Anisotropic low-temperature piezoresistance in (311)A GaAs two-dimensional holes
B. Habib, J. Shabani, E. P. De Poortere, M. Shayegan, R. Winkler
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Anisotropic low-temperature piezoresistance in (311)A GaAs two-dimensional holes
B. Habib, J. Shabani, E. P. De Poortere, M. Shayegan
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
R. Winkler
Department of Physics, Northern Illinois University, De Kalb, IL, 60115, USA (Dated: October 24, 2018)We report low-temperature resistance measurements in a modulation-doped, (311)A GaAs two-dimensional hole system as a function of applied in-plane strain. The data reveal a strong butanisotropic piezoresistance whose magnitude depends on the density as well as the direction alongwhich the resistance is measured. At a density of 1 . × cm − and for a strain of about 2 × − applied along [01¯1], e.g., the resistance measured along this direction changes by nearly a factorof two while the resistance change in the [¯233] direction is less than 10% and has the oppositesign. Our accurate energy band calculations indicate a pronounced and anisotropic deformationof the heavy-hole dispersion with strain, qualitatively consistent with the experimental data. Theextremely anisotropic magnitude of the piezoresistance, however, lacks a quantitative explanation. The piezoresistance effect – the change of a device’selectrical resistance as a function of applied stress – is ofgreat technological importance as it is utilized in makingforce, displacement and pressure sensors [1]. It has alsobeen used recently in measuring sub-nanometer displace-ments of tips in atomic force microscopes [2, 3], and pro-posed data storage devices based on piezoresistive read-back [4].Strain affects the resistance of a solid in two ways. Itchanges the physical dimensions of the resistor and canalso modify its energy band structure. In metals, it is theformer effect that is dominant and the the gauge factor,defined as the fractional change in resistance divided bythe solid’s fractional change in length, is dictated by thePoisson’s ratio which is typically around 2. On the otherhand, for certain semiconductors the modification of theband structure is a more prominent factor affecting thepiezoresistance. In multi-valley semiconductor systems,for example, strain changes the relative population of thevalleys. If the valleys have anisotropic effective masses,the charge transfer induces a large resistance change incertain crystallographic directions [5]. Gauge factors of ∼ ∼ p ) dependentpiezoresistance effect in a (311)A GaAs 2D hole system(2DHS). We utilize a simple but powerful technique toapply quantitatively measurable in-plane strain in-situ [8] and measure resistance parallel and perpendicular tothe applied strain. We deduce a gauge factor of ∼ p = 1 . × cm − and T = 0 . . Ga . As/GaAs interface is separated from a 17 nm-thick Si-doped Al . Ga . As layer (Si con-centration of 4 × cm − ) by a 30 nm Al . Ga . Asspacer layer. We fabricated L-shaped Hall bar samplesvia photo-lithography and used In:Zn alloyed at 440 ◦ Cfor the ohmic contacts. Metal gates were depositedon the sample’s front (10nm Ti; 30nm Au) and back(100nm Ti; 30nm Au) to control the 2D hole density.Magneto-resistance data taken at p = 2 . × cm − and T = 0 . . × cm /Vs and4 . × cm /Vs in the [01¯1] and [¯233] directions respec-tively. This mobility anisotropy in the (311)A growthdirection stems from the quasi-periodic ridges along the[¯233] direction [9, 10].We apply tunable strain to the sample (thinned to ∼ µm ) by gluing it on one side of a commercial mul-tilayer piezoelectric (piezo) actuator with the sample’s[01¯1] crystal direction aligned with the poling directionof the piezo (Fig. 1 inset) [8]. When bias V P is applied tothe piezo, it expands (shrinks) along the [01¯1] for V P > V P <
0) and shrinks (expands) along the [¯233] direction.We have confirmed that this deformation is fully trans-mitted to the sample, and using metal strain gauges gluedto the opposite side of the piezo, have measured its mag-nitude [8, 11]. Based on our calibrations of similar piezoactuators, we estimate a strain of 3 . × − V − alongthe poling direction. In the perpendicular direction, thestrain is approximately − .
