] Available: https://subjectguides.esc.edu/researchskillstutorial/sourcetypes. [Accessed: 5 November, 2022].
↑2	“Signal and Lead-in Phrases,” [online] Available: https://owl.purdue.edu/owl/research_and_citation/using_research/quoting_paraphrasing_and_summarizing/signal_and_lead_in_phrases.html.[Accessed: 5 November, 2022].
↑3	Suny Empire State College “Types of information sources,” subjectguides.esc.edu, 24 October, 2022. [online] Available: https://subjectguides.esc.edu/researchskillstutorial/citationparts. [Accessed: 5 November, 2022].
↑4	“What is Common Knowledge,” [online] Available: https://academicintegrity.unimelb.edu.au/plagiarism-investigation-and-penalties. [Accessed: 10 November, 2022].

The post Reference & Citations overview appeared first on Your studyhack.

Einstein, Politics and Science

Florius — Wed, 20 Mar 2024 20:28:49 +0000

Add Your Heading Text Here

The first world war (WWI) started in august 2014, and the German army proceeded very fast through Belgium. This rapid progression created a problem in the cities that were already captured, as the main army moved on to other locations. Harsh measures were taken to avoid any kind of resistance. In the city of Leuven in Belgium, a large part was burned down by the German army as an act of punishment, because apparently, some Belgian fired at the German soldiers. One of the many buildings that got burned down, was the university hall. It contained the library with manuscripts dating back to the middle ages.

The Germans had no respect for culture and civilians. What made it even worse is that many intellectuals and scientists reacted to the propaganda.

A pamphlet of a few pages, was created, that defended an undefendable crime. Some of the signatures were from Planck, Nernst, Wien, etc. The best of the German scientists. Science used to be neutral and international, suddenly this changed; they considered themselves Germans, and only then scientists. During the war, poison gas was developed, but the Germans were the first to use it. Eventhuogh It was initially discovered by Haber to produce ammonium for agriculture. After the war, German scientsits were considered bad people. They had not stayed neutral. This was the reason why they were boycotted after the war. They were prevented from collaboration with scientists from allied countried.

There were exceptions, such as the Netherlands, and Sweden. That is why Nobel prizes were still awarded to Germans (i.e. Planck and Haber).

There was one German who did not sign the propaganda; Einstein. He wrote his own pamphlet, depicting science as neutral. He dared to stand up against his government. Eventhough he was excempt from the boycot, and he was allowed to go to conferences, he decided to go only when other Germans were invited as well.

At this same time, also something else important happened. Quantum mechanics was important within the physics community, but his relativity theory had immediate reactions from everyone. During WW1 he expanded his theory, and proposed the general theory of relativity in 1915. He knew that he needed experimental evidance, therefore, he proposed to look for an eclipse. This was not possible during the war, so he had to wait till 1919, after WW1. The royal society performed two expeditions, one in Brazil and the other in some African islands. The bending of light was verified shortly after, just as Einstein predicted. He became world famous and the symbol of a new world. The theory itself was not well understood, but it gave a new vision of the world, and that was exactly what people needed so briefly after the war. Even with the fact that again it was an indication of supremacy of German science.

Einstein was the most iconic figure in physics in the 20th century. But he was also a very singular person, difference from every other physicist. Einstein did not have a normal education. He was born in Germany from a Jewish family. His father went bankrupt and they moved to Italy. There he abandonded the Germany nationality, and he became a stateless person. Later on in live he went to Berlin and worked in a patent office, reviewing the patents that were submitted. In 1905, he published 4 article, but only in 1921 did he get the Nobel prize, for “his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect”.

At this point, things started to unravel in politics and in his life. A lot of people were against Jews and science, and this created an atmosphere in which Einstan wasn’t happy. He started to accept invitations from all over the world, of which some were quite long journeys, to avoid being in Germany. This was possible because in this period, he was the most famous scientist in the world. by the 5th Solvay conference, in 1927, Einstein was considered to be of the former generation. For many of the German physicists, Einstan was a past generation.

Besides a physicist, he was also considered a master in pacifism. He was asked to collaborate with an international institute to publish a book on pacifism. This volume was formed by correspondence between Freud and Einstein.

Then, in 1933, new events started to occur. Nazis took over power, and one of the first laws was to prohibit Jews from participating in government activities. As the universities were part of the government, a lot of Jews in academia had to leave. Even Haber, the most famous chemist in Germany had to leave. Einstein was also forbidden to enter Germany again. He had a house, a bank account, and more in Germany, but he could not claim anything. A family in Antwerp, Belgium, offered him a refuge for the summer. During this time, he received a lot of people, that wanted to give him a job. He also had contact with the Belgian royal family, as the Belgian queen was a German. At this time in Belgium, there was a military trial for 2 boys that refused to join the army. Einstein used his connections with the king to talk about this trial. He declared to be a pacifist, but in view of what was happening in Germany, it would be better to be prepared for war. He relocated to the USA, where he would spend the rest of his life. Einstein, never was much interested in the USA. The first time he went there 91921) he was quite negative (no culture), but he admitted that it was a nation with a lot of willpower and possibilities. In the 1930s he agreed to teach in the USA. But it was quite different from how it was in Europe; he had to adapt to the American way of doing science. His first paper was rejected by the peer review (which wasn’t a thing in Germany). This is where I want to end solely on Einstein and broaden the horizon.

Physics from around the world

In Germany, the Nazi’s came to rule, and there was this movement for German physics, which started basically with Philip Lenard and Johannes Stark., both Nobel laureates. They said that all major scientists came from Aryan race, and that it had to do with a certain attitude. But that is not true, I want to highlight a few aspects to avoid misguiding from western bias.

In 1967, Basalla published a paper regarding the development of western science. He said that science was a western thing, and that it spread unidirectional to other countries across the world (from Europe to asia, latin america, etc). His model was composed by 3 phases:

Non-scientific nations: This coincides withthe 16th century to the 19th century, when western countries were interested in exotic countries, for their minterals, botanic information and more. A good example was Darwin who traveled across the world, collected geo specimens, and more. But in his research he never mentioned a local scientist that he met or that helped him.
Colonial science: Many of the western countries entered in a form of colonialism. They had to control the country, for their resources. They created institutions for geology, or hospitals, where western people actually worked. They did hire locals to work for them, but at a lower level. Not very much happened, western people worked there, but they still published in their own country and in their own language.
Third phase: This phase is the transplantation of existing organizations. After the second world war, many western nations were exhausted, as they used up a lot of resources to fight. They couldn’t keep control over their colonies, which became independent. Thus, the locals reused the colonial institutes as their own.

This model is not always true or complete. Not all countries have been colonies; such as Japan that was never a colony of anyone. The second major problem is that it suggests that the colonial phase is beneficial for the colony. That is not the case; the rulers were actively trying to stop any scientific development of the locals.

