Magnetotransport properties of the Quantum Spin Hall and Quantum Hall states in an inverted HgTe/CdTe and InAs/GaSb quantum wells

A topological insulator is a material that behaves as an insulator in the bulk, but has conducting states at their edges or surfaces, meaning that electrons conduct dissipationless along the edges or surfaces of the materialtopo; topo2. These surface/edge electronic states have spin-orientations following the direction of the electron’s momentum, thus protected by the time-reversal (TR) symmetryTR. Quantum spin Hall (QSH) states in HgTe/CdTe ref1; ref2; ref3; ref4 and InAs/GaSb InAs1; InAs2; ref2; InAs4 quantum wells arising from the band inversion (where the top of the valence band is at an energy level higher than the bottom of the conduction band) are a few of such candidates where topological properties are inevitable. The band inversion in these systems is tunable to achieve a topological insulating behaviortune; tune2. The Fermi level in such quantum wells are accompanied by a completely inverted band structurebhz; Hamil2. Also, the QSH states in these materials are characterized by a topological $Z_{2}$ invariant that predicts their topological phase, (whether the system hosts the trivial insulating states or the QSH time-reversal symmetry protected edge states).

Theoretically, the system is described by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian which is a four-band model, having two electron bands (spin up and spin down) and two hole bands (spin up and down).bhz; Hamil2; TR; Hamil3; ref1; Hamil5; QSH

Here, $\sigma_{i}$, with $i=x,y,z$ being the Pauli’s spin matrices and $\sigma_{0}$ is the $2\times 2$ identity matrix. The first term $\epsilon({\bf k})=C-dk^{2}$ consists of the symmetry-breaking term, ‘$d$’ between electron and hole bands. The second term $M({\bf k})=M_{0}-bk^{2}$ consists of two parameters, viz., ‘$M_{0}$’ that controls the band inversion and ‘$b$’ that symmetrically controls the band curvatures. The last two terms control the electron-hole coupling. This basic theory of the QSH leads to the search of materials with band inversion at the TR invariant k-points, two example of which are InAs/GaSb and HgTe/CdTe quantum wells. As a consequence of the band inversion, such a system hosts gapless surface or edge states as demonstrated in Fig. (1). The spin configuration of the edge states (which always occur in pairs) in such a system is demonstrated in Fig. (2) (often called as QSH system).

It is well-known that an application of a magnetic field is one way of breaking the TR symmetry in such a systemTRB. In the presence of a magetic field, these inverted quantum systems shows a peculiar behaviour of the zeroth landau levels. Precisely, below a critical magnetic field $B<B_{c}$, the uppermost valence Landau level has instead electron-like characteristics and the lowermost conduction Landau level has hole-like characteristicsll1; ll2; ll3. In the regime $B<B_{c}$ the quantum well still hosts counterpropagating spin polarised edge states. In other words, QSH still exists in the regime $B<B_{c}$. Beyond the critical magnetic field ($B>B_{c}$), there is a band ordering that trace back to the normal Landau levels same as that of a Quantm Hall (QH) system.

In this paper, we present a theoretical study that reveals some results out of the peculiar electron-like nature of the valence band and the hole-like nature of the conduction band. Specifically, we present a calculation of the magnetotransport coefficients, namely the longitudinal and Hall conductivity in such a system. Based on the results of the longitudinal and Hall conductivity, we look for some conditions by which one can define a clear distinction between the two possible quantum states in the system, viz. , the QSH and QH states. In short, the main finding of this work is centered around the transition from QSH to QH state. We show that, there is a regime of higher Hall conductance around the low Fermi energy level when the system is in the QH regime, while the same is less dominant in the QSH regime. The boundary that partitioned this jump in the conductance, is the critical magnetic field which goes as $B_{c}=4\hbar M_{0}/(4b-\mu_{B}\hbar(g_{e}+g_{h}))$. This value of the critical field comprehends the results reported in the previous works critical1; Hamil2.

This paper is further organized as follows: In section II, we present the basic formalism of the inverted band structure in InAs/GaSb and HgTe/CdTe bilayer. In section III, we present the analytical calculations for the different transport coefficients. Numerical results and discussion are presented in section IV. We summarise this paper in section LABEL:Sum.

