Tuning of Fermi Contour Anisotropy in GaAs (001) 2D Holes via Strain

TL;DR

This paper demonstrates that applying small in-plane strain to GaAs (001) 2D holes significantly alters their Fermi contour anisotropy, enabling in situ control for studying quantum phenomena.

Contribution

It introduces a method to tune Fermi contour anisotropy in 2D holes via in-plane strain, revealing large anisotropy effects in a high-mobility GaAs system.

Findings

Small strain induces large Fermi wavevector anisotropy (up to 3.3).

Fermi anisotropy correlates with strain on the order of 10^{-4}.

Method enables in situ control of anisotropy for quantum studies.

Abstract

We demonstrate tuning of the Fermi contour anisotropy of two-dimensional (2D) holes in a symmetric GaAs (001) quantum well via the application of in-plane strain. The ballistic transport of high-mobility hole carriers allows us to measure the Fermi wavevector of 2D holes via commensurability oscillations as a function of strain. Our results show that a small amount of in-plane strain, on the order of $10^{-4}$, can induce significant Fermi wavevector anisotropy as large as 3.3, equivalent to a mass anisotropy of 11 in a parabolic band. Our method to tune the anisotropy \textit{in situ} provides a platform to study the role of anisotropy on phenomena such as the fractional quantum Hall effect and composite fermions in interacting 2D systems.

Full text

High quality two-dimensional (2D) holes in GaAs exhibit various quantum mechanical phenomena, rendering them an attractive platform for fundamental resarch as well as applications for novel electronic and spintronic devices. For example, the strong spin-orbit interaction in GaAs leads to the intrinsic spin Hall effect, Wunderlich.PRL.2005 ; Bernevig.PRL.2005 and 2D holes in narrow quantum wells (QWs) have shown long spin coherence times, making them promising cadidates for spin qubits. Korn.NJP.2010 Also, high-mobility hole carriers and their zero-field spin-splitting, tunable by external electric field, Heremans.SC.1994 ; Lu.PRL.1998 ; Papadakis.Science.1999 ; Papadakis.PhysE.2001 ; Fischer.PRB.2007 are essential components for mesoscopic spin devices. Heremans.SC.1994 ; Lu.PRL.1998 ; Rokhinson.PRL.2004 ; Nichele.PRL.2015 In addition, recent magnetotransport measurements reveal that the inverted (“Mexican hat”) dispersion of the hole excited subband hosts an exotic annular Fermi sea. Jo.PRB.2017

The band-structure modifications resulting from the application of in-plane strain to GaAs 2D holes also provide exciting opportunities for fundamental studies and device applications. Previous studies demonstrated the tuning of the spin-splitting and the Fermi contour shape by applying in-plane strain to the GaAs (311)A 2D holes. Habib.PRB.2007 ; Shabani.PRL.2008 ; Habib.SST.2009 However, in such samples, despite the nearly isotropic Fermi contour, low symmetries in the wafer orientation and the interface corrugations cause severe anisotropic mobilities along the $[01\bar{1}]$ and $[\bar{2}33]$ directions. Heremans.JAP.1994 ; Wassermeier.PRB.1995 These complications can be eliminated by growing the 2D holes on a GaAs (001) substrate. There have indeed been magnetotransport studies on GaAs (001) 2D holes in the presence of strain, and these have reported an effective mass change for one of the spin-split subbands Kolokolov.PRB.1999 as well as a control of the interaction-induced anisotropic phase in high magnetic fields. Koduvayur.PRL.2011 Yet, quantitative measurements of strain-induced Fermi contour anisotropy have not been reported. In this Letter, we demonstrate tunable Fermi contour anisotropy via the application of in-plane strain to GaAs (001) 2D holes, and measure the Fermi wavevector anisotropy as a function of strain using commensurability oscillations.

Figure 1 highlights our study to induce anisotropic Fermi contours in 2D holes confined to a symmetric GaAs QW grown on a GaAs (001) substrate. In Fig. 1(a) we show the calculated Fermi contours at a density of $p=1.3\times 10^{11}$ cm*-2* under various strain values applied along the $[\bar{1}10]$ direction. fnote1 ; Shayegan.APL.2003 ; Shkolnikov.APL.2004 The self-consistent numerical calculations are performed based on the $8\times 8$ Kane model augmented by the Bir-Pikus strain Hamiltonian. Bir.Book ; Winkler.Book Owing to the bulk inversion asymmetry of the zinc-blende structure of GaAs, a finite spin-splitting is expected in our symmetric QW, as indicated by two Fermi contours (solid and dashed curves). fnote2 Without strain ($\epsilon=0$), the Fermi contours show a four-fold symmetry in reciprocal space; the smaller contour is close to circular and the larger contour is slightly warped. When tensile strain ($\epsilon>0$) is applied along $[\bar{1}10]$, the Fermi contours become elongated along $[\bar{1}10]$. For compressive strain ($\epsilon<0$), the distortion of the Fermi contours occurs in the opposite direction. Remarkably, the induced Fermi contour anisotropy is quite large even for a small amount of strain, of the order of $10^{-4}$. This pronounced distortion is attributed to the large hole effective mass and the strong heavy-hole and light-hole mixing in the band structure of GaAs 2D holes; Kolokolov.PRB.1999 ; Platero.PRB.1987 ; Winkler.Book ; Sun.JAP.2007 the strain effect in GaAs 2D electrons is negligible. fnote3