The Basalla model is much too naive. The phase of independence needs a lot of conditions to be met, that allows the country to be scientific with a quick development. Many conditions were already there during the colonial era. Some of these conditions are that you receive education in your home country; gain respect from your work in your own country; You should be able to find intellectual stimuation in your surroundings; and more. Most of these conditions also applied to western Eruope in the 16th and 17th century.

Japan

Japan is a very interesting conutry. In the 17th century, there was a decision by the regime to close the borders; no one could leave or neither the country. The port city of Nagasaki was the only contact with the outside world. This lasted until 1868, when there was a new regime that opened the borders. Modernization started rapidly. While before there was suspicion against the west, after 1868, they more or less adopted the American mode, by sending their best students to Germany and the USA. This Meiji government did import western technology, but not science as a way of thinking. The Taisho government (1912-1926) had more modernization, even introducing the scientific spirit. When Einstein was invited to lecture on the relativity theory, he was asked to explain what it is to be a scientist. This kind of modernization meant to not use the old Japanese way of living, and some did not like that. If you introduce western way of thinking, you would also introduce western ideologies (such as communism).

When Einstein published his relativity, a lot of European scientists did not accept his idea directly. In Japan, they were much more enthusiastic, because they did not know most other European scientists, only Einstein. Very rapidly, the Japanese were already collaborating and being visible in the world. The development of science in Japan does not follow the Basalla model, but nevertheles was very successful. The effort of Japanese scientists was not aided by the western countries, and they were at the forfront within 2 generations.

India

India is a very rich cultural country. Always a place of movement of people, and trading companies. The brutish took possession of it in the 19th century, and it became a colonial empire. The indian culture was never unscientific (mathematics and scientific thinking was there for a long time), and it was a well-developed nation.

The indian empire was huge, and if it has to be a colony, it needs to be govened by the motherland, which was too far away. There was a movement within the Indians, that they wanted some independence, that they wanted their own organizations, their own board of science (they were definitely not underdeveloped). They did not want supervision from London. If you were in favor with the British, you might get a scientific career, but you wouldn’t be a real Indian anymore. Or if you were Indian, you wouldn’t be allowed to do the same thing as the British were doing.

Universities were founded, but it were mostly british people who go there; but how much was actually controlled by the British?

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History of Physics (1895 – 1945)

Florius — Sun, 17 Mar 2024 19:19:28 +0000

Physics has always been an interesting topic, but it was during a span of only 50 years that physics completely changed the world. The movie “Oppenheimer” showed in great detail the history of what happened and how they got to the atomic bomb. But this is only part of the story. This revolution, that ended with the bomb in 1945, started already in 1895, and it is what I will try to explain here.

1. Introduction

Physics is a discipline that goes back to antiquity (600 BC). It start from Greek philosophers that considered physics as the understanding of nature; what is natural, or what is the normal course of things. This disciple didn’t change for 2000 years, until 1687, and Isaac Newton was the most important person in this revolution. Physics became the science of phenomena of the natural world. The job of physics was to describe phenomena and try to link them together in a theoretical overview, without talking about the essence of things. Essentially, it became the discipline of experiments, which was unthinkeable for the ancient Greeks, as it no longer is the natural behaviour, but you force nature to see what happens with experiments. Mechanics is the basis of modern physics and it follows the definition very well. This new definition is narrower and it is still what is used in today’s physics as well. After Newton, science became important, and with the era of enlightenedment, science became part of university education in the 1800’s. [latexpage]

The big revolution of physics was in the beginning of the 20th century. So in the 1950s most important theories were already done, and these physicists moved on to other disciplines, giving rise to new discoveries (Watson and Crick, 1953 with the double helix DNA), thanks to the different approach.

In the 19th century physics was science of particles and fundamentals. The most important science was Chemistry, since you could work in the industry. Most students went into chemistry, as you can get lucrative discoveries and there was lots of work. Pure physics was a basic science, about the laws of mechanics and heat. Even Max Planck’s professor adviced him not to go into physics. Then in the beginning of 1900s, with Einstein and Quantum Mechanics (QM) and Relativiy, it became the most important science. Suddenly physics became a central discipline that introduced new discoveries and new laws. Now the chemists had to learn about physics, about quantum chemistry. The biologists went to physics as well to understand about molecular biology; and everyone was admired and inspired by it. This type of physics, also introduced a new way of thinking. QM and relativity redefined science. Is science describing nature? No, it is just a description of what you see in nature; it just explains the phenomena and experiments. There is a separation between scientific theory and relaity. Theory may give a description of it, but it is not the real thing.

This revolution happened in the span of only 50 years. In 1895 we have the first Nobel prize for physics. In 1905 Einstein, who was a lot younger than the other physicists published 3 important papers. They had the fifth Solvay conference in 1927, where they discussed QM (that was explained with waves). And in the 1930s, nuclear phycics became the new standard, as everything was said about QM. It was in 1933 that the nazi’s came to power, banning Jews from all official positions, and many jobless physicists spread out to all over the world, of which many went to the USA.

But the whole revolution came to a halt with the creation of the atomic bomb (Manhattan project), which started in 1942. It was so expensive and impressive, but on the other hand also very dangerous. Governments began to implement science policies, as the money had to be paid by the taxpayers. These policies defined a new view of physics, it was no longer free of investigation, but guided by money and society. It also shifted from just a single person to large teams to investigate something. Physicists now had to write funding proposals to fund their project.

Three different revolutions due to physics in the 1900s

Conceptual revolution: Ohysics became a new field with new phenomena that needed to be studied without empirical data. Previously, most people went for experiments, but now more and more theoretical physicists came. QM and and relativity were also conceptually new. It changed the way people looked at the reality.
Professional revolution: The identity of a physicists changed. Before this time, scientists were like philosphers who might do some experiment. You could think of an experiment in the morning and finish it in the afternoon. Yuo were able to publish whatever you want, without peer review. Nowadays, it takes years and you need the right equipment. With the Manhattan project, we saw that it was a large team, with scientists fro all over the world who migrated to the USA. Physicists in the USA were different from their European counterparts. In the USA it was much more practical (they knew how to gain money from it), compared to the philosophers in Europe.
Social revolution: A change from fundamental physics with electrical and opical phenomena, to radar technology and the atomic bomb in the second world war. Science became a matter of social debate, not just about funding, but also whether it is beneficial to society. Anti-nuclear movements had impact on politicians, and they in turn had impact on scientists. If the public funding stops, so does physics, as it started to cost a lot of money.

2. Classical Physics

Boltzmann was the one who first coined the term “Classical Physics” in 1899, when he wrote:
“The goal of physics was for all times reduced to establish the force laws between two atoms, and then to solve the equations of these interactions under the relevant conditions. How all of this has changed. I am the only one left who still embraced the old doctrines with unreserved enthusiasm. I therefore present myself to you as a reactionary, one who has stayed behind and remains enthusiastic for the old classical doctrines as against the men of today.”
That was his own understanding of Newton’s definition. Everything must be reduced to interactions between particles and that should be the final aim of physics. As it was about evolution with only the strong ones will remain, and the poor ones will tay behind. Boltzmann didnt feel accepted by the other physicists, as everyone else was thinking about a revolution that he didn’t like, he eventually comitted suicide.

