Wigner Crystallization

Introduction

The study of electron transport is at the very heart of condensed matter physics. Band theory explains the physical properties of numerous materials, such as simple insulators, metals, and semiconductors. In metal, the electrons can move freely even at zero temperature because the conduction band is partially filled. On the other hand, insulators and semiconductor have fully filled valence band separated by an energy gap from an empty conduction band.

According to band theory, electrons in a perfectly periodic array of ions experience no collisions at all. However, some transition metal oxides with partially filled Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} bands (predicted to be a metal) are poor conductors or even insulators. The reason for the absence of carrier mobility is electron localization. The first fundamental mechanism of electron localization is the random scattering of mobile electrons caused by impurities or defect (disorder), which is called Anderson localization. The transition from metal into insulator occurs when the mean free path becomes smaller than the De Broglie wavelength. Based on scaling theory of localization by Abraham, Anderson, et.al. [3], the metal-insulator transition (MIT) exists for non-interacting electrons in three dimensions (3D). In addition, the system is an insulator if the Fermi energy is smaller than the characteristic amplitude of the disorder potential. According to this theory, there is no true metallic behavior in two dimensions (2D) and one dimensions (1D) system with non interacting electron.

The second fundamental mechanism in MIT is caused by electron-electron interaction. One of the simplest examples for interaction driven localization is Wigner crystallization. In this system, each electron is confined not by an ionic potential, but due to the formation of a deep potential well produced by repulsion from other electrons embedded in a positive charge background.

The Jellium Model

The Hamiltonian of N electrons that interact with one another and with a uniform positive charge background is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=\hat{H}_{\mathrm{el}}+\hat{H}_{\mathrm{back}}+\hat{H}_{\mathrm{el-back}},\,}

where

• H_el is the electronic Hamiltonian consisting of the kinetic and electron-electron repulsion terms:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_{\mathrm{el}}=\sum_{i=1}^N\frac{p_{i}^2}{2m}+\sum_{i<j}^N\frac{e^2}{|\mathbf{r}_i-\mathbf{r}_j|}}

• H_back is the Hamiltonian of the positive background charge interacting electrostatically with itself:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_{\mathrm{back}}=\frac{e^2}{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}'\ \frac{n_b(\mathbf{R})n_b(\mathbf{R}')}{|\mathbf{R}-\mathbf{R}'|}}

• H_el-back is the electron-background interaction Hamiltonian, again an electrostatic interaction:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_{\mathrm{el-back}}=\int_{\Omega}\mathrm{d}\mathbf{r}\int_{\Omega}\mathrm{d}\mathbf{R}\ \frac{n(\mathbf{r})n_b(\mathbf{R})}{|\mathbf{r}-\mathbf{R}|} }

• The electron number density operator:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}(\mathbf{r})=\sum_{i=1}^{N}\delta(\mathbf{r-r_{i}})}

• The uniform charge density of the background:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle en_{b}(\mathbf{r})=en} .

In other words, we replace the actual structure of the background on which the electrons live (e.g the atomic lattice) by homogeneous jelly-like continuum of positive charge background. Hence, we call it "Jellium model". We get rid of the complications associated with the structure of the host material, and only focus on the distinctive effects of the electron-electrons interaction. This model is very simplistic for traditional metallic systems, but it is quiet realistic in the semiconductor systems described in the next [hotlink[section]].

If we do Fourier transform of the electronic density operator:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{n}_\mathbf{q}=\sum_{i=1}^{N}e^{-i\mathbf{q.r_i}}} ,

and the interaction potential

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{\mathbf{q}}(\kappa)=\int v(\mathbf{r},\kappa)e^{-i\mathbf{q.r}}d\mathbf{r}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa^{-1}} is the regularization characteristic range by using Yukawa interaction as a tool and take the limit to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa=0} after the end of calculation.

