Spin-orbit interaction in a dual gated InAs/GaSb quantum well

TL;DR

This study explores how electric fields influence spin-orbit interactions in a dual gated InAs/GaSb quantum well, revealing tunable spin splitting and detailed characterization of Dresselhaus and Rashba effects.

Contribution

It provides the first detailed measurement of spin-orbit coupling parameters in a dual gated InAs/GaSb quantum well, demonstrating electric field control of spin splitting.

Findings

Linear Dresselhaus strength is 28.5 meVÅ.

Rashba coefficient varies from 75 to 53 meVÅ with electric field.

Spin splitting exhibits nonmonotonic behavior in two-carrier regime.

Abstract

Spin-orbit interaction is investigated in a dual gated InAs/GaSb quantum well. Using an electric field the quantum well can be tuned between a single carrier regime with exclusively electrons as carriers and a two-carriers regime where electrons and holes coexist. Spin-orbit interaction in both regimes manifests itself as a beating in the Shubnikov-de Haas oscillations. In the single carrier regime the linear Dresselhaus strength is characterized by $\beta =$ 28.5 meV$\AA$ and the Rashba coefficient $\alpha$ is tuned from 75 to 53 meV$\AA$ by changing the electric field. In the two-carriers regime the spin splitting shows a nonmonotonic behavior with gate voltage, which is consistent with our band structure calculations.

Equations

$$H=\frac{({\hat{p}}_x^2+{\hat{p}}_y^2)}{2m^{\ast }}{\sigma }_0+\alpha ({\hat{p}}_y{\sigma }_x-{\hat{p}}_x{\sigma }_y)/\hslash +\beta ({\hat{p}}_x{\sigma }_x-{\hat{p}}_y{\sigma }_y)/\hslash +\gamma ({\hat{p}}_y{\hat{p}}_x{\hat{p}}_y{\sigma }_x-{\hat{p}}_x{\hat{p}}_y{\hat{p}}_x{\sigma }_y)/{\hslash }^3+\frac{1}{2}g{\mu }_{B}B{\sigma }_z$$

$$\begin{array}[]{c}H=\frac{(\hat{p}_{x}^{2}+\hat{p}_{y}^{2})}{2m^{*}}\sigma_{0}+\alpha(\hat{p}_{y}\sigma_{x}-\hat{p}_{x}\sigma_{y})/\hbar+\beta(\hat{p}_{x}\sigma_{x}-\hat{p}_{y}\sigma_{y})/\hbar\\ +\ \gamma(\hat{p}_{y}\hat{p}_{x}\hat{p}_{y}\sigma_{x}-\hat{p}_{x}\hat{p}_{y}\hat{p}_{x}\sigma_{y})/\hbar^{3}+\frac{1}{2}g\mu_{B}B_{z}\sigma_{z}\end{array}$$

$$\hat{p}=-i\hslash \nabla +e\mathbf{A}.$$

$$\begin{array}[]{rr}a^{\dagger}=&\displaystyle\frac{\lambda_{c}}{\sqrt{2}\hbar}\left(\hat{p}_{x}+i\hat{p}_{y}\right),\\[11.38109pt] a=&\displaystyle\frac{\lambda_{c}}{\sqrt{2}\hbar}\left(\hat{p}_{x}-i\hat{p}_{y}\right),\end{array}$$

$${\sigma }_{xx}=\frac{e^2}{{\pi }^2\hslash }\sum _{n,\uparrow \downarrow }\left(n\pm \frac{1}{2}\right)\exp \left(-\frac{(E_f-E_{n,\uparrow \downarrow }{)}^2}{{\Gamma }^2}\right).$$

$${\rho }_{xx}={\sigma }_{xx}/({\sigma }_{xy}^2+{\sigma }_{xx}^2)\approx {\sigma }_{xx}/{\sigma }_{xy}^2\approx {\sigma }_{xx}{\left(B_z/e{n}_s\right)}^2$$

Full text

Combining InAs and GaSb in a quantum well gained much interest because of the type-II broken-gap band alignment Kroemer (2004). As a result, the GaSb valence band maximum is higher in energy than the InAs conduction band minimum, opening a range of energies where electrons in the InAs coexist with holes in the GaSb. The spatial separation of these electron and hole gases allows for tunability of the band alignment using an electric field. Therefore, a rich phase diagram can be mapped out using dual gated devices Liu et al. (2008); Qu et al. (2015). Although spatially separated, strong coupling between the materials allows for electron-hole hybridization which opens a gap in the energy spectrum when the density of electrons equals that of holes de Andrada e Silva et al. (1994); Cooper et al. (1998), driving the band structure topologically non-trivial Liu et al. (2008).

