Double-layer Quantum Hall Ferromagnets

Introduction

The discovery of quantum Hall effect (QHE) was one of the most remarkable achievements in condensed matter physics in the second half of the last century. Together with superconductivity, QHE has extended our knowledge of quantum mechanics in many-body systems. In this effect, a 2D electron gas when subjected to a strong magnetic field exhibits a quantized Hall resistance and, at the same time, a nearly vanishing dissipative resistance in a range of magnetic field strength. The quantized Hall resistance can be characterized by integer quantum numbers in the integer quantum Hall effect (IQHE) or by some rational fractions in fractional quantum Hall effect (FQHE). In the IQHE regime, electrons occupy only the first few Landau levels because of the enormous degeneracy of these levels at high magnetic field. In the FQHE, the underlying physics is the Coulomb interaction and the correlation among electrons. It turns out that Coulomb interaction plays an important role not only in FQHE but also in QHE when we consider the spin dynamics of the system. In free space, the Zeeman splitting is exactly the same as the cyclotron splitting which is of the order of 100 K. But in real material, such as GaAs, the Zeeman splitting is reduced by about 2 orders of magnitude compared to the cyclotron splitting due to the reduction in effective mass and the gyromagnetic ratio of electrons. Therefore, at low temperature, the system lies in a special situation in which the orbital motion is fully quantized 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 \displaystyle k_B T \ll \hbar \omega_c} but the low-energy spin fluctuations are not forbidden 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 \displaystyle k_B T \sim g^* \mu_B B} . Interestingly, in the ground state of quantum Hall system, all the spins are still aligned ferromagnetically not due to the Zeeman splitting but due to the Coulomb interaction. Coulomb interaction is independent of spin therefore we can expect the same effect in double-layer quantum Hall system with the layer indices play the role of spin. In this report, we start out with an introduction to quantum Hall effect and quantum Hall ferromagnet. We then discuss two quantum many-body effects in double-layer quantum Hall ferromagnet by using pseudo spin treatment in which the layer indices play the role of spin. The first one is the interlayer phase coherent effect which associated with the interaction between electrons in different layers. The second one is the tunneling between the two layers and the effect of the in plane component of the magnetic field, which is related to the broken 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 \displaystyle U(1)} symmetry of the system.

Quantum Hall Effect

The advance in technology makes it possible to prepare 2D electron gas system with extremely low disorder and high mobility. The motion of an electron in such system in the presence of a uniform magnetic field perpendicular to the plane can be studied conveniently by choosing the Landau gauge for the vector 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 \displaystyle \bold A(\bold r)=xB \hat{\bold y}} .

Using this gauge, the physics of the system is invariant under the translation in y axis. The wave function for the motion in this axis is therefore simply plane wave characterized by the wave vector 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 \bold k} . By separating variables one can see that the wave function for the motion in 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 \bold x} axis is the same as that of a harmonic oscillator whose frequency equals to the cyclotron frequency and whose motion centers at

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 X_{\bold k}=-k l^2}

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 l= \sqrt {\frac {\hbar c} {eB}}} is the magnetic length.

The energy levels corresponding to the motion in 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 \bold x} axis are quantized as

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_{k,n}=(n+\frac1{2})\hbar \omega_c}

and are called Landau levels. Landau levels degenerate because they do not depend on the wave vector 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 \bold k} . The numbers of states in each Landau levels can be calculated by imposing the periodic boundary condition in 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 \bold y} axis. Let’s consider a rectangular sample with dimensions 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 \displaystyle L_x} ; 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 \displaystyle L_y} and the left edge is at 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 \displaystyle x=-L_x} and the right edge is at 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 \displaystyle x=0} , then the condition for the centers of motion to be inside the sample

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 \displaystyle -L_x \leqslant X_c \leqslant 0}

gives

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 \displaystyle 0 \leqslant k \leqslant \frac {L_x}{l^2}} .

The total number of states in each Landau levels is then:

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 \displaystyle N = \frac {L_y}{2 \pi} \int_0^{\frac {L_x}{l^2}} dk = \frac {L_x L_y}{2 \pi l^2}=N_\phi}

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 \displaystyle N_{\phi}=\frac {BL_xL_y}{\phi_0}} is the number of flux quanta penetrating the sample;

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 \displaystyle {\phi_0}=\frac {hc}{e}} is the quantum of magnetic flux. So there is exactly one state per Landau level per flux quantum.