The expression of the potential in each dimensions are:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{\mathbf{q}}=\begin{cases} \frac{4\pi e^{2}}{q^{2}} & 3D,\\ \frac{2\pi e^{2}}{q} & 2D, \textrm{and}\end{cases}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{\mathbf{q}}(a)=-e^{2e^{q^{2}a^{2}}}Ei(-q^{2}a^{2})\quad1D.} ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ei(x)} is the exponential-integral. In one dimension we use the trick by modelling the 1D system as an infinitely long 3D cylinder of small radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , hence the transverse motion of electrons is effectively "frozen" in the lowest energy state described by a Gaussian wave function[1].

The Wigner Crystal

The Coulomb energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{C}} is proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/a} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is the distance between the electrons. Hence, the Coulomb energy is proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{1/d}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is the density and d is the dimension of the system. The Fermi energy is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{F}\sim n^{2/d}} , therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{s}\sim n^{-1/d}} and at very low density , the electron can freeze and form a Wigner Crystal. A Wigner crystal (WC) is believed to be extremely fragile to both quantum and thermal fluctuations. To minimize the potential energy, the electrons form a triangular lattice (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{B}=-3.921\epsilon_{0}} ) in 2D, and BCC lattice in 3D, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{0}=e^{2}/a} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is the nearest distance between electrons. It does not form a square lattice (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{B}=-3.900\epsilon_{0}} ) at zero temperature because of unstable shear modes in 2D systems. However, one cannot rule out the possibility that in some range of temperatures the square lattice (or other) is favored over the triangular lattice by entropy considerations [6]. As density increases, quantum fluctuation dominates the system. Above the critical density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{c}} , a Wigner crystal cannot be found even at zero temperature.

When the density is very low, the distance between the electrons is sufficiently large. Hence, quantum effects are negligible and a classical approach can be applied. The values of the melting temperature Tc depend on the dimension, interaction range, and the filling factor. We can approach the continuum model by using a lattice model with very small filling.

We will review some known facts about Wigner crystallization for both classical and quantum systems. In classical models (Table 1), the transition temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{C}} for short range interactions in lattice system (Ising model) is of the same order as the nearest neighbor (NN) interaction energy (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{C}=1.128} ) . The unit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{C}} is the coupling constant between nearest neighbors particles (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J/k_{B}} in Ising model and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{2}/k_{B}a} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is Ising coupling, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{B}} is Boltzmann constant, e is electron charge, and a is the lattice size spacing). There are only a few models that have exact analytical solutions, such as 1D and 2D Ising models. Other models can only be solved numerically, e.g. 3D Ising model. For long range interaction, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{C}} is one or two orders of magnitude lower than the nearest neighbor coupling . In a very early computer simulation, Brush, Shalin, and Teller [11] observed a transition in a 32 particles continuum Coulomb gas model (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_{c}\approx 125} ). The current estimate is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_{c}\sim172-178} in 3D Wigner crystal and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_{c}\sim125} in 2D Wigner crystal .

The Classical Monte Carlo Approach

We consider models given by the Hamiltonian:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\frac{1}{2}\sum_{ij,i\neq j}V_{ij}(n_{i}-<n>)(n_{j}-<n>)}

where spinless electrons interact via a repulsive long-range potential of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r_{ij})=1/r_{ij}^{\alpha}} on a positive charge background for neutrality. Here, n is the occupation number of the lattice site (n=0 when there is no electron and n=1 when there is an electron on that site). We first examine in half-filled <n>=1/2 lattice where the system can be viewed as anti-ferromagnetic Ising model with the spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{i}=2(n_{i}-1/2} ).

• Classical Monte Carlo/Metropolis algorithm

We use the simple Metropolis algorithm with Boltzmann statistics. Electric charge, lattice spacing, and dielectric constant are taken to be one to make the temperature T dimensionless. The system is a hypercubic half filled lattice with a fixed number of particles.