Interestingly, the magnitude of this hybridization gap is spin dependent due to SOI Zakharova et al. (2001); Halvorsen et al. (2000); Xu et al. (2010). Therefore, a spin polarized state is seen at energies close to the hybridization gap Nichele et al. (2016) and at higher energies a dip in the spin-splitting is expected Li et al. (2008). The latter has yet to be observed. Here, we experimentally study SOI through the difference in density of the spin-orbit split bands of an InAs/GaSb quantum well. This zero-field density difference ($\Delta n_{\tiny\textit{ZF}}$) is extracted from magnetoresistance measurements. First, SOI is investigated in the regime where the GaSb is depleted from carriers. Rashba and Dresselhaus SOI strengths can be extracted from measurements of $\Delta n_{\tiny\textit{ZF}}$. Second, SOI is investigated just above the hybridization gap where $\Delta n_{\tiny\textit{ZF}}$ almost vanishes, consistent with band structure calculations.

A 20 µm wide and 80 µm long Hall bar device is defined using chemical wet etching techniques. A top gate is separated from the mesa by a 80 nm thick SiN${}_{\textrm{\scriptsize x}}$ dielectric layer. The Hall bar is fabricated from the same wafer used in Ref. Nguyen et al. (2015); Qu et al. (2015). The quantum well consists of 12.5 nm InAs and 5 nm GaSb between 50 nm AlSb barriers. The doped GaSb substrate acts as a back gate. All measurements are done at 300 mK using standard lock-in techniques with an excitation current of 50 nA.

Figure 1 presents the longitudinal resistance of the Hall bar device as a function of top gate voltage $V_{\mathrm{tg}}$ and back gate voltage $V_{\mathrm{bg}}$. The measurement is performed in 2 T perpendicular magnetic field and therefore shows quantum oscillations resulting from the changing electron density. Quantum oscillations corresponding to holes are less pronounced as the mobility of holes in this system is much lower than the mobility of electrons Qu et al. (2015). For lines parallel to these oscillations, such as line I in Fig. 1a, the electron density is constant while the electric field changes. Regions of high resistance, indicated by the dashed white and green lines, correspond to having the Fermi level inside an energy gap. A detailed description of the phase diagram obtained from measurements on the same wafer was reported by Qu et al. Qu et al. (2015).

The green solid line in Fig. 1 divides the phase diagram in two regimes. Right of this line is the electron-only regime, where the GaSb is depleted. The system effectively is an asymmetric InAs quantum well with a trivial band alignment and a Fermi level residing in the conduction band (see inset of Fig. 1a). In this regime we investigate $\Delta n_{\tiny\textit{ZF}}$ along line I, where the electron mobility is highest while only the lowest subband remains occupied. The regime at the left of the green line is the two-carriers regime where electrons and holes coexist. Line II is chosen to evaluate $\Delta n_{\tiny\textit{ZF}}$ close to the hybridization gap (highlighted by the dashed green line). Before discussing the spin-orbit interaction in the two-carriers regime (along line II) we first study the electron-only regime (line I).

Figure 2a shows magnetoresistance traces for 10 points along line I. The density of electrons is fixed (see Fig. 2c) while the electric field is changed. We first consider trace 1. Clear oscillations in the longitudinal resistance $R_{xx}$ are observed as a function of perpendicular magnetic field $B$ modulated by a beat pattern. These Shubnikov-de Haas (SdH) oscillations appear for each single spin band and are periodic in $1/B$ with a frequency that relates to the carrier density via $n=e/h\cdot f$ Nitta et al. (1997); Onsager (1952). The beat modulation observed in trace 1 is caused by two slightly different SdH frequencies $f_{1},f_{2}$. This is also evident from the fast Fourier transform (FFT) of the magnetoresistance trace $\mathcal{F}[R_{xx}(1/B)]$ presented in the first curve of Fig. 2b (see supplementary info for details on the Fourier procedure Sup ). These two SdH frequencies indicate two distinct densities $n_{1},n_{2}$. They must correspond to different spin species because their sum $n_{1}+n_{2}$ equals the Hall density $n_{H}$ (see Fig. 2c). Subsequently, one spin species has a larger density than the other, $n_{1}>n_{2}$, implying that the system favors one spin-orbit eigenstate over to the other. The difference, $\Delta n_{\tiny\textit{ZF}}=n_{2}-n_{1}$, is a measure for the zero-field spin splitting energy, $\Delta E_{\scriptsize\textrm{ZFSS}}=\Delta n_{\tiny\textit{ZF}}\left(m^{*}/\pi\hbar^{2}\right)^{-1}$.