In general, applying an electric field along the 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 \displaystyle \bold x} axis will result in a shift in the centers of the motions 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 \displaystyle X_k} and a 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 \displaystyle \bold k} -dependence in Landau levels. If the electric field is non-uniform and there are disorders in the system, all the states in the bulk are localized due to Anderson’s localization. In the other words, deep inside the sample, the Landau levels are still 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 \displaystyle \bold k} -independent, the group velocity is therefore equal to zero. But near the edges 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 \displaystyle x=-L_x} 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 \displaystyle x=0} of the sample Landau levels does depend on 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 \displaystyle \bold k} , furthermore, the group velocity

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 \displaystyle \bold v_k=\frac {1}{\hbar} \frac {\partial \epsilon_k}{\partial k} \hat {\bold y}}

has the opposite sign on the two edges of the sample. Therefore, there are edge currents running in opposite directions along the 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 \displaystyle \bold y} axis. The electric transport in the system can be analyzed in analogy with the Landauer formalism for the transport in narrow wide. The edge currents correspond to the left and the right moving states between two Fermi points. The net current can be calculated by adding up the group velocities of all the occupied states:

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 \displaystyle I=-\frac{e}{L_y}\int_{-\infty}^{+\infty}dk{\frac {L_y}{2 \pi}{\frac {1}{\hbar}}{\frac {\partial \epsilon_k}{\partial k}}}n_k}

where we assume that in the bulk only a single Landau level is occupied 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 \displaystyle n_k} is the occupied probability. At zero temperature, we have

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 \displaystyle I=-\frac {e}{h} \int_{\mu_R}^{\mu_L}d \epsilon}

The definition of the Hall voltage drop 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 \displaystyle (+e)V_H \equiv (+e)(V_R - V_L)=\mu_R - \mu_L}

Hence

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 \displaystyle I=-\frac {e^2}{h}V_H}

If there 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 \displaystyle \nu } Landau levels are occupied in the bulk, then

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 \displaystyle I=- \nu \frac {e^2}{h}V_H}

Here, the applied voltage is in 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 \displaystyle \bold x} axis and the net current is along 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 \displaystyle \bold y} axis, therefore what we are calculating here is indeed the Hall resistance. So, we derive to what are observed in QHE

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 \displaystyle \sigma_{xx} = 0}

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 \displaystyle \sigma_{xy} = - \nu \frac {e^2}{h}}

where the integer quantum 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 \displaystyle \nu } is also known as Landau level filling factor.

Quantum Hall Ferromagnet

Even though FQHE is not necessary for the discussions in this report, it is worth to mention that in that class of QHE, the quantum 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 \displaystyle \nu } takes in some rational fraction

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 \displaystyle \nu = \frac 1{3};\frac 2{5};\frac 3{7};\frac 2{3};\frac 3{5};\frac 1{5};\frac 2{9};\frac 3{13};\frac 5{12};\frac {12}{5};...} .

The underlying physics in FQHE is the Coulomb interaction and the correlation between electrons. One may think that such interaction has nothing to do with integer quantum Hall system. However, the study of ferromagnetism in the system with 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 \displaystyle \nu = 1} has shown that Coulomb interaction also plays an important role. In a fully ferromagnetic state, all the spins are lined up parallel to each other, hence the spin part of the wave function is symmetric under the particle exchanges. Therefore, the spatial part of the wave function must be fully antisymmetric and vanish when any pair of particles approaches each other. Such condition keeps the particles away from each other thus lowers the Coulomb interaction. It turns out that for filling factor 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 \displaystyle \nu = 1} , the Coulomb interaction is about 2 orders of magnitude greater than the Zeeman splitting and hence strongly stabilizes the ferromagnetic state. Indeed, at zero temperature, the ground state of the system with 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 \displaystyle \nu = 1} is spontaneously fully polarized even in the absence of an external magnetic field. The spin wave excitations can be studied by using the method similar to that used in Heisenberg model despite the fact that spins in Heisenberg model are localized whereas quantum Hall ferromagnet is a system of itinerant spins. The spin wave dispersion shows a gap at 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 \displaystyle \bold k=0} equal to the Zeeman splitting and starts out quadratically at small 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 \displaystyle \bold k} . At large wave vectors, the energy saturates at the Coulomb interaction scale. Effective action theory can also be used to reproduce these results. In this theory, first we write down the Lagrangian of the system and then derive the equation of motion. The Lagrangian that realizes the correct precession of spin in magnetic field and also the global spin rotation (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 \displaystyle SU(2)} symmetry)can be written as

