Screening effects of superlattice doping on the mobility of GaAs two-dimensional electron system revealed by in-situ gate control
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Screening effects of superlattice doping on the mobility of GaAs two-dimensional electron systemrevealed by in-situ gate control
T. Akiho and K. Muraki
NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan (Dated: February 8, 2021)We investigate the screening effects of excess electrons in the doped layer on the mobility of a GaAs two-dimensional electron system (2DES) with a modern architecture using short-period superlattice (SL) doping.By controlling the density of excess electrons in the SL with a top gate while keeping the 2DES density constantwith a back gate, we are able to compare 2DESs with the same density but different degrees of screening usingone sample. Using a field-penetration technique and circuit-model analysis, we determine the density of statesand excess electron density in the SL, quantities directly linked to the screening capability. The obtained relationbetween mobility and excess electron density is consistent with the theory taking into account the screening bythe excess electrons in the SL. The quantum lifetime determined from Shubnikov-de Haas oscillations is muchlower than expected from theory and did not show a discernible change with excess electron density.
I. INTRODUCTION
High-mobility two-dimensional electron systems (2DESs)in AlGaAs/GaAs heterostructures are the basic platform totest new concepts and study emergent phenomena in low-dimensional systems. The modulation doping technique thatseparates the channel and the doping layer for carrier sup-ply [1–5] and advances in molecular-beam epitaxy that en-able the residual impurity concentration to be decreased arethe key ingredients in realizing clean 2DESs. Over the years,improvements in sample quality, manifested as higher mobil-ity, have led to the discovery of new transport phenomena[6–10] and correlated phases including the fractional quan-tum Hall effects (FQHEs) [11, 12]. However, it has recentlybeen recognized that not only mobility but also the screen-ing of the long-range disorder potential caused by modulationdoping is essential for the observation of fragile FQHEs suchas the one at an even-denominator Landau-level filling factor ν = / x Al x Ga − x As ( x = . .
25) alloy [14] has been shown tobe effective, where the electrons in the doped layer delocalizeand screen the Coulomb potential from ionized donors.The concept of SL doping where a δ -doped donor layer islocated within a narrow GaAs layer flanked by narrow AlAslayers was originally introduced by Friedland et al. to re-duce remote-impurity scattering and thereby enhance mobil-ity [15]. In the AlAs/GaAs/AlAs SL, the energy level of theX-band formed in the AlAs layers is lower than that of the Γ -band formed in the GaAs layer (inset of Fig. 1). Conse-quently, mobile electrons supplied from the donor layer ac-cumulate not only in the GaAs quantum well (QW) severaltens of nanometers away but also in the neighboring AlAs lay-ers. The SL doping technique was later applied to ultra-high-quality samples with mobility exceeding 10 × cm /Vs,where its impact on the FQHEs has been demonstrated [1–5]. Recently, effects of excess electrons in the SL on mo-bility and the quantum scattering lifetime have been studiedtheoretically [16, 17]. Experimentally, gating of samples withSL doping has been attempted to examine the influences of the parallel conducting layer on mobility [15, 18–20]. How-ever, uncontrollable charge redistribution and hysteresis thataccompany the SL doping [18] have made it difficult to ex-tract quantitative information such as the density of excesselectrons in the SL.In this study, we vary the excess electron density in the SLin a controlled manner by appropriately choosing the temper-ature at which the gate voltage is swept. This enabled the insitu control of disorder screening. We determined the elec-tron density in the SL as a function of gate voltage by usinga field-penetration technique [21] and circuit-model analysis,which also allowed us to estimate the quantum capacitance,or the density of states (DOS), in the SL. We show that theobtained relation between mobility and excess electron den-sity can only be explained by theory taking into account the V FG = 0.5 V T = 1.6 K R xx [ W ] R xy [ k W ] B [T] n QW G a A s A l A s
75 nm110 nm X Γ n = 4 2 1 10
30 nm A l A s A l G a A s A l G a A s G a A s n SL FIG. 1. Magnetotransport properties, R xx (red) and R xy (black),of the Hall bar measured at 1.6 K. Insets show the schematic layerstructure of the sample and Γ -conduction band edge profile. TheX- and Γ - conduction band edge profiles near the AlAs/GaAs/AlAssuperlattice are shown on an enlarged scale in the dashed box. screening by excess electrons. II. EXPERIMENT AND ANALYSISA. Sample characterization