But it has always been sort of an evolution; Newton was also reacting against whatever was before him. For him, everything was made up of particles, even photons and matter. Laplace is the one who revived Newtonian physics, as most of what we now know from Newtonian physics, is formulated by Laplace. Some of his famous writings are “Exposition du systeme du monde (1796), Mecanique Celeste (1799) and Theorie analytique des probabilites (1812). It was in the latter one he write:
“Some perfect genius who knows the location and movement of all the particles in the world, can preduct all futures (and past) events.”
Classical physics has three characteristics:

Corpuscular mechanics: Meaning small particles.
Mathematical formalism: It should be accurately explained in mathematical formulae.
Causality: No random phenomena. Everything can be explained by the preceding conditions of the world.

As a result of Laplace’s work, the French were very well versed in mathematics, such as Coulomb, Fresnel, Biot, Ampere, Fourier, Poisson, Carnot. Also many people worked on optics, with interference and polarization.

On the other side of the border, you had germany, who just lost the war against France and they tried to catch up with promoting philosophy studies at the highest level of university (PhD). Physics and mathematics were considered part of philosphy, it became more important in a deeper sense.

During this time, there were two main problems that people tried to solve. The first one was the problem of matter. Very often problems were divided into ponderabilia (with mass) and imponderabilia (without mass, such as het light electricity). You had Faraday who, without having any education, was working with the concept of magnetism. He had concepts of lines of force, but could explain what those lines of force were. Ostwald had another direction, as he was trying to do chemistry on a physics base. He noticed there is heat in chemical reactions, and he wanted to explain all of nature as a function of energy. We also have the famous dutch physicist Lorentz, who was trying to make aether theories to both ponderabilia and imponderabilia. However, no one knew what this aether really was.

The second problem was unification of theories. On one hand you had theories on radiation and optics, on the other hand you had particles. It was almost embarrashing that you had 2 disciplines that describe everything, but they could be discussed completely independent from each other. Maxwell had some success, as he was able to unify electricity, magnetism and optics in one theory, “Treatise on Electricity and Magnetism (1873)“. This was also the problem that Boltzmann was facing. Everyone else seemed to go in the direction of aether, which is something difficult to understand. He was saynig that the physicists were giving away their idea of particles and replacing it with something which had no empirical data at all.

Since the problems accumulated durnig the 19th century, people were looking for inspiration to solve he problem. The atmosphere of the time was neo-romanticism, which means that there are things in the world beyond your understanding. Maybe “matter” was not the ultimate reality. They believed that their science was not enough to understand the world, and they had to look for new ideas or concepts that could explain what is beyond. The older generation of scientists were not really prepared for new laws, however the young people were. One such physicist was inspired by this neo-romanticism. He was looking for concepts that were completely unknown before, and his name was Albert Einstein. [Helge Krahg, Higher specilations, 2015)].

3. Quantum revolution

At the end of the 19th century, there were approximately 1000 physicist all around the world. Within this tight community, Germany was the leading nation from around 1860. Not just in physics, but also other sciences, this continued to be the case until 1930. Even though they not had as much money as the British physicists, the German community was more productive than any other. German scientists have always been very organized, which may be the reason for their success. Until 1925, almost half of the citations were German papers. The end of the 19th century, and in the beginning of the 20th, many discoveries were many in a short period of time; it was like every year a new research field was discovered, that again produced new discoveries. [Richard Staley, Einstein’s Generation (2008)]. Some examples are:

X-rays

W. Rontgen worked on cathode rays at home that produced imaging. He published his paper in 1986, including a picture of the hand of his wife. Within a few months, people came with applications, which was a great success, until they found that X-rays damage the human body.

radioactivity

Henri Becquere was studying Radium salt, and by accident he saw on a photographic paper the capture of a medallion that he kept in the office. He discovred that Radium send some kind of waves, and that it was radioactive.

Plum pudding model

J. Thomson discovered particles with electrical charge, as was already theoretically predicted. He came up with an atomic model that involved Aether (positive charge) with small charged particles (negative charge) moving around like in a soup. It is different than our current model.

One of the major problems was the Blackbody radiation, which was first defined by Kirchhoff, who shown that it only depends on temperature. Several people tried coming up with equations. Boltzmann’s law with $T^4$ dependency, after that we have the Wien’s law (also called Displacement law). Max Planck found a way to derive the Wien displacement law, but it was not enough, as it did not hold for all frequencies. He restarted and used a quantum hypothesis, so the energy levels of the oscillators were on different levels. He had actually hoped that the concept of discrete levels would go away, through some limit, but it didnt’t. He found a formula that looked right, but without a theoretical foundation on why those levels exist, and nothing in between. Planck hoped it was not a revolution, he wished he could explain it by classical mechanics, but most other physicists were happy, as they had an equation they could work with. There was only 1 person who realized the radical, nonclassical nature of Planck’s theory, but at this time he was not a physicist yet.

3.1 Einstein

Einstein worked after graduation in a patent office, so he knew about all the experimental physics from the patents he saw. He was a young man without a career in physics, so he had a different approach for it, and he didn’t care about his image if he was wrong (as he was not well known at this time). He truly came onto the scene in 1905, when he published 4 major papers. One of them was about the quantum hypothesis, a new approach on the nature of light. He believed that light consisted of quanta, referring to the photoelectric effect as proof, which fitted with his theory. But this effect was already well known, therefore it wasn’t really interesting for the physics community. This new theory wasn’t noticed by others, as it was nothing new for experimental physicists. He was known for publishing 4 papers in one year, but nothing particularly in need of attention. Hence the revolution did not start here. The breakthrough came in 1907 when he applied quantum theory to specific heat, which was researched by chemists. There were many people studying it, and they could use his theory, especially for low temperatures. Because this was a field with lots of research, the number of papers on quantum theory increased rapidly, and it became a major research topic.

3.2 Solvay conference (1911)

In 1911, the first Solvay conference took place in Brussels. Solvay was a chemistry industrial, who created artificial soda. The founder of the company had a very deep admiration for science, so he gave money to science and wanted to support it. He created the conference. The idea of the conference was that only the best of the best and leaders of the field should come together. These conferences were on physics and chemistry. He wrote to Nernst (a chemist), who was working on thermodynamics, to know who were the “best” to invite. As Nernst was working on specific heat, he was fully aware of the new quantum theory, and he invited some of the major scientists at that time. The conference was about specific heat theory, not about quantum theory, but the discussion went that direction anywyas. Nernst invited 50% experimental physicists and 50% of theoretical. While in fact only 1% of the world did theoretical physics, and any other ordinary conference would be 99% filled with experimentalists. [Helge Krahg, Higher specilations, 2015)].