Figures 1(b) and 1(c) show the schematic of our experimental setup. Our sample contains 2D holes confined to a 175-Å-wide, modulation-doped, GaAs QW grown on a GaAs (001) substrate by molecular beam epitaxy. The symmetric QW is flanked by a 960-Å-thick Al0.24Ga0.76As spacer layer and a C $\delta$-layer on each side, resulting in a density $p\simeq 1.3\times 10^{11}$ cm*-2* and mobility $2\times 10^{6}$ cm2/Vs at 0.3 K. A $4\times 4$ mm2 piece of the wafer is thinned to $\sim 120~{}\mu$m using mechanical lapping followed by chemical-mechanical polishing. fnote4 A Ti/Au gate is deposited on the back-side of the wafer to tune the density and also to shield any spurious external field from the piezo-actuator. The thinned sample, etched into the Hall bar geometry, is glued on one side of the stacked piezo-actuator using two-component epoxy, and cooled to 0.3 K for the magnetotransport measurements. A strain gauge is also glued on the other face of the piezo to monitor the relative change of the strain as a voltage bias $V_{\mathrm{P}}$ is applied to the piezo. Shayegan.APL.2003 ; Shkolnikov.APL.2004 After sample cool-down, a finite built-in strain develops due to the different thermal contractions of the sample and the piezo. We determine the $\epsilon=0$ condition when the measured Fermi wavevector equals $k_{0}=\sqrt{2\pi p}$. (see Ref. 29.)

In order to measure the Fermi wavevector, we fabricate a grating of negative electron-beam resist with period $a=200$ nm on the surface of the wafer. The grating induces a periodic strain on the GaAs surface which in turn results in a small periodic modulation of the 2DHS density via the piezoelectric effect . Endo.PRB.2000 ; Kamburov.PRB.2012a When a small perpendicular magnetic field $B$ is applied, the holes move along the cyclotron orbits, whose shapes resemble the Fermi contours, but rotated by $90^{\circ}$. When the cyclotron diameter along the current direction becomes commensurate with $a$, the magnetoresistance exhibits commensurability oscillations, also known as Weiss oscillations. Weiss.EPL.1989 ; Winkler.PRL.1989 ; Gerhardts.PRL.1989 ; Beenakker.PRL.1989 The minima in the trace are directly related to the 2D holes’ Fermi wavevector along $[\bar{1}10]$, $k_{[\bar{1}10]}$, through $2R_{c}/a=i-1/4$ ($i=1,2,3...$) where 2$R_{c}=2\hbar k_{[\bar{1}10]}/eB$ is the cyclotron diameter along $[110]$, $e$ is the electron charge, and $\hbar$ is the Planck constant. Weiss.EPL.1989 ; Winkler.PRL.1989 ; Gerhardts.PRL.1989 ; Beenakker.PRL.1989 ; Endo.PRB.2000 ; Kamburov.PRB.2012a [As an example, the shapes of the cyclotron orbits for $\epsilon>0$ are depicted by black curves in Fig. 1(c).]

Figure 2(a) shows the measured magnetoresistance traces at different $\epsilon$. The green trace represents the $\epsilon=0$ case, satisfying $k_{[\bar{1}10]}=k_{0}$. The blue traces are for tensile strain ($\epsilon>0$) while the red ones are for compressive strain ($\epsilon<0$). The commensurability features appear symmetrically with respect to $B=0$, and the minima positions corresponding to $i=2$ and 3 are used to determine $k_{[\bar{1}10]}$. The $i=1$ minimum is expected at higher field ($\simeq 0.8$ T for $\epsilon=0$), but it is masked by the Shubnikov-de Haas oscillations. The evolution of the mimima positions as a function of strain is clearly seen in Fig. 2(a) following the dashed lines; with increasing $\epsilon$ the minima move away from $B=0$, implying an increasing $k_{[\bar{1}10]}$. In Fig. 2(b), black solid and dashed curves represent real-space cyclotron orbits of holes in different $\epsilon$ regimes; $\epsilon>0$ (top), $\epsilon=0$ (middle), and $\epsilon<0$ (bottom). In principle, there are two cyclotron orbits corresponding to the two spin-subbands. However, we measure a single $k_{[\bar{1}10]}$ from the commensurability minima, implying that our experiments do not resolve the finite but small spin-splitting. The measured $k_{[\bar{1}10]}$ normalized to $k_{0}$ is shown in Fig. 2(c) as a function of $\epsilon$. The red and blue circles are obtained from two different cool-downs. Although the built-in strains are different between the two cool-downs, both sets of data contain $k_{[\bar{1}10]}=k_{0}$, i.e., $\epsilon=0$. This enables us to determine the built-in strain, and thereby the absolute values of $\epsilon$. It is clear that the overlapping $k_{[\bar{1}10]}$ values from two measurements agree very well each other. When $\epsilon$ is increased from $-1.8$ to $1.7\times 10^{-4}$, $k_{[\bar{1}10]}/k_{0}$ changes from 0.85 to 1.38, i.e., there is a $\simeq 60\%$ increase of the Fermi wavevector along the $[\bar{1}10]$ direction.