The sample consisted of a 30-nm-wide GaAs QW sand-wiched between Al . Ga . As barriers, grown on an n -typeGaAs (001) substrate. The QW, with its center located 207nm below the surface, was modulation doped on one side,with Si δ -doping ([Si] = 1 × m − ) at the center of theAlAs/GaAs/AlAs (2 nm/3 nm/2 nm) SL located 75 nm abovethe QW (inset of Fig. 1) [3]. The Si δ -doping in the thinGaAs layer provides mobile electrons not only in the QW 75-nm away but also in the neighboring AlAs layers [2–5, 15].The mobile electrons in the AlAs layers provide screeningof the disorder potential created by the ionized Si donors.The wafer was processed into a 120- µ m-wide Hall bar withvoltage-probe distance of 100 µ m and fitted with a Ti/Au frontgate. The n -type substrate was used as a back gate. Mea-surements were done at temperatures of 0 . . V FG ) of 0.5 V, which is supposed to increase thedensity of mobile electrons in the SL. Interestingly, despite thepresence of mobile electrons in the SL, there are integer quan-tum Hall effects at Landau-level filling factor ν = ,
2, and4, where the longitudinal resistance ( R xx ) drops to zero andthe Hall resistance ( R xy ) is quantized. From the Shubnikov-de Haas oscillations, we obtained a sheet carrier density of1 . × m − , which agrees within 3% with the value de-duced from the slope of R xy . This suggests that, even if theSL contains conduction electrons, they apparently do not con-tribute to transport. We confirmed similar results for V FG upto 0.8 V.Figure 2 shows the V FG dependence of the sheet carrier den-sity deduced from R xy at 0.2 T. The blue solid curve was ob-tained by sweeping V FG at 1.6 K. As shown above, at 1.6 Kthe measured R xy reflects exclusively the carrier density inthe quantum well ( n QW ), and not that in the SL. As we de-creased V FG from 0.8 V, n QW remained almost constant for − . < V FG < . n QW . A distinct change in n QW occurs only at V FG < − . V FG , we observed a pronounced hysteresis in re-gion II, where n QW changed at a faster rate, with an over-shoot near the boundary with region I. The rate of changed n QW /d V FG = . × m − V − for the up sweep, shownby the blue dashed line, was consistent with the geometricalcapacitance between the front gate and the center of the QWcalculated from the distance (207 nm) and the permittivity of -1.2 -0.8 -0.4 0.0 0.4 0.8 1.6 K equilibrium V F G [ V ] n Q W [ m –2 ] V FG –1.2 V –1.3 V Time [hour]1.50.5 n Q W [ m –2 ] FIG. 2. V FG dependence of n QW obtained from R xy at B = ± . n QW obtained by sweeping V FG at 1.6and 4.3 K, respectively. The dashed blue line is a linear fit for theup sweep. Open circles show n QW in the equilibrium state (see maintext for details). Black dashed line is a fit using a double exponentialfunction. Inset shows the time evolution of n QW at V FG = − . − . AlGaAs ( ε = V FG inregion II, electrons accumulate only in the QW, resulting in ametastable state in which the QW (SL) is overpopulated (un-derpopulated) with respect to its equilibrium density. We con-jecture that this results from the difficulty to inject charge intothe SL once it becomes close to depletion and poorly con-ducting. The red curve in Fig. 2, obtained by sweeping V FG at 4.3 K, shows similar behavior, while the boundary betweenregions I and II shifts to a more negative V FG , with a smallerovershoot in the up sweep.Turning our attention to region I, we notice that n QW is notconstant, but varies slightly with V FG . This indicates that partof the electric field from the gate penetrates the SL populatedwith electrons. This is reasonable, as the SL is not a perfectmetal; it has only a finite DOS, that is, a finite screening ca-pability. In turn, by analyzing the change in n QW with V FG asshown later, we can quantify the DOS, and hence the screen-ing capability, of the SL. (See Ref. [21] for the principle ofthis field-penetration technique.)Even though we used a very slow sweep rate of 0.67mV/sec to set V FG , in region II n QW gradually increased ona scale of several minutes to several tens of hours after V FG was set at a constant value (inset of Fig. 2). This was thecase even for down sweeps for which the system is closer toequilibrium. Similar temporal behavior was reported in Ref.