Because this was the first conference with such high amount of theoretical people, they were able to interact and come up with new concepts, while the experimentalists could think of ways to solve the problems with expierments. Several people received Nobel prizes, and many more of them would receive them later on. This is the moment Einstein started to gain popularity. [Jungnickel and R. McCormmach, Intellectual Mastery of Nature, vol. 2 (1986)].

3.3 Copenhagen

Relativity theory was different from quantum theory in that it captured the heart of the masses and philosphers, as it had a lot to do with paradoxes. QM on the other hand was more technical and more confined with specialists. In the beginning not many understood it, and it was not always accepted as something revolutionary. There were always people hoping it would go away, or that it could be replaced by classical physics. The point of no return in QM would be the Copenhagen interpretation (1927). Einstein was the leading physicists between 1905 and 1925, but he was more of a loner, not a typical professor with students. On the other hand you have Niels Bohr, who was a typical physicist of this period. He was Danish, and was with Einstein part of the younger generation. He was partly Jewish, which had its own advantages and disadvantages.

At a very young age, he published a paper on electron theory, and received funding to go to the UK for an internship, working for Thomson and Rutherford. When he returned to Copenhagen, he wrote a paper on the atomic mode. Rutherford had an atomic model comprosed by a limited number of electrons circling around a positive nucleus, but his model was not stable. Bohr was trying to reconcile electromagnetism in this model, and he made 3 postulates, to make his model fit. These postulates had no real reason or argument on where they came from, but they were inspired by quantum theory.

The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what classical electromagnetism suggests. These stable orbits are called stationary orbits and are attained at certain discrete distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant.
Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency $$v$$ determined by the energy difference of the levels according to the Planck relation: $$\Delta E = hv$,$ where $h$ is Planck’s constant.

The strength of Bohr’s theory was not in its theoretical foundation, but its experimental confirmation, over a wide range of phenomena. The Balmer and Lyman series could explain why you had to subtract 2 numbers, as it was the subtraction between 2 energy levels. The red line in the hydrogen spectrum has a double structure. Bohr was unaware of it, but it turned into another confirmation, but it required a major extension of the theory, made by A. Sommerfeld. Bohr did not only publish his paper based on H en He, but he also tried to apply his model to all atoms, but it was not so successful. He did try to explain why the table of elements was the way it was. Since Mendelejev, the table was just made pure on the resemblance of properties of elements. In this time, that was okay. But in the 1900s, chemists were less convinced that it based on this alone, and they could make other tables, that were based on grouping. The Bohr model could explain the table’s structure, with the position of electrons. The more electrons there are, the more layers there are. Bohr then published the first physical interpretation of the Mendelejev table, that shwoed that the position of elements was based on atomic structure.

After the first world war (WW1), physics started to rise again. A lot of interest went to Einstein and his relativity theory, but in Copenhagen Bohr was able to fund his institute of theoretical physics. He became a host of quite a lot of physicists, that went to Copenhagen because of the easy-going nature of Bohr. He was always open for discussion and he would debate with anyone. At the same time, Germany had financial difficulties, and it was being boycotted because of the war. German scientists could not collaborate with sciencists from Allied forces. For many years this continued, but it didnt work out completely. Several countries stayed neutral during WW1, such as The Netherlands, Switzerland, and Denmark. The Copenhagen institute was a way to meet german physicists. As example, Heisenberg was the assistant of Bohr in 1925. A lot of the people in Copenhagen were not students, but well-established physicists that went there to work and discuss with Bohr.

Of course he was able to attract younger students as well. So many of these physicists that were responsible for QM were young. That is why Pauli refered to QM as “Knabenphysik” (Boys physics). Nobody of them had a PhD yet, and were still at the beginning of their career and working on topics that were on the forefront of science. Many of these bright young physicists felt, arrogantly, that QM belonged to them and that most elder physicists were just incapable of mastering the theory. They thought that they could do it better than the preceding generation, and tried to make the theory very difficult, so that the older generation would not be able to master it anymore. An example of the older generation is Einstein (around 50 years old now), who did not understand anymore what was going on with GM.

3.3.1. Heisenberg & Schrodering

Heisenberg started his research based on the research of Einstein, who had developed his relativity theory based on descriptionism attitude: “The physicist does not have to present a model of reality, but describe the observation”. Einstein introduced 4D reality, but never claimed that reality is 4D, he just needed 4D to explain his relativity theory. Heisenberg wanted to treat QM with just observables. He created a model based on matrices. Many physicists wre skeptical because of the theory’s lack of visualisability and its unfamilair mathematical formalism

On the other hand you have Schrondering, who was already somewhat older. he attempted QM with a different approach, namely with one based on waves. He went even as far as to prove that his wave mechanics, and heisenberg’s matrix mechanics were equivalent.

Back then, most physicists were experimentalists and didnt know much about matrix mechanics, so Schrodinger’s equation had great advantages, and therefore muche easier to use. In the end, they took Schrodinger’s side, instead of the heisenberg’s model.

Then Heiseberg came to a second big event. he shwoed that no matter how accurate the instrument used; QM limits the precision when two properties are measured at the same time. He concluded that to make a measurement at such small dimensions, you must use some kind of radiation with high energy. Therefore, the act of measurement itself would impart so much energy to the atom that it would cause significant disturbance or disruption. So you can either make a determination of momentum or a determination of position, but never at the same time.

This understanding lead heisenberg to think about causaility that was proposed by Laplace. The assumption that you can see the future and past by knowing the position and momentum of all particles was wrong, because you cannot know both. If you determine one observable in extreme degree of precision, nature will prevent you from knowing the other. Nature will interfere in a sense. Heisenberg discovered the uncertainty relations. Bohr understood that uncertainty principle had to do with the duality of matter as particles or as waves). It was complementarity, and a physicist has to know when to use a particle or when to use a wave. This lead to the Copenhagen interpretation, which is a statistical interpretation of wave mechanics, with the uncertainty equation and the complementarity of particles and waves.

This was the end of the evolution of quantum mechanics. People moved on to a new field of interest, namely nuclear physics

4. Nuclear Physics

Enrico Fermi, an Italian physicist, recognized by his peers as a genius. He was born in Rome, at a young age, he obtained his PhD and travelled to Germany and the Netherlands. When he returned to Rome, he became a professor of theoretical physics (the first in Italy). He married a Jewish girl, and with his work, he succeded in making Rome one of the important places in European physics. He won the Nobel prize, but did not return to Italy after collecting his Nobel prize in Sweden. He immediately moved to the USA, because of the fascist regime and the anti-Jew laws that were introduced in Italy. Fermi was one of the few European physicist that had a good career in the USA, as most others were not accepted in a positive way (they were foreigners, communist view, who did not speak English well).