$\displaystyle L=-\hbar Sn\int {d^{2}{\mathbf {r}}\{{{\dot {m}}^{\mu }({\mathbf {r}})}A^{\mu }({\mathbf {m}})-\Delta m^{z}({\mathbf {r}})}\}-{\frac {1}{2}}\rho _{s}\int {d^{2}{\mathbf {r}}\partial _{\mu }m^{\nu }\partial _{\mu }m^{\nu }}+\int {d^{2}{\mathbf {r}}\lambda ({\mathbf {r}})(m^{\mu }m^{\mu }-1)}$

where

$\displaystyle S={\frac {1}{2}}$ is the spin length

$\displaystyle n={\frac {\nu }{2\pi l^{2}}}$ is the particle density

$\displaystyle {\dot {m}}^{\mu }({\mathbf {r}})={\frac {dm^{\mu }({\mathbf {r}})}{dt}}$

$\displaystyle {\mathbf {m}}$ is a vector field of unit length which describes the local orientation of the order parameter (the magnetization)

$\displaystyle {\mathbf {A}}$can be determined by requiring that it leads to the correct precession of the magnetization

$\displaystyle {\mathbf {\Delta }}=\Delta {\hat {\mathbf {z}}}$ is the external magnetic field

$\displaystyle \rho _{S}$ is a phenomenological spin stiffness

$\displaystyle \lambda$ is the Lagrange multiplier that imposes the unit length constraint.

… NEED MORE WORK

Double-layer Quantum Hall Ferromagnet

Double-layer quantum Hall ferromagnet consists of two layers of 2D electron gas. Although the layer separation is comparable to the distance between electrons in the same layer, the interlayer electrons interaction is still smaller than the intralayer electrons interaction, therefore in this case the system does not have a full $\displaystyle SU(2)$ symmetry. The Lagrangian is now written as

$\displaystyle L=-\hbar Sn\int {d^{2}{\mathbf {r}}\{{{\dot {m}}^{\mu }({\mathbf {r}})}A^{\mu }({\mathbf {m}})-\Delta m^{z}({\mathbf {r}})}\}-{\frac {1}{2}}\rho _{s}\int {d^{2}{\mathbf {r}}\partial _{\mu }m^{\nu }\partial _{\mu }m^{\nu }}+\int {d^{2}{\mathbf {r}}\lambda ({\mathbf {r}})(m^{\mu }m^{\mu }-1)}-$

$\displaystyle -\int {d^{2}{\mathbf {r}}\beta m^{z}m^{z}}+\int {d^{2}{\mathbf {r}}ntm^{z}}$

...NEED MORE WORK

Broken $\displaystyle SU(2)$ symmetry; XY model

...NEED MORE WORK

Linear instead of quadratic dispersion due to the $\displaystyle \beta$ term

...NEED MORE WORK

Interlayer Phase Coherent

Interaction

$|\psi \rangle =\prod _{\mathbf {k}}\{c_{{\mathbf {k}}\uparrow }^{+}+c_{{\mathbf {k}}\downarrow }^{+}e^{i\phi }\}|0\rangle$

$H={\frac {1}{2}}\int {d^{2}{\mathbf {r}}|\nabla \phi |^{2}}+...$

... NEED MORE WORK

Interlayer Tunneling and Tilted Field Effects

Broken $\displaystyle U(1)$ symmetry

$H_{eff}={\frac {1}{2}}\int {d^{2}{\mathbf {r}}|\nabla \phi |^{2}}-\int {d^{2}{\mathbf {r}}ntcos(\phi )}$

Tunneling

$H_{T}=-t\int {d^{2}{\mathbf {r}}\{\psi _{\uparrow }^{+}({\mathbf {r}})\psi _{\downarrow }({\mathbf {r}})+\psi _{\downarrow }^{+}({\mathbf {r}})\psi _{\uparrow }({\mathbf {r}})\}}$

$H_{T}=-t\int {d^{2}{\mathbf {r}}S^{x}({\mathbf {r}})}$

In the present of magnetic field

$H_{eff}={\frac {1}{2}}\int {d^{2}{\mathbf {r}}|\nabla \phi |^{2}}-\int {d^{2}{\mathbf {r}}ntcos(\phi -Qx)}$

...NEED MORE WORK