38 times the strain in the pol-ing direction [8, 12]. In this paper we specify strain valuesalong the poling direction; we can achieve a strain rangeof about 2 . × − by applying − ≤ V P ≤
300 Vto the piezo [13]. Finally, the back-gate on the sampleis kept at a constant voltage (0 V) throughout the mea-surements to shield the 2DHS from the electric field ofthe piezo actuator.Figure 1 summarizes our measured piezoresistance.Three trends in the data are clear. First, the changein resistance ( R ) with strain is density dependent andis larger for lower densities. Second, the piezoresistanceis more prominent along the [01¯1] direction. Indeed, at p = 1 . × cm − , R along this direction changes by [233]_[011]_ [311]+ -V P Piezo Sample -300 -200 -100 0 100 200 3000.20.40.60.81.01.2 1.6x10 cm -2 cm -2 cm -2 T=0.3 K R ( k Ω ) V P (V) [011][233]__Strain (10 ) -4 FIG. 1: (Color on-line) Piezoresistance measured along the[01¯1] and [¯233] directions for three densities. To obtain theresistivity (in units of kΩ per square), multiply R by 0.3.Inset: Experimental setup. The poling direction for the piezois along [01¯1]. The strain values indicated in the top scalehave an uncertainty in the absolute value and hence specifythe change in applied strain (see Ref. [13]). a remarkable factor of two for an applied strain of only ∼ × − . Third, in the entire strain range we havestudied, R along [01¯1] is always larger than in the [¯233]direction.In order to understand the data of Fig. 1, we performedself-consistent calculations of the Fermi contours of the2D holes residing in the heavy-hole (HH) valence bandas a function of strain. Note that in this system the spindegeneracy of the HH band is lifted due to the presenceof strong spin-orbit coupling [14]. The spin subbands aretermed heavy HH (HHh) and light HH (HHl), reflectingthe magnitude of their effective masses. We used the 8 × p = 1 . × cm − . Figure 2 shows that the anisotropyof the HHh band is strongly enhanced with the appli- p=1.6x10 cm -2 k [233] (10 m -1 ) _ k [ ] ( m - ) _ HHhHHl Strain2x10 -4 -4 [011] [233] R / R Strain (10 -4 )_ _0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.40.00.30.60.91.21.51.82.12.4 -3 -2 -1 0 1 2 30.81.01.2 FIG. 2: (Color on-line) Calculated Fermi contour plots forthe HHl and HHh spin subbands; k [01¯1] and k [¯233] are thewavevectors along the two perpendicular ([01¯1] and [¯233]) di-rections. Inset: Comparison between the experimentally mea-sured (symbols) and calculated (lines) piezoresistance. cation of strain. This results in the anisotropy of theHHh effective mass as well. Qualitatively, a larger Fermiwavevector in a particular direction typically implies alarger mass in the same direction. Hence, applying ten-sile strain along the [01¯1] direction leads to a larger massalong [01¯1] and a smaller mass along [¯233]. This changein effective mass qualitatively explains the change in re-sistance in the two directions.To make a direct comparison with the experimentaldata, we calculated the resistance for our sample param-eters taking the strain dependence of the anisotropy intoaccount. We assume a constant and isotropic relaxationtime ( τ ) and sum over spin subbands, n , with energy dis-persion E n ( k ). We evaluate the velocity, ∂E n / ( ~ ∂k ) at E F and T = 0 to obtain the conductance ( σ ) [19], σ ij = τ e X n Z d k (2 π ) ∂E n ~ ∂k i ∂E n ~ ∂k j δ [ E F − E n ( k )] . (1)Since it is difficult to estimate the value of τ , we normal-ize the calculated resistance to its value at zero strainand compare the calculations to the data of Fig. 1. Forthe comparison shown in the inset of Fig. 2, the exper-imental resistance was also normalized to the resistanceat zero applied piezo bias, R . As the figure shows, thecalculations certainly match the qualitative trend seen inthe data. It should be noted that the calculations arebased on the particular sample structure and density butcontain no fitting parameters.A puzzling aspect of Fig. 2 (inset) data is that in [011][233]__-300 -200 -100 0 100 200 3000.20.40.60.81.01.2 p = 1.9x10 cm -2 E = - . k V / c m E = E = . k V / c m T = 0.3K R ( k Ω ) V P (V) ┴ ┴ ┴ FIG. 3: (Color on-line) Electric