4.1 The discovery of the neutron

The discovery of the neutron was a very important step in understanding the nucleus. It was very difficult to detect, and at at first there wasnt any need for it. Scientists only dealth with protons (+) and electrons (-). However, Fermi noted that some things dont add up very well. Some scientists envisioned there were additional protons in the nucleus, but than the total net charge would not be zero. The discovery of the neutron happened in 1932, when experiments were done by bombarding Beryllium wth alpha particles from a Polonium source. Some scientists thought this highly penetrating radiation consisted of high energy photons. But how can this kind of radiation have so much energy to kick out these heavy protons? Chadwick found one solution; the radiation was not consisting of photons, but of neutral heavy particles, with a mass similar to a proton. He did not think there would be an application for it, as these particles were neutral. However, Szilard had another idea; if neutrons could be pushed out from one nucleus, it can be used to bombard another nucleus, and fission can be induced, which releases an incredible amount of energy. This made nuclear weapons possible. [Niels Borh and John Archibald Wheeler, The Mechanics of Nuclear Fission, (1939)]

Nuclear fission marks a pivotal moment in the history of nuclear physics, but its roots lie in earlier discoveries and theories. With Chadwick’s discovery of the neutron, it opened new pathways for understanding atomic structure and nuclear interactions. At the Solvay Conference of 1933, while beta decay wasn’t the central focus, it was a topic of discussion among physicists.

Fermi played a significant role in advancing the understanding of beta decay. He proposed a theory in 1933 that described beta decay as a process in which a neutron in the nucleus transforms into a proton, releasing energy that manifests as an electron (beta particle) and a neutrino (Italian for little neutron). This theory introduced the concept of weak nuclear interaction, a fundamental force distinct from gravity and electromagnetism.

Fermi’s work extended beyond theory; he recognized the potential of neutrons as tools for inducing nuclear reactions. By bombarding nuclei with slow neutrons, Fermi and his team increased the likelihood of interaction. In 1938, Fermi’s research took a significant turn when it became clear that the ‘trans-uranium elements’ he and his team had been seeking were not present in their samples. This realization paved the way for further investigations, eventually leading to the discovery of nuclear fission later that year by Hahn and Strassmann. When the idea of splitting the atom, to produce energy and neutrons was published, many people understood that it was a dangerous discovery, especially with what was happening in europe at the time. Most of the physicists stopped publishing about nuclear physics during the way. In the end of 1945-46 a massive amount of papers were published regarding the discoveries during WW2.

4.2 The second World War

During the second world war, research was only possible in the UK and USA, as the americans were not interested in being involved in the war initially. Szilard tried to convince not only physicists, but also politicians to do something about it. He even went to Einstein, who was well familair with the belgium royals (as the best uranium was found in the Belgian colony, the Congo). It took until 1941, with the attack on pearl harbor, that the USA entered the war.

The Manhattan project started, and a metallurgical laboratory was founded in Chicago where Fermi worked. It was a military project, and at the end of the war, tens of thousands of people were employed. The civilian leader of this project was Oppenheimer, a theoretical physicist. One of the important steps was tomake a reactor, to determine if the chain reaction was possible. This was the work of Fermi. Several universities were involved in this research, but the central place was in a new site, completely isolated from the world. This village was called Los Alamos. It grew in population, up to 2000, all relatively young people. The sucessful test of the plutonium bomb occured on July 16, 1945 at the so-called Trinity test side in the New Mexico desert.

When the war finished on in Europe, the Japanese were still attacking. On August 6, 1945, the uranium bomb was dropped on the Japanese port city of Hiroshima. Three days later, the plutonium bomb destroyed Nagasaki. Almost 200,000 people died in those two blasts. The American public’s attitude was anger and suspician against the scientist who contributed on the atomic bomb. In only a few years after dropping the bomb, the view of scientists went from a figure of stature and wisdom to an enemy. [G.S.Allen, Master Mechanics & Evil Wizards, (1992)].

4.3 The aftermath

Lots has changed during this period. Projects became larger and larger. We’ve gotten space research, big machines were build, like the cyclotrons in Berkeley. And importantly, the USA became a scientific superpower, which was mainly due to money. Previously, most universities were independent institutes. Robert Millikan introduced a strategy to concentrate money on just a few research universities (top 10). This money came from philantropists, such as the Rockefeller foundation. Among these universities were Caltech, Harvard, Princeton, MIT, etc. Instead of students going to europe to do their PhDs there, now people came to the USA to study there.

During this time, Europe was recovering from the war .They were in need of money and support. The rebuilding was founded by the Americans, however they did it with the intention to know what was happening in Europe. CERN is such an example. It was an european high energy institute that collaborated with Ammericans, so they could see the state of knowledge in Europe, and make sure that physicists don’t go to Russia. Unfortunately, not much has changed since then. I truly hope that we can learn from the past and make better choices for the future.

5. Links & Suggested reading

Emilio Segré, From Falling Bodies to Radio Waves. Classical Physicists and Their Discoveries (1984)
Helge Kragh, The Quantum Generations. A History of Physics in the Twentieth Century (1999)
David C. Cassidy, A Short History of Physics in the American Century (2011)
¡Jon Agar, Science in the Twentieth Century and Beyond (2012)

The post History of Physics (1895 – 1945) appeared first on Your studyhack.

What are thyristors and how do they work?

Florius — Sat, 09 Mar 2024 20:13:09 +0000

In this article I will discuss what thyristors are and how they work. I will start with the semiconductor structure of a thyristor and use the characteristic V-I curve to explain the three different modes. Subsequently, I will use an two transistor model to explain the thyristor from a different point of view. Lastly, I will briefly explain how to turn off typical silicon controlled rectifiers. [latexpage]

1. Thyristors

Thyristors are four layer devices, with three terminals: Anode (A), Cathode (K) and Gate (G). Figure 2 shows an illustration of the structure of a thyristor. The anode is formed from the $p$ and $p^+$ layers. The thickness of this part is roughly 30 to 50 μm. The $n^-$ layer is only lightly doped compared to the rest, with a doping level of around $10^{14}$ /cm$^2$. However the width is ranging from 50 to 1000 μm. When the thyristor is forward biased (meaning $V_{AK}$ is positive), the junction $j_2$ is reversed biased and the other two junctions are forward biased. The depletion layer formed at $j_2$ is absorbed by the $n^-$ layer. The width corresponds to the forward blocking voltage of the thyristor.

The other $p$ layer is doped at around $10^{17}$ /cm$^3$ and forms the gate of the thyristor. The last $n^+$ layer has a doping of $10^{19}$ /cm$^3$ and forms the cathode. In the next sections, I will explain the workings of this device with the different modes.

On the right hand side of Figure 2 is the simplified structure of the thyristor. This figure is what you will find in most articles on thyristors, but remember that this is only a simplified figure of the left hand side.

Some might confuse a thyristor with a power diode, but they are not exactly the same. A thyristor has a gate terminal that can control the device, and because of the extra layer(s), it also has more junctions than a diode (which has only 1 junction). While diodes are used for rectification, freewheeling and feedback; thyristors are mainly used for controlled rectification AC regulation, inversion and DC-DC conversion. Furthermore, diodes are used as protection for thyristors.