The metropolis algorithm steps include:

1. Choose an initial state with an electron Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{l}} on a site i

2. Calculate the resulting energy change Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E} if the electron Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{l}} move to a site j

3. Generate a random number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<n_{r}<1}

4. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E<} 0 or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{r}<e^{-\Delta E/k_{b}T}} the movement is accepted

5. Go to the next electron Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{m}} and repeat the procedure.

In our simulation on half filled lattice, each movement in step 2 is only one site distant. If one uses lower filling factor in the lattice system, electron can jump over different distance.

Ewald Summation

• Ewald summation to calculate potential from finite size system with periodic boundary condition

In the following we describe some technical details of the computational procedure used in Monte-Carlo simulations, in presence of long-range interactions.

Ewald Potential. In order to compute the effective potential of long range interaction $1/|{\vec {r}}_{ij}|^{\alpha }$ in hypercubic lattice, we use Ewald-type summation $V({\vec {r}}_{ij})$ with the help of the integral representation of

${\frac {1}{|{\vec {r}}|^{\alpha }}}={\frac {1}{\Gamma (\alpha /2)}}\int _{0}^{\varepsilon }t^{{\frac {\alpha }{2}}-1}e^{-r^{2}t}dt+\int _{\varepsilon }^{\infty }t^{{\frac {\alpha }{2}}-1}e^{-r^{2}t}dt$

where $\Gamma (\alpha )$ is Gamma function. We switch the first term of the integral to a momentum sum because the sum does not converge rapidly in the real space. Next, we use the representationwhere $f({\vec {r}})$ is any arbitrary function and on the right-hand side the summation is over the vectors of the reciprocal lattice. We then integrate out r and change the variable of the integration in the first term $t\rightarrow 1/t$. The final expression of the potential takes the form $V({\overrightarrow {r}})$ Next, we use the representation:

$\int d^{d}r\sum _{\vec {n}}\delta ({\vec {r}}-[L{\vec {n}}+{\vec {r}}_{ij}])f({\vec {r}})=\int d^{d}r\sum _{{\vec {G}}_{l}}[e^{i{\vec {G}}_{l}.{\vec {r}}}-\delta ({\vec {r}})]f({\vec {r}})$,

where f(\vec{r}) is any arbitrary function and on the right-hand side the summation is over the vectors of the reciprocal lattice. We then integrate out r and change the variable of the integration in the first term t\rightarrow1/t. The final expression of the potential takes the form:

$V({\overrightarrow {r}})={\frac {1}{\Gamma ({\frac {\alpha }{2}})}}\sum _{\vec {n}}\varepsilon ^{\alpha }\phi _{{\frac {\alpha }{2}}-1}(\varepsilon ^{2}|{\overrightarrow {r}}+{\overrightarrow {n}}L|^{2})+{\frac {1}{\Gamma ({\frac {\alpha }{2}})\Omega }}\sum _{{\vec {k}}\neq 0}\pi ^{\frac {3}{2}}\varepsilon ^{\alpha -3}\phi _{\frac {1-\alpha }{2}}({\frac {|{\vec {k}}|^{2}}{4\varepsilon ^{2}}})e^{-i{\vec {k}}.{\vec {r}}}-{\frac {2\varepsilon ^{\alpha }}{\Gamma ({\frac {\alpha }{2}})}},$

Simulation in half filled hypercubic Wigner crystal

Our results give support to the physical pictures of Wigner crystals proposed by experimentalists [Tsui,et.al.]. The existence of an almost frozen liquid at $k_{B}T_{C}\lesssim k_{B}T\ll E_{C}$ proves to be a general phenomenon of any system with sufficiently long range interactions. We focus on the simplest possible models that can be studied in detail. Our system is a lattice gas model with Coulomb-like interactions in the form $V(r)\sim R^{-\alpha }$.