[18]. The transient time increased with decreasing tempera-ture and decreasing V FG . At V FG = − . n QW at C b1 C b2 C AlAs1 V FG C QW C AlAs2 C i SL V F G [ V ] C S L [ F / m ] m S L / m -0.4 0.0 0.4 0.8 T = 1.6 K 0.3 (a) * n S L [ m –2 ] n S L / N S i -1.2 -0.8 -0.4 0.0 0.4 0.8100515 T = 1.6 K (b) V F G [ V ] FIG. 3. red(a) Quantum capacitance C SL of the AlAs layers com-prising the SL calculated from the fitting of n QW in the equilibriumstate. The right axis shows the corresponding effective mass for elec-trons in the SL. Inset shows the equivalent circuit model used to cal-culate C SL . We assume C AlAs1 = C AlAs2 ≡ C SL (see main text for de-tails). (b) Calculated V FG dependence of n SL red( = n AlAs1 + n AlAs2 ).The right axis indicates n SL normalized by the doping density N Si = × m − . each V FG , which is essential for evaluating the screening ef-fect. First, we set V FG at 4.3 K to facilitate the equilibrationand waited until n QW reached a steady value. Then, we de-creased the temperature to 1.6 K and determined n QW from thelow-field R xy . By repeating this process for different V FG , weobtained n QW as a function of V FG , which we plot as open cir-cles in Fig. 2. At V FG = n QW valueas that for the V FG sweep. However, the difference betweenthe two methods became significant at lower V FG . We there-fore employed the data obtained by the equilibration methodfor V FG ≤ V FG sweep for V FG > B. Circuit model
To deduce the excess electron density in the SL and therebyquantitatively characterize the screening effect, we analyzedthe charge equilibration among the front gate, redtwo AlAslayers [AlAs1(2)] comprising the SL, and QW using the cir-cuit model shown in the inset of Fig. 3(a). In addition tothe geometrical capacitances between the neighboring ele-ments among these ( C b1 , C i , and C b2 ), the model containsthe quantum capacitances of the QW ( C QW ) and the AlAslayers [ C AlAs1(2) ]. The quantum capacitance is expressed as C α = e D α , where D α = g α m ⋆ α /2 π ¯ h is the DOS ( m ⋆ α is theelectron effective mass, α denotes QW or AlAs1(2), e is theelementary charge, g α is the degeneracy, and ¯ h = h /2 π is thereduced Planck constant). C b1 , C i , and C b2 are calculated fromthe layer thicknesses and permittivity and C QW is known fromthe effective mass m ⋆ QW = . m of GaAs ( m is the electronmass in vacuum) and the twofold spin degeneracy. This leaves C AlAs1(2) the only unknown parameters in the model. For themodel to be solvable, we need to assume that the two AlAslayers have the same density of states at the Fermi level, thatis, C AlAs1 = C AlAs2 ( ≡ C SL ). We confirmed this assumption tobe acceptable by noting that the calculated chemical potentialdifference between the two AlAs layers (up to 7 meV) wassmaller than the disorder-broadened tail (a few tens of meV).We calculate d n QW / d V FG as a function of V FG using theequilibrium relation between n QW and V FG obtained above.By numerically solving the circuit model with the d n QW / d V FG value at each V FG as an input, we can deduce C SL as a functionof V FG , as shown in Fig. 3(a). C SL decreases with decreasing V FG , reflecting the disorder-broadened tail of the DOS. Inter-estingly, C SL is not constant even at V FG > V FG . Since quantum capacitance is propor-tional to the DOS at the Fermi level, the obtained C SL canbe translated into the effective mass m ⋆ SL that would producethe same DOS for parabolic dispersion through the relation D SL = g SL m ⋆ SL /2 π ¯ h . In thin AlAs layers, quantum confine-ment and strain split the three-fold valley degeneracy in bulkinto one and two, with the former becoming lower in energyfor a thickness below 5 . . g SL =
2, taking into account the spin degeneracy. Theeffective mass m ⋆ SL evaluated in this way is shown on the rightaxis of Fig. 3(a). In bulk AlAs, the effective masses in thetransverse and longitudinal directions of the ellipsoid Fermisurface are 0 . m and 0 . m , respectively [24]. For AlAsQWs thinner than 6.0 nm, the 2DES occupies the lower non-degenerate valley, where experiments report a transverse massof (0 . . m [25–27]. The obtained m ⋆ SL / m , which ap-proaches the expected value (0 . .
3) with increasing V FG , isreasonable.Once C SL is obtained as a function of V FG , one can cal-culate the electron density in the AlAs layers [ n AlAs1(2) ] andSL ( n SL = n AlAs1 + n AlAs2 ). Figure 3(b) shows the V FG de-pendence of n SL . The right axis of the figure indicates theexcess electron density normalized by the doping density( N Si = m − ). While n SL varies almost linearly with V FG ,the slope decreases slightly at V FG < − . C. Effects on mobility
Now let us investigate the effect of screening on mobil-ity. The symbols in Fig. 4(a) show the 1.6-K mobility ( µ )measured at the same carrier density ( n QW = . × m − )but with the sample prepared to have different n SL / N Si val-ues. The open circles show data obtained by re-adjusting n QW with the back gate after equilibrating the system at 4.3 K for m [ m / V s ] (m BI + m EES ) –1 –1 m BI –1 (m BI + m STD ) –1 –1 –1 n QW = m –2 n S L / N S i (a)(b) n S L / N S i T = 1.6 K m EES ( n SL / N Si << 1) m [ m / V s ] m EES (1 – n SL / N Si << 1) m STD
Fit
FIG. 4. (a) Mobility vs. n SL / N Si . Open circles (squares) are ex-perimental results for n QW = . × m − measured at 1.6 K with V FG set at 4.3 K (room temperature). The error bar represents theuncertainty in n SL / N Si for the data taken at V FG = − .