The advantages of thyristors:

Only a small amount of gate drive is required, since it is a regenerative device.
They exist with high voltage and high current ratings.
ON-state losses are reduced in thyristors.

The disadvantages of thyristors:

Once the thyristor is on, the gate no longer has control over it (however, special GTO-thyristors exist which are fully controllable by the gate terminal).
external circuits are required to turn it off.

2. Characteristics of thyristors

The working of a thyristor can be discussed in three different modes, namely the reverse blocking mode, the forward blocking mode and the forward conducting mode. Figure 3 shows the static V-I characteristic of a thyristor. The anode to cathode current ($I_{AK}$) is plotted against the anode to cathode voltage ($V_{AK}$). Furthermore, there is a reverse breakdown voltage ($V_{BR}$) and a forward breakover voltage ($V_{BO}$). Different gate currents (i.e. $I_{g1}, I_{g2}, I_{g3}$) can be applied to the thyristor.

2.1 Reverse blocking mode

Figure 4 shows the situation when a thyristor is in reverse blocking mode. The anode (A) is made negative with respect to the cathode (K) (negative $V_{AK}$). The gate terminal (G) is kept open. Due to the PN layering, there are 3 junctions, $j_1, j_2, j_3$. Out of these 3 junctions, $j_1$ and $j_3$ are reverse biased, and $j_2$ is forward biased. Only junction $j_2$ is in forward bias. Because of the reverse bias, the thyristor does not conduct,. However, there is still a very small current flowing from the cathode to the anode, which is called the reverse leakage current.

As can be seen by Figure 3, in this reverse blocking mode, the voltage can increase (to the left), but only small amount of current flows. At a certain voltage, called the reverse breakdown voltage ($V_{BR}$), the reverse current increases dramatically. At this moment, with high voltage and high current, there is a large power dissipation in the thyristor. Because of that, the thyristor’s junction temperature has exceeded its specified limit and it will break down. That is why reverse voltage should never exceed $V_{BR}$!

I mentioned that the gate terminal is kept open during reverse blocking mode. If you’d apply a positive voltage on the gate and cathode, junction $j_3$ is forward biased. Now $j_3$ and $j_2$ are both forward biased, and more leakage current will flow, adding to the dissipation.

2.2 Forward blocking mode

Figure 5 shows the situation when a thyristor is in forward blocking mode. The anode is made positive with respect to the cathode (positive $V_{AK}$). Because of this configuration, junctions $j_1$ and $j_3$ are now forward biased and $j_2$ is reverse biased. That means, that the forward voltage is still on hold by the junction $j_2$. A small current still flows, called the forward leakage current. In this mode, the thyristor is forward biased, but it does not turn on. You can increase the voltage up to the forward breakover voltage ($V_{BO}$), and once it reaches it, the thyristor turns on (it won’t get damaged unlike at the $V_{BR}$ side). At this stage, it goes from forward blocking to forwarding conducting mode.

The forward breakover voltage is obtained due to the blocking capability of junction $j_2$, while the reverse blocking capability is due to junctions $j_1$ and $j_3$ combined. Because the latter is stronger than just junction $j_2$; the reverse blocking voltage is higher than the forward blocking voltage of a thyristor.

2.3 Forward conducting mode

There are several techniques for which the thyristor can go from forward blocking mode to forward conduction mode:

When $V_{AK} > V_{BO}$
When voltage is applied to the gate terminal
When the gate-cathode junction ($j_3$) is exposed to light
when $\frac{dv}{dt}$ exceeds its specified limits

1. $V_{AK} > V_{BO}$

When $V_{AK} > V_{BO}$, the thyristor is driven in forward conduction mode, no matter the voltage on the gate (even if grounded). As mentioned before, during the forward blocking mode, junction $j_2$ is in reverse bias, but when $V_{AK}$ gets too large, avalanche breakdown of the junction takes place, and a large current flows through the thyristor and the voltage between anode and cathode reduces to a small value. This current is limited to the external load in the system, hence: $I_{AK} = \frac{V}{Load}$

2. Gate triggering

A positive gate voltage is applied with respect to the cathode, and it drives the thyristor from forward blocking mode into forward conducting mode. When this voltage is applied, an additional current starts flowing in parallel to the forward leakage current. The avalanche breakdown at junction $j_2$ now requires less $V_{AK}$ voltage to start. The thyristor can be brought into forward conduction mode below $V_{AK} < V_{BO}$, as is shown in figure 3 by the dashed lines $I_g1, I_g2, I_g3$. As the gate current increases, the lower the anode to cathode voltage is. Once the thyristor goes into forward conducting mode, the gate has no control over the conduction, and the current is limited similar to what was discussed before.

3. Light activated SCR

Similar to part 2, but instead of putting a voltage on the gate, if the gate-cathode junction, $j_3$, is exposed to a beam of light, current starts to flow due to the photons of light. This current drive the thyristor into conduction. These systems are a special type of thyristor, called the Light activated Silicon Controlled Rectifier (LASCR).

4. $\frac{dv}{dt}$ exceeds limits

During the forward blocking mode, a small forward leakage current flows through the thyristor; and something equivalent to an internal capacitor is formed inside the thyristor; one from anode to gate and the other from gate to cathode. A transient current can be induced in these internal capacitors by rapid changes in the voltage across the thyristor. This current adds to the foroward leakage current, and similar to gate triggering, it lowers the anode to cathode voltage required to turn it on. It can trigger the thyristor at unwanted moments, and to avoid that, you must keep the voltages changes below its permissible value, given in the datasheet. One way to stop this, is by adding a small resistance between gate and cathode that acts as an external path for this transient current.

2.4 Latching current vs Holding current

In thyristors, specifically in the context of silicon-controlled rectifiers (SCRs), the terms “latching current” and “holding current” refer to two important operating parameters. The definitions appear similar, but they are very different. I will try to explain their differences below:

Latching current is effective at the time the thyristor turns ON, whereas the holding current is effective at the time it turns OFF.
Latching current is the minimum current that is required at the time of triggering to turn ON the thyristor. Whereas once the thyristor is already in the ON state, it current should not be below the holding current, otherwise it turns OFF.
Lastly, Latching current is always greater than Holding current.

2.4.1. Calculation example

In this example, the thyristor has a latching current of 20 mA (it requires 20 mA to turn on), and is fired with a pulse of 50 μs. We will see if the current is large enough to turn on the thyristor.

The current through the RL circuit for a step input is given as:

$i(t) = \frac{V_s}{R}\left(1-e^{-t\frac{R}{L}}\right)$

By filling in the values given to us in Figure 6, we can calculate the current through the thyristor to be:

$i(t) = \frac{100}{20}\left(1-e^{-50\times 10^{-6}\times\frac{20}{0.5}}\right) = 10 mA$

The current through the thyristor is only 10 mA, and does reach the latching current required to turn on the thyristor. One possible way, is to increase the width of the pulse to increase the current.