Upon moving from point 1 to 10 along line I, two trends are observed. First, an extra frequency peak emerges in the FFTs at $(f_{1}+f_{2})/2$. This originates from the asymmetry between adjacent beats in the SdH oscillations, visible both in amplitude and number of oscillations of beats A and B in Fig. 2a Sup . Second, the spacing between the outer peaks in the FFT spectrum decreases, evident from decreasing $\Delta n_{\tiny\textit{ZF}}$ over line I (Fig. 2c). This arises from an increasing number of oscillations in both beats A and B (see Sup ), which also pushes the beat nodes to lower magnetic fields. Before we extract the actual SOI strengths and show its electric field dependence, we first elucidate the origin of the emerging center frequency peak.

The center frequency, interestingly, does not correspond to an actual density. The sum of the densities $n_{1}$ and $n_{2}$ (corresponding to the outer peaks in the FFT) still equals the Hall density. There are, however, mechanisms involving scattering between Fermi-surfaces that can result in extra frequency components. Such mechanisms are magnetic inter subband scattering (MIS) Sander et al. (1998); Rowe et al. (2001), magnetophonon resonances (MPR) Tsui et al. (1980); Gurevich and Firsov (1961) and magnetic breakdown (MB) Averkiev et al. (2005); Symons et al. (1998); Shoenberg (1984).

We exclude MIS and MPR. By changing electron density all the frequency peak positions shift with equal strength Sup . However, the oscillation frequency of MIS and MPR is determined by the subband spacing and a specific phonon frequency, respectively. Both do not depend on the electron density. In contrast, for MB the spurious peak always appears in between $f_{1}$ and $f_{2}$. Magnetic breakdown explains this spurious central peak as carriers tunneling between spin polarized Fermi-surfaces at spin-degeneracy points. The interplay of Dresselhaus and Rashba SOI in our heterostructure could lead to such an anisotropic Fermi surface Averkiev et al. (2005); Ganichev et al. (2004). In order to confirm this hypothesis, we extract the individual Rashba and Dresselhaus contributions by comparing our data to quantum mechanical Landau level simulations that include the MB mechanism.

The quantum well in this electron-only regime is modeled by a Hamiltonian with spin-orbit interaction in 2D electron systems subject to a perpendicular magnetic field $B$, as given by Winkler (2003); Fabian et al. (2007):

[TABLE]

Where $p_{i}\rightarrow p_{i}+eA_{i}$ is the canonical momentum, $\sigma_{i}$ Pauli spin matrices, $\alpha,\beta,\gamma$ the Rashba, linear Dresselhaus and cubic Dresselhaus coefficients, respectively, $\hbar$ the reduced Planck’s constant, $\mu_{B}$ the Bohr magneton. An electron effective mass of $m^{*}=0.04m_{0}$ is measured from the temperature dependence of the SdH oscillations Sup and a g-factor of $-11.5$ is used in the calculations Mu et al. (2016); foo . We solve for the Landau level energies in a perpendicular magnetic field $B_{z}$ and extract the resistivity as a function of magnetic field (see supplementary information for details Sup ).

The parameters $\alpha,\beta,\gamma$ in the model are estimated and fine tuned to match the node positions and the number of oscillations in a beat of the measured SdH traces. Figure 3 a,b shows the measured SdH data together with the simulated data for traces 1 and 10. Trace 1 is fitted with $\alpha_{1}=75$ meVÅ, $\beta_{1}=28.5$ meVÅ, $\gamma_{1}=0$ meVÅ3 and trace 10 is fitted with $\alpha_{10}=53$ meVÅ , $\beta_{10}=28.5$ meVÅ ,$\gamma_{10}=0$ meVÅ3. The node positions and amplitude modulation of the simulated data agrees well with the measured SdH oscillations.