We started from the BHZ Hamiltonian that describe the QSH states in these systems as given in Eq. (1). The energy spectrum here is $\varepsilon_{\lambda}({\bf k})=\epsilon({\bf k})+\lambda\sqrt{M({\bf k})^{2}+A^{2}k^{2}}$, where both the conduction band ($\lambda=+1$) and the valence band ($\lambda=-1$) are spin degenerate. The topological invariant number for the Hamiltonian in Eq. (1) is found to be $\mathcal{C}=({\rm sign}(M_{0})+{\rm sign}(b))/2$ which does depend on the band inversion parameter, ‘$M_{0}$’, expectedly. In the presence of a magnetic field, the momentum vector transformed according to ${\bf k}\rightarrow{\bf k}+e/\hbar{\bf A}$. For an applied magnetic field ${\bf B}=B\hat{z}$ normal to the plane containing the quantum well, the BHZ Hamiltonian can be expressed in terms of the ladder operator ‘$a$’ and ‘$a^{\dagger}$’ as below

Here, we have considered the Landau gauge ${\bf{A}}=(0,xB,0)$. The Hamiltonian operator has a two non-zero blocks and it operates seperately on the two spin degrees of freedom. Here, $h_{0}^{+}=-(b+d)\frac{2}{l_{c}^{2}}(a^{\dagger}a+\frac{1}{2})+M_{0}+C+\frac{g_{e}\mu_{B}B}{2}$ and $h_{0}^{-}=(b-d)\frac{2}{l_{c}^{2}}(a^{\dagger}a+\frac{1}{2})-M_{0}+C-\frac{g_{h}\mu_{B}B}{2}$, with $g_{e}/g_{h}$ being the Lande g-factor of the conduction/valence band. The two operators ($a$ and $a^{\dagger}$) when acting on the oscillator wave function $|n\rangle$ give $a|n\rangle=\sqrt{n}|n-1\rangle$ and $a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle$, respectively. The oscillator wavefunction is itself a function of $x-k_{y}l_{c}^{2}$, where $k_{y}$ is good quantum number, since the Hamiltonian commutes with momentum operator $p_{y}$. The Landau levels wavefunction, $\Psi_{n,k_{y}}^{s,\lambda}(x,y)$ for $n\geq 1$ in terms of the oscillators wave function $|n\rangle$ are as follows

Here $s=+/-$ and $\lambda=+/-$ are the good quantum numbers that denote the spin-up/spin-down electronic or hole state and the conduction/valence bands of the system, respectively.

The corresponding Landau levels are

with $\mu_{Z}=\frac{\mu_{B}(g_{e}+g_{h})B}{4}$ and the different coefficients in Eq. (II) are given by the genral formula

where $\Delta_{sn}=\sqrt{\frac{2A^{2}n}{l_{c}^{2}}+(M_{0}+\frac{sd-2bn}{l_{c}^{2}}+\mu_{Z})^{2}}$. The normalizing constants are $\mathcal{A}_{\uparrow,n}=1+\alpha_{\uparrow n}^{2},\;\mathcal{A}_{\downarrow,n}=1+\alpha_{\downarrow n}^{2}$. The eigensystem of the two Landau levels for $n=0$ are found to be as follows

The landau level spectrum for both $M_{0}<0$ and $M_{0}>0$ is shown in Fig. (2 [d & e]). For a given Fermi level (given in solid dashed line of Fig. (2 [e)]) and assuming that it is located just below the level $\varepsilon_{1}^{\downarrow,+}$, we can divide the spectrum into two regions. Region I, where the QSH states still persist (that the pair of edge states consists of both the up and down spin states: Refer Fig. (2 [b])). With further increase of the magnetic field, we reach a critical magnetic field $B_{c}=4\hbar M_{0}/(4b-\mu_{B}\hbar(g_{e}+g_{h}))$, after which the edge states are just normal QH states (Refer Fig. (2 [c])). Here the TR symmetry is broken such that the edge states now consist of a pair of electronic states of the same spin configuration propagating in the same direction. We determined this critical field by equating the two Landau level states with $n=0$. For a given values of the involved parameters, this value of the critical magnetic field that separates the QSH and the QH states is found consistent with that reported in references[critical1; Hamil2].