3. Equivalent two transistor model

Using the two transistor model is a good way to explain a thyristor. Figure 7 shows how to convert a 4 layer structure (a) into a two transistor model (c).

When you have a 4-layer thyristor, you can imagine the top 3 layers and the bottom 3 layers as separate parts. Each part is a transistor; transistor $T_1$ is pnp, while transistor $T_2$ is npn. The base of $T_1$ is connected to the collector of $T_2$. Similarly, the base of $T_2$ is connected to the collector of $T_1$. Both of these transistors are in common base configuration. When the thyristor is forward biased, and the gate is left open, the current $I_D$ flows from anode to cathode. Current $I_D$ is a combination of collector current, emitter current and leakage current of both transistors combined.

Through some mathematics, you can conclude that the current $I_D$ can be calculated by:

$I_D = \frac{I_{CO1} + I_{CO2}}{1-(\alpha_1 + \alpha_2)}$

where $(I_{CO1} + I_{CO2})$ are the total reverse leakage currents of the junction $j_2$. $\alpha_1$ and $\alpha_2$ are common base gains of transistor 1 and 2, respectively. Initially they are small, but when the forward voltage increases, the gains also increase. When it reaches unity, $I_D$ goes to infinity. This is the moment the thyristor goes to forward conduction mode, and the current is only limited by the external load.

4. Turning off

Now that we know how a thyristor works, it might be beneficial to also know how to turn it off. There are special Gate Turn Off thyristors, but in this last chapter I briefly want to focus on the normal silicon controlled rectifier and how to turn it off.

We know that a thyristor turns off when the current falls below the holding current, this can be done by either natural or forced commutation. For natural commutation, you lower the supply voltage; or even go negative, this will put the thyristor in reverse bias. With forced bias, additional external components can temporarily apply an impulse of negative current turning it off as well. This external circuitry has to create an impulse of a certain duration to let the excess carriers near junction $j_2$ recombine. This time is called gate recovery time ($t_{gr}$). If the impulse is shorter than this, the thyristor may turn on again.

3-phase IGBT-inverter – Working principles

Florius — Sat, 24 Feb 2024 22:16:26 +0000

In this article the 3-phase IGBT inverter and its functional operation are discussed. In order to realize the 3-phase output from a circuit employing dc as the input voltage, a 3-phase (IGBT) inverter has to be used. The inverter is build of switching devices, thus the way in which the switching takes place in the inverter gives the required output. In this article the concept of IGBT’s and the working principle of the inverter is explained. [latexpage]

1. Insulated Gate Bipolar Transistor

The Insulated Gate Bipolar Transistor, also called IGBT, is something of a cross between a metal-oxide-semiconductor field-effect transistor (MOSFET) with a bipolar junction transistor (BJT). It can be used as a semiconductor switching device. The IGBT takes the insulated gate technology of a MOSFET (hence the first part of the name) and combines it with the output performance characteristics of a BJT, (hence the second part of the name). As a result, the IGBT has the output switching characteristics of the BJT, but can be voltage controlled, such as a MOSFET. It takes the best of both worlds.

It can be used in small amplifying circutis, replacing BJTs and MOSFETs. But because it has a very low “on-state” resistance, lower then the other two, it has very low power loss ($I^2R$). Due to this, and the fact that it is capable of handing large currents (1000+ Amps) with basically zero gate current, it is mainly used in power electronics, such as power supplies and inverters. Even though high power BJTs exist, their switching speed is much lower than these IGBTs, making this the ideal device to use.

IGBT has three terminals, namely Gate (G), Collector (C) and Emitter (E), as is depicted in Figure 1. The terminal names imply that their names have been taken from the other 2 type of transistors. The Gate inputterminal is taken from the MOSFET, while the Collector and Emitter parts are the output terminal from the BJT.

1.1 Construction of an IGBT

IGBT is made of four layers of semiconductors that form a PNPN structure. At the bottom there is a metallic electrode for the Collector terminal. At the top the emittor electrode is attached between the P and N layer. Furthermore, there is an additional silicon oxide layer that seperates the gate electrode from the semiconductor.

The main difference between the construction of an IGBT and a MOSFET, is the additional $p^+$ substrate layer, known as the injection layer. This layer is heavily doped with an intensity of $10^{19}$ per cm$^2$. The thickness of the $n^-$ is proportional to its voltage blocking capacity. The $p$ layer on top of that, is known as the body of the IGBT.

In the left figure, you find the typical Non-Punch Through (NPT) IGBTs, which have a symmetrical breakdown voltage characteristics i.e. their forward and reverse breakdown voltage are equal. Due to this, they are used in high voltage AC circuits.

Another type is the Punch Through (PT) IGBTs, that have an additional $n^+$ buffer layer that is included in the stack, which creates an asymmetric voltage blocking capability i.e. their forward and reverse breakdown voltages are different. These PT IGBTs are unidirectional, and are therefore used in DC circuits such as inverters and high voltage chopper circuits.

1.2 Equivalent structure

The equivalent circuit shows an N-channel MOSFET that drives a PNP bipolar transistor in a darlington configuration. Intuitively, you can see that the voltage across the IGBT is one diode voltage higher than when using a single N-channel MOSFET on its own. You can measure that the on-state voltage across the IGBT is at least one diode drop (~0,7 V). However, when equally comparing an IGBT vs a MOSFET of the same size, at same temperature and same current, the IGBT will have a lower on-state voltage. The reason for that is, that in IGBTs the current flow consists of both holes and electrons, compared to MOSFET, which is a majority carrier device only. Thus, the injection of minority carriers in IGBTs result in a reduction of on-state voltage and it is one of the major advantages of IGBTs over power MOSFETs.

The downside of using IGBTs is the slower switching speed. When turning off the device, the flow of electrons can be stopped rather quickly by reducing the gate-emitter voltage. But the holes, are still left in the drift region and takes longer to recombine. The rate of recombination can be controlled by adding an additional $n^+$ buffer layer. This buffer layer will quickly absorb holes during the turn off phase. IGBTs with this additional buffer layer are called Punch-Through IGBTs, as discussed in the previous section.

The second disadvantage is the possibility of latchup, where the IGBT can no longer be shut down by the gate voltage. It happens when you are using the device outside of its specs. As example, when too much current runs thruogh the device, it can destroy it due to overheating, because it can not be turned off. Similarly, when switching large voltages in short periods, the IGBT can also derail. Normally, by staying within safe operating ranges, you will not trigger latchups

1.3 Working principles of an IGBT

The Insulated Gate Bipolar Transistor is a three-terminal semiconductor device that can be controlled with the Gate terminal, while the conduction path is between the Collector-Emitter terminals. As it is a unidirectional device, it cannot conduct current in reverse and only has two modes; forward blocking, where the IGBT behaves like a reversed biased diode and the conducting mode, which allows the current to flow between Collector and Emittor.