Curiously, only good fits are obtained when setting the cubic Dresselhaus term $\gamma$ to zero. In 2D systems, $\beta$ is related to $\gamma$ via $\beta=\langle k_{z}^{2}\rangle\gamma$, where $\langle k_{z}^{2}\rangle\approx(\pi/d)^{2}$ is the expectation value of the transverse momentum Winkler (2003); Fabian et al. (2007) in a quantum well of thickness $d$. So $\gamma$ should be non-zero. Currently we do not understand this discrepancy. A recent experimental study on a similar material system also found that the cubic Dresselhaus term could be neglected Herzog et al. (2017).

Now we consider all traces (1-10) and show that the two trends of Fig. 2 (emerging center FFT peak and approaching outer FFT peaks) are reproduced by changing only the Rashba SOI strength. Figure 3c shows the FFTs of the simulated traces where $\alpha$ is linearly interpolated between $\alpha_{1}$ and $\alpha_{10}$ while fixing $\beta=28.5$ meVÅ and $\gamma=0$ meVÅ3. Linear interpolation is used because the electric field changes linearly along line I, and Rashba SOI strength depends linearly on electric field Bychkov and Rashba (1984a, b); Winkler (2003). All simulated FFTs and the SdH traces Sup match the measured data very well, clearly reproducing the emerging central peak and the approaching outer peaks.

In the remainder of this paper we switch to the two-carriers regime, located left of the solid green line in Fig. 1. Electrons in InAs are present alongside with holes in GaSb (n+p). Here we study the influence of the hybridization of electrons with holes on $\Delta n_{\tiny\textit{ZF}}$ by investigating magnetoresistance traces on the points 1-13 along line II.

Before continuing with the measured magnetoresistance traces, it is insightful to examine the expected band structures at points 1 and 13, as illustrated in Fig. 4b. The first point of line II is located near the boundary between the two-carrier and single carrier regimes. A small amount of holes with a large amount of electrons is present. At point 13, close to the hybridization gap, the electron and hole densities are roughly equal, hence the Fermi level $E_{f}$ is close to the hybridization gap. Note also that $k_{\textrm{\scriptsize cross}}$ decreases from 1 to 13, since the electric field changes.

Figure 4a shows the magnetoresistance traces 1-13 along line II. Starting from trace 1 towards trace 13 we find series of traces with or without beating, depicted in blue and red respectively. For traces 1 to 3, at large electron density, beating is observed from which we extract $\Delta n_{\tiny\textit{ZF}}=1.7\cdot 10^{14}$ m*-2* 111We cannot directly extract the spin-orbit strength from this $\Delta n$ by comparing to the single-carrier case, since the effective mass in this region is unknown.. Remarkably trace 4 and 5 do not show any beating, therefore no zero-field density difference can be extracted. For traces 6 to 10, the beating revives showing strong beating. Finally, traces 11-13 show no beating. Figure 4c depicts the extracted $\Delta n_{\tiny\textit{ZF}}$ along line II, which shows a non-monotonic behaviour as a function of gate voltage along line II.

In order to understand this non-monotonic $\Delta n_{\tiny\textit{ZF}}$ near the hybridization gap (points 1-10) we performed band structure calculations of our InAs/GaSb quantum well Sup . The $\Delta n$ extracted from these calculations is plotted in Fig. 4d, which qualitatively agree with the observed dip in $\Delta n_{\tiny\textit{ZF}}$ at points 4 and 5 (Fig. 4c.). In order to understand the simulated $\Delta n$, the band structure near the hybridization gap is depicted in the inset of Fig. 4d (zoom-in on Fig. 4b indicated by the red box). The black and red lines represent different spin bands. The bands cross at the black arrow, indicating the vanishing of $\Delta n$, such as observed in the experiment. We found this feature to be robust for different electric fields and crystal directions Sup . We therefore attribute the observed dip in $\Delta n_{\tiny\textit{ZF}}$ to the band structure near the hybridization gap.

Note that only qualitative comparison between experiment and calculations is possible as Fermi-energy is varied in the simulation, while in the experiment the band structure ($k_{\textrm{\scriptsize cross}}$) and Fermi-energy are expected to change. The fact that $\Delta n_{\tiny\textit{ZF}}$ in Fig. 4d does not completely vanish is because the crossing of the spin bands in the [110] occurs at a slightly different energy than in the [100] direction.