We consider models given by the Hamiltonian:

$H={\frac {1}{2}}\sum _{ij,i\neq j}V_{ij}(n_{i}-<n>)(n_{j}-<n>)$

Hwhere spinless electrons interact via a repulsive long-range potential of the form

$V(r_{ij})=1/r_{ij}^{\alpha }$

on a positive charge background for neutrality. Here, $n$ is the occupation number of the lattice site ($n=0$ when there is no electron and n=1 when there is an electron on that site). We first examine in half-filled $<n>=1/2$ lattice where the system can be viewed as anti-ferromagnetic Ising model with the spin $s_{i}=2(n_{i}-1/2$).

2.1 Classical Monte Carlo Simulation

We use the simple Metropolis algorithm with Boltzmann statistics. Electric charge, lattice spacing, and dielectric constant are taken to be one to make the temperature T dimensionless. The system is a hypercubic half filled lattice with a fixed number of particles.

The metropolis algorithm steps include:

1. Choose an initial state with an electron $e_{l}$ on a site i

2. Calculate the resulting energy change $\Delta E$ if the electron $e_{l}$ move to a site j

3. Generate a random number r such that $0<n_{r}<1$

4. If $\Delta E<0$ or $n_{r}<e^{-\Delta E/k_{b}T}$ the movement is accepted

5. Go to the next electron e_{m} and repeat the procedure.

In our simulation on half filled lattice, each movement in step 2 is only one site distant. If one uses lower filling factor in the lattice system, electron can jump over different distance.

2.1.1 Simulation Steps

For N particles, one Monte Carlo time includes N metropolis cycles/updates. We use sequential updating rather than random updating to get smaller autocorrelation time [20].

A typical MC simulation consists of two parts:

1. Thermalization: Initial sweeps before the system reaches equilibrium state. There is no measurement during these sweeps.

2. Production: The system reaches equilibrium state and the measurements are performed.

The first step of the simulation is setting the initial condition. At low temperature, the system starts from crystallized condition (checkerboard pattern). At high temperature, the initial condition of the particles are randomly arranged. The decorrelation steps (purge sweeps between measurement) was performed to reduce the integrated autocorrelation time \tau_{int} [20]. The autocorrelation function of any f is defined by:

${\hat {C}}(t)=\left\langle (f(t)-\left\langle f(0)\right\rangle )(f(t)-\left\langle f(0)\right\rangle )\right\rangle$,

the variance of f is a special case of autocorrelations ${\hat {C}}(0)=\sigma ^{2}(f)$. We need to calculate the autocorrelation time to determine the right error bar $\sigma$ of any statistical parameter, $\sigma ^{2}({\bar {f}})=\sigma _{naive}^{2}({\bar {f}})/\tau _{int}$, where the estimated uncorrelated random variables error bar $\sigma _{naive}({\bar {f}})=\sigma (f)/{\sqrt {N}}$ (the central limit theorem), and $\tau _{int}$ is the integrated autocorrelation time, where

${\hat {c}}(t)={\hat {C}}(t)/{\hat {C}}(0)$.

Finite size effect and scaling

Finding Tc from order parameter curves

Classical Electron Liquid visualization

Algorithm and visualization of Electron liquid with disorder and electric field (linked to the project with high school students)

Phase diaram of electron liquid

How the phase change with temperature and $r_{s}$.

Experimental Support of Wigner crystallization

The picture of Wigner crystallization can be realized by ultra low density 2D electron or hole liquid in semiconductor devices. At the lowest density and the absence of disorder, one expects the electrons to freeze into a Wigner crystal, driving the system into an insulator. As we increase the density, the kinetic energy becomes sufficiently large, leading to the formation of a metal.

Over the last few years, Tsui and collaborators have searched for more direct evidence for Wigner crystallization, by looking at the cleanest samples at the lowest accessible densities $n\sim 10^{11}cm^{-2}$ in zero magnetic field ($\mu \sim 10^{6}cm^{2}/Vs$ in GaAs, $\mu \sim 10^{4}cm^{2}/Vs$ in Si).

• References