field dependence of piezore-sistance in a 20 nm square well sample grown on a (311)Asurface at constant density. The gate biases are: (V F = -0.2 V, V B = 0 V) for E ⊥ = -4.5 kV/cm, (V F = -0.375 V, V B = 84 V) for E ⊥ = 0, and (V F = -0.5 V, V B = 144 V) for E ⊥ = 3.2 kV/cm. the [01¯1] direction, the experimental change in resistanceis typically larger than predicted by the calculations.The [¯233] direction, on the other hand, exhibits a muchsmaller change in the measured resistance than in thecalculations. It is tempting to relate this mismatch tohow the quasi-periodic corrugations [9] parallel to [¯233]might affect the directional dependence of τ with strain.However, our piezoresistance measurements on a 20 nmsquare quantum well sample, as a function of perpen-dicular electric field ( E ⊥ ) at a constant density, indicateotherwise [Fig. 3]. The square well sample is glued tothe piezo with the poling direction in the [01¯1] crystal di-rection. E ⊥ is tuned while keeping the density constantwith the help of front ( V F ) and back ( V B ) gate biases[20]. At E ⊥ = 0, the confining potential of the squarewell and the carrier wavefunction are symmetric. For E ⊥ = 0, the potential becomes asymmetric, resulting inthe carriers being ‘pushed’ closer to the interface. Hence the resistance increases with the increase in the magni-tude of E ⊥ because of interface roughness, as shown inFig. 3 at a particular piezo bias. If the direction de-pendent mismatch of the piezoresistance were related tothe corrugations along [¯233], we would expect a largerpiezoresistance mismatch with an increase in the mag-nitude of E ⊥ . On the contrary, Fig. 3 shows that thepiezoresistance is nearly independent of E ⊥ [21].Moreover, a similar mismatch was also observed in(100) GaAs 2DHSs [22], where the change in resistancewas reported to be larger than the calculated valuesin the [011] direction, while along [01¯1] the measuredchange was much smaller than predicted by the calcula-tions. The mobility anisotropy in (100) GaAs 2D holesat zero strain is only about 1.2-1.3 between the two per-pendicular ([01¯1] and [011]) in-plane directions. This ismuch smaller than the (311)A samples where the mo-bility anisotropy is typically about 2 to 3 between the[01¯1] and [¯233] directions. It is unlikely, therefore, thatthe corrugations in (311)A samples are responsible forthe mismatch between the rate of change of resistancealong the [01¯1] and [¯233] directions. We emphasize thatthis mismatch is independent of the poling direction ofthe applied strain: Our measurements on a different het-erostructure sample from the same wafer as the one usedin Fig. 1 but glued on a piezo actuator with the polingdirection along the [¯233] direction also exhibit a largerchange in resistance in the [01¯1] direction compared to[¯233].In conclusion, we observe a large piezoresistance ef-fect in (311)A GaAs 2DHSs. The data indicate a strongdependence on density, and on the in-plane directionalong which the resistance is measured. The maxi-mum strain gauge factor we measure (along [01¯1] andfor p = 1 . × cm − ) is ∼ p = 5 × cm − [23] and for GaAs 2DHS grown in the (100) direction at p = 7 . × cm − [22].We thank the ARO, DOE and NSF for support, andM. Grayson for stimulating discussions. 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De Poortere, and M. Shayegan, Phys. Rev. Lett. , 186404 (2006).[12] The strain in the perpendicular direction is taken intoaccount in our calculations described later in the paper.[13] In our experiments, we can determine the relative changes in values of strain accurately, but not the piezo-bias corresponding to zero-strain precisely. This is be-cause, thanks to a mismatch between the thermal ex-pansion coefficients of GaAs and the piezo-stack, at lowtemperatures the sample can be under finite strain evenat V P = 0. This residual strain is cooldown-dependentand we do not know its precise value for our data. Basedon our experience with cooldowns of samples glued tosimilar piezo-stacks, we expect a residual strain up toabout ± × − .[14] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer,Berlin, 2003).[15] G. L. Bir and G. E. Pikus,