In the forward blocking mode, there is no current flowing between the collector and emittor, it is also called the off-state. To get the IGBT in the forward blocking mode, we need to set a positive voltage on the collector with respect to the emitter and put zero voltage on the Gate terminal. This will effect the 3 junctions that exist in the IGBT. Junction 1 and 3 will get into forward bia. However, junction 2 is in reverse bias, preventing the flow of current.

When we increase the gate voltage $V_G$, so it is more positive than the emitter voltage, an inversion layer (a capacitive effect) is created near the metal gate layer. This layer creates a channel, that shorts the $n^+$ and $n^-$ regions. The doped part $n^+$ has an excess of electrons that get pushed into the $n^-$ region, also called the drift region. The amount of electrons will lower the potention for the holes (shown as red circles) to flow from the $p^+$ substrate into this region as well, as they get attracted by the negative electrons. This ultimately leads to a plasma of holes, which attracts more electrons, etc. At this point everything is forward biased and it is in conducting mode.

If, at this moment, you’d remove the voltage from the gate. The capacitance on the gate will keep the inversion layer and the IGBT continues to conduct. Only if you let the gate capacitance discharge, by grounding it, will the IGBT stop conducting.

2. Transfer characteristics

When an IGBT is fully “on,” it is said to be saturated. If it exceeds its maximum current rating, it becomes over-saturated and may fail. To illustrate, let’s consider the IGBT NGTB15N120IHLWG. The datasheet typically specifies its maximum ratings at the highest temperature. For instance, at 100 degrees Celsius, it exhibits a saturation voltage ($V_{CEsat}$) of 1.8V at 15A. Interestingly, at 25 degrees Celsius, it can handle a current of 30A.

The saturation voltage is the voltage drop across the collector-emitter of the IGBT, similar to a diode. This drop results from the resistance inside the IGBT, albeit low, and is dissipated as heat. According to the datasheet specifications at 100 degrees Celsius, it would dissipate 27W of energy. Even with a total dissipation of 62.5W, there is still some room for switching or handling higher currents. While heatsinks can assist in lowering the heat drawn from the IGBT, they do have their limits. The operating junction temperature range ($T_J$) indicates whether a heatsink is necessary.

Another critical parameter is the gate voltage ($V_G$), responsible for turning the IGBT on or off. Similar to a MOSFET, an increase in gate voltage leads to higher current. There is a minimum gate voltage required, known as the threshold voltage ($V_{TH}$), to turn on the device. It is not uncommon to have a gate voltage $V_G = 15V$. Normally, the higher the gate voltage, the higher the current through the device.

Figure 5 depicts a typical IGBT characteristic curve, highlighting the safe operating region (in yellow) just before over-saturation begins. Beyond this point, the IGBT enters a runaway behavior, where the current remains constant despite an increase in $V_{CE}$, which can be detrimental to the circuit as more heat is being dissipated. This graph provides valuable insight into the safe operating current for a given gate voltage. However, many datasheets may not include such an extended version and instead only present the safe operating region for various temperatures, as can be seen in Figure 6.

One of the great things about IGBTs, is that they can deal with short-circuits for short periods of time, some IGBTs can handle up to 10μs. The short-circuit withstand time can vary widely from device to device, so it is recommended to build in protection circuits. Overcurrent protection is normally implemented by current measurement by adding shunt resistors in the inverter leg and phase output to cover shoot- through faults and motor winding faults. Microcontrollers must than shut down the IGBT within the short circuit withstand time.

3. IGBT three phase inverter

IGBTs are used in many different power electronic devices, particularly in power conversion systems like motor drives and industrial equipment. Due to their ability to switch high voltages and currents efficiently, a power inverter is a very interesting topic for IGBTs. An inverter converts direct current (DC) into alternating current (AC). This conversion is essential in applications where AC power is needed, such as solar inverters, wind turbines and in the rail industry where you have high voltage DC lines (while the traction motors are AC). Figure 7 shows an high-level block diagram of an IGBT inverter, which will be explained further down below.

The block diagram consists of several important blocks:

IGBT module
Gate drivers
Current sensing technique
3 phase AC induction Motor

The IGBT module consists of six IGBTs in three sets of two in series. Every IGBT’s collector and emitter is connected to the positive and negative side of the DC power supply on one end, and on the other side it is connected to the load. When we switch on the IGBT, it is conducting and current is flowing from high voltage side via the IGBT through the load.

To activate the IGBTs, a gate driver is essential. These drivers typically incorporate galvanic isolation between their input and output sides, often achieved through opto-couplers. Opto-couplers utilize light waves to transmit electrical signals across the isolation barrier, ensuring safety and preventing damage to the control circuitry.

In the setup shown in Figure 7, two IGBTs are placed that resembles a half H-bridge configuration. One IGBT is activated during the first half of a period by the top gate driver. This allows current to flow from the collector (connected to a positive voltage source, VDC+) to the emitter (linked to one of the load’s terminals, U/V/W). Subsequently, during the second half of the period, the top gate driver deactivates the first IGBT, while the bottom gate driver switches on the other IGBT. Now, current travels from the load’s terminal (U/V/W) on the collector side, to the emitter (associated with a negative voltage source, VDC-). This sequential switching mechanism facilitates bidirectional control of current flow through the load.

At this moment, it would make a square wave, switching between max VDC+ and VDC-. One way to solve this, is by using PWM, that allows precise control over the average output voltage supplied to the motor. By varying the width of the pulses in the PWM signal, the effective voltage across the motor terminals can be adjusted. For instance, if the pulses are narrow, the average voltage across the motor terminals will be lower, and if the pulses are wide, the average voltage will be higher. Figure 8 illustrates this idea.

PWM also enables control over the frequency of the generated AC waveform. This control is achieved by adjusting the rate at which the PWM pulses are switched on and off. It’s crucial for aligning the operating frequency of the inverter with the requirements of the AC motor it’s powering.

In addition to this, inverters incorporate an over-current protection, which monitors the amount of current flowing through the system. Various techniques are used for this purpose, including the use of shunt resistors in the inverter leg (as discussed earlier), transducers for measuring current through wires, or in-phase resistors as depicted in Figure 7. Since a constant current flows through the shunt regardless of which IGBT is active, it can also detect line-to-line or line-to-ground shorts. This signal goes through an opamp into a microcontroller or microprocessor that can control the gate drivers and turn it safely off.

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Tunneling effect at semiconductor/oxide interfaces

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2.1 Transmission coefficient

3. MOSFET tunneling

Direct tunneling

Fowler-Nordheim tunneling

3.1 Advancements in nanoelectronics

4. References

Giant (GMR) and Tunnel (TMR) magnetoresistance

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Spin-Transfer Torque: An Introductory Overview

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2.4 Dzyaloshinskii-Moriya interaction

2.4.1 Bulk DMI

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