The lack of beating of traces 11-13 is not captured with the simulation. There are two possible reasons for this deviation. First, a strong asymmetry in SdH amplitudes of the two spin species ($A_{\textrm{\scriptsize up}}\gg A_{\textrm{\scriptsize down}}$) determines the visibility of the beating pattern. The single spin band SdH oscillation amplitude depends on effective mass $m^{*}$ and scattering time according to $A_{\textrm{\scriptsize SdH}}\sim(eB/m^{*})^{3}\exp(-\pi/\omega_{c}\tau)$ Luo et al. (1990). Both effective mass and scattering time for the two spin bands become very dissimilar when approaching the hybridization gap Sup , as a result that the beating visibility is reduced to below the experimentally detectable visibility. Second, Nichele et al. Nichele et al. (2016) shows there is an energy window with only one single spin band present. In such spin polarized state no beating can occur. Here, we cannot discriminate between these two reasons that explain the lack of beating in traces 11-13.

In conclusion, we presented a study of the spin-orbit interaction in an InAs/GaSb double quantum well. The Fermi-level and band structure are altered by top and bottom gates. In the electron-only regime we find a electric field tunable spin-orbit interaction, and extract the individual Rashba and Dresselhaus terms. In the two-carriers regime we observe a non-monotonic behavior of the spin splitting which we trace back to the crossing of the spin bands due to the hybridization of electrons and holes.

I Fourier transforms

The Fourier transforms in this manuscript are obtained using the method described here. Starting from a magnetoresistance curve, first a magnetic field range is chosen. The lower bound is fixed at 0.15 T. The upper bound is chosen such that the interval ends at 40% of a beat maximum. Truncating the signal in this way causes minimal deviation from the true frequency components. Next, the background resistance is estimated using a 6${}^{\textrm{th}}$ order polynomial fit, which subsequently is subtracted from the signal. The remaining signal is interpolated on a uniform grid in $1/B$ and padded with zeros on both sides. No extra window function is applied. A fast Fourier transform converts the signal to the frequency domain $R(\omega)$ and the power spectrum is obtained using $P(\omega)=R(\omega)\times R^{*}(\omega)$. All Fourier transforms are normalized such that the maximum is 0.8 a.u.

II Number of oscillations in a beat

III Peaks shift upon changing electron density

IV Dingle plots of the effective mass

V Details on the Landau level simulation

This section describes the calculations used to simulate the magnetoresistance traces to extract the Rashba and Dresselhaus coefficients as shown in Fig. 3a of the main text. We closely follow the method presented in Ref. Luo et al. (1990) and Ch. 4 of Ref. Winkler (2003).

The Hamiltonian in the momentum basis is presented in Eq. 1 of the main text, here repeated for convenience:

[TABLE]

For the perpendicular magnetic field $B=(0,0,B_{z})$, the symmetric gauge $\mathbf{A}(x,y)=\frac{B_{z}}{2}(-y,x,0)$ is used. The canonical momentum can be written as

[TABLE]

Raising and lowering operators are defined as

[TABLE]

where $\lambda_{c}=\sqrt{\hbar/eB}$ is the magnetic length. The raising operators act on the Landau levels, i.e. $a^{\dagger}|n,\uparrow\rangle=\sqrt{n+1}\,|n+1,\uparrow\rangle$. The momentum operators are rewritten in the raising and lowering operators, which are then substituted into the Hamiltonian. We take a basis of $N=400$ Landau levels in order to capture magnetic fields $\gtrsim 0.1$ T for the electron density $n_{s}\simeq 17.6\cdot 10^{15}$ m*-2*. Solving the Hamiltonian results in the Landau level energies at a particular magnetic field $E(n,B_{z})$.

Following Luo et al. Luo et al. (1990) the conductance is written as:

[TABLE]

We assume a fixed Fermi energy at $E_{f}=(\pi\hbar^{2}n_{s})/m^{*}$. To obtain the resistivity we use the approximation that for quantizing magnetic fields $\left(\sigma_{xy}^{2}\gg\sigma_{xx}^{2}\right)$ the transverse resistivity $\rho_{xx}$ is given as Luo et al. (1990):

[TABLE]

VI SdH traces of the Landau level simulations

VII Fourier transforms in the two-carrier regime

VIII Band structure calculations for multiple electric fields

IX Effective mass & wave function in the quantum well