Figure 3(a) shows the measured $k_{[\bar{1}10]}/k_{0}$ and its comparison with the calculations as a function of $\epsilon$. The circles, squares, and triangles represent the measured $k_{[\bar{1}10]}/k_{0}$ in three different cool-downs where different built-in strains were attained. The black solid and dashed lines represent the calculated $k_{[\bar{1}10]}/k_{0}$ for the two spin-subbands, and the thick orange curve shows their averaged values. The data shown by circles and squares are slightly lower than the orange curve. The small discrepancy indicates that the measured $k_{[\bar{1}10]}$ deduced from the commensurability features may not reflect exactly the averaged $k_{[\bar{1}10]}$ of the two spin-subbands, or the calculated Fermi contours are slightly different from those in our sample. Compared to the circles and squares, the measured data plotted by triangles show very large $k_{[\bar{1}10]}/k_{0}$ values, implying that a large built-in strain develops for this cool-down. Shabani.PRL.2008 ; fnote5 Because $k_{[\bar{1}10]}$ cannot be tuned over a sufficiently large range to reach $k_{0}$ by applying $V_{\mathrm{P}}$ to the piezo, limited within $\pm~{}300$ V, we are unable to determine the absolute magnitude of $\epsilon$ for this cool-down directly from the measured data. If we assume a built-in strain of $\simeq 4.7\times 10^{-4}$, however, the measured data match very well with the orange curve. Overall, the experimental data in Fig. 3(a) agree well with the averaged Fermi wavevector of the calculations to within 6%.

Figure 3(b) shows the Fermi wavevector anisotropy, defined by $k_{[\bar{1}10]}/k_{[110]}$ where we use the average values of the two spin-subbands for each direction, based on calculations results for $p=1.3\times 10^{11}$ cm*-2*. A remarkably large Fermi wavevector anisotropy, as high as 3.3, is achieved when $\epsilon=5.5\times 10^{-4}$ is applied. Note that this is equivalent to an effective mass anisotropy of 11 in a parabolic band, much larger than the intrinsic mass anisotropy ($\simeq 5$) of semiconductors such as Si and AlAs. Sze.Book ; Shayegan.PSSb.2006 Moreover, the tunability of the anisotropy in 2D holes allows for systematic studies of anisotropy effect on transport properties, which is not viable with the fixed, anisotropic 2D electrons in Si and AlAs.

Before closing, we note that recent experiments have demonstrated that parallel magnetic fields can induce anisotropic Fermi contours in high-quality GaAs 2D hole (and electron) systems and that the induced anisotropy can be determined via commensurability oscillations measurements. Kamburov.PRB.2012b ; Kamburov.PRB.2013 ; Mueed.PRL.2015 However, the induced anisotropy is primarily caused by the coupling between the in-plane and out-of-plane motions of the carriers, making the theoretical understanding of the data challenging. In contrast, the strain-induced anisotropy is originated from the band structure modifications at zero magnetic field. We also emphasize several different aspects between the two methods. First, the in-plane strain can both expand and contract the Fermi contour along a particular direction, while parallel-field can only elongate the Fermi contour in the direction perpendicular to the applied field direction. Second, the out-of-plane motions of the carriers driven by the in-plane field is affected by the QW width, thus the induced anisotropy is strongly depedent on the QW width as well as the out-of-plane effective mass, Kamburov.PRB.2014 while the strain does not bring in such complexity to the 2D holes. Third, a large parallel field inevitably leads to a large Zeeman spin-splitting of 2D holes. To this end, 2D holes exhibiting tunable anisotropy under strain provide an ideal platform to study the role of anisotropy in phenomena such as the fractional quantum Hall effect and composite fermions in interacting 2D systems. Kamburov.PRL.2013 ; Kamburov.PRB.2014 ; Mueed.PRB.2016 ; Balagurov.PRB.2000 ; Gokmen.NatPhys.2010 ; Haldane.PRL.2011 ; KYang.PRB.2013 ; Mueed.PRB.2016 ; Balram.PRB.2016 ; Jo.arXiv.2017 ; Ippoliti.arXiv.2017