To calculate the longitudinal and Hall components of the conductivity tensor, we employ the Kubo formalismMahan. The collisional contribution accords to the longitudinal component of the conductivity tensor. The diffusive conductivity which has a vanishing contribution is due to the zero value of the expectation value of the diagonal elements of the velocity operator.kubo.

Collisional conductivity: The calculation of the collisional conductivity involves an assumption that the fermions are elastically scattered by the charged impurities which is presumed to be distributed uniformly over the quantum well. This assumption limits to low temperature for the calculation to remain valid. The Kubo formula for the expression of the collisional conductivity is given by Van; Carol; vasilo; Peet; wang

Here $|\xi\rangle=|\lambda,s,n,k_{y}\rangle$ is the set of all quantum numbers that defines the eigenstates of the system, $\Omega$ is the surface area containing the quantum well, $f(\varepsilon_{\xi})=1/(\exp((\varepsilon_{\xi}-\mu)\beta)+1)$ the Fermi-Dirac distribution function with $\beta=1/(k_{B}T)$ and $x^{\xi}=\langle\xi|x|\xi\rangle=k_{y}l_{c}^{2}$ being the expectation value of the $x$ component of the position operator. Finally, the transition probability between two states $|\xi\rangle$ and $|\xi^{\prime}\rangle$ is given by

Here the quantity $n_{\rm im}$, is the impurity density in the quantum well and and $U(q)=1/(2\epsilon_{0}\epsilon(q^{2}+k^{2}_{s})^{1/2})$ the Fourier transform of the considered screened Yukawa-type impurity potential taken as $U(r)=e^{2}e^{-k_{s}r}/(4\pi\epsilon_{0}\epsilon r)$. The constants $\epsilon$ and $k_{s}$ are the dielectric constant of the medium and the screened wave vector, respectively. The form factor $F_{\xi,\xi^{\prime}}=\langle{\xi}|e^{i{\bf q}\cdot{\bf r}}|{\xi^{\prime}}\rangle$ is derived in the Appendix LABEL:AppA. By writing a general form of the wavefunction as $\Psi_{n,k_{y}}^{s,\lambda}(x,y)=e^{ik_{y}y}\begin{pmatrix}p_{n}|n-1\rangle&iq_{n}|n\rangle&r_{n}|n-1\rangle&is_{n}|n\rangle\end{pmatrix}^{\prime}/\sqrt{\mathcal{A}_{s,n}}$, where ‘$\prime$’ indicates the transpose, the collisional conductivity for $n\geq 1$ as given in Eq. (LABEL:A6) of the appendix is

Here in this paper, we will scale the longitudinal and Hall conductivity in units of $\sigma_{Q}=e^{2}/h$. Following Eq. (LABEL:n_0_coll) of the Appendix (LABEL:AppA), we have for $n=0$

Hall conductivity: The expression for the Hall conductivity $\sigma_{yx}$ is given by vasilo; Peet; wang

The matrix elements of the components of the velocity operator are given in Eqn (LABEL:vell) of Appendix LABEL:AppB. By virtue of the Kronecker delta symbols in Eqs. (LABEL:vy_exp & LABEL:vx_exp), it is confirmed that the transitions are allowed only between the adjacent Landau levels $n^{\prime}=n\pm 1$. The contribution due to the inter-spin branch transitions is found to be negligibly small compared to the contribution from the intra-spin branch transitions. Also, we observed that the contribution of the transitions from conduction to valence bands and vice versa of the type $\varepsilon_{n}^{s,\pm}\rightarrow\varepsilon_{n+1}^{s,\mp}$ are again found to be negligibly small. The Hall conductivity in Eq. (14) thus has terms that should only involve transitions of the type $\varepsilon_{n}^{s,\pm}\rightarrow\varepsilon_{n+1}^{s,\pm}$.

The form of the Hall conductivity expressed as a summation involving the quantum numbers ($n,s,\lambda$) is derived in Eq. (LABEL:hall_con_form_1 & LABEL:hall_con_form_2) of Appendix [LABEL:AppB], the results of which will be discussed in the subsequent section (IV).

For analysing the Longitudinal and Hall conductivity in the above section III, we take the value of the different parameters refering to table I below.