70 V. The redline is the total mobility calculated using RI-limited mobility withexcess-electron screening ( µ EES ) shown in (b) and fitted to exper-imental data using BI-limited mobility ( µ BI ) as a parameter. Thevalue of µ BI used for the fit is shown by the cyan line. The blackline shows the total mobility for the standard model without excesselectron screening ( µ STD ) shown in (b). (b) RI-limited mobility cal-culated with various models. The dashed green and blue curves wererespectively calculated using the analytical formula of µ EES for thetwo cases, n SL / N Si ≪ − n SL / N Si ) ≪ V FG ( − . V FG range, n SL / N Si varied between 0.13 and 0.93 [see Fig. 3(b)].As equilibration was not obtained for V FG < − . n SL / N Si .The open squares in Fig. 4(a) show data obtained by apply-ing V FG ≤ − . n QW at 1.6 K with the back gate. We confirmed that the SL wasalready depleted (i.e., n SL / N Si =
0) for V FG = − .
88 V bynoting that at 1.6 K the 2DES was depleted at zero back gatevoltage. As Fig. 4(a) shows, µ decreased by 37% as n SL / N Si decreased from 0.93 to 0.We characterize the n SL / N Si dependence by consideringtwo main sources of disorder in modulation-doped GaAs2DESs, i.e., background ionized impurities (BIs) and remoteionized impurities (RIs). For the mobility limited by RIs, weused the excess electron screening (EES) model proposed bySammon et al. [16]. The dashed blue and green curves in Fig. 4(b) show the mobility µ EES calculated using the EESmodel for the two limits, n SL / N Si ≪ − n SL / N Si ≪ µ EES = e ¯ h k d × . n SL N Si − . (1)Here, k F = ( π n QW ) / is the Fermi wave number and d = ( µ BI + µ EES ) − based on Matthiessen’s rule,where µ BI is the only parameter and we assumed it to beconstant. The µ BI value obtained from the fit is shown bythe cyan line in Fig. 4(a). The EES model well explainsthe overall n SL / N Si dependence of the measured µ , provid-ing good agreement for n SL / N Si ≥ .
13. For n SL / N Si < .
13, the agreement between the experiment and model be-comes less satisfactory, which is because we tried to fit bothregimes of n SL / N Si ≪ − n SL / N Si ≪ µ STD ) [28], assuming independentscattering by ( N Si − n SL ) ionized donors. The black lines inFigs. 4(b) and 4(a) show µ STD and the resultant total mo-bility ( / µ BI + / µ STD ) − , respectively. The independent-scattering model predicts a mobility way below the experi-mental result, which in turn demonstrates the importance ofthe screening by excess electrons.We also examined the possibility of excess electrons in theSL affecting µ BI . Sammon et al. reported that, for strongscreening, the contribution of BIs to µ BI is canceled out by theimage-charge effect when they are located farther than 0 . d from the center of the QW [17]. We calculated µ BI by inte-grating contributions from BIs over different spatial ranges,0 . d and d . The difference between the two cases is less than1%, thus corroborating our assumption of constant µ BI . D. Quantum lifetime
Finally, let us investigate the effect of screening on quantumlifetime ( τ q ) deduced from Shubnikov–de Haas (SdH) oscilla-tions, a quantity often argued to be a better indicator of samplequality than mobility in terms of FQHEs. Figure 5(a) showsthe SdH oscillations measured at 0.27 K at a constant car-rier density ( n QW = . × m − ) under different screen-ing conditions of n SL / N Si = .
93, 0 .
18, and 0 (corresponding V FG of 0, − .
20, and − .
88 V, respectively). Under the well-screened condition ( n SL / N Si = . τ q by using the functional form of SdH oscillations, given as[29] ∆ R = R exp (cid:18) − πω c τ q (cid:19) χ ( T ) , (2) t qEES (t qBI + t qEES ) –1 –1 t qBI –1 n S L / N S i B [T] 250200150100500 0.60.50.40.30.20.10.0 D R / R X ( T ) B –1 [ T –1 ] (b)(a) t q [ s ] –10 –12 –11 n = 9 n S L / N S i T = 0.27 K R xx [ W ] n QW = m –2 FIG. 5. (a) Shubnikov-de Haas oscillations under strong (red), inter-mediate (black), and weak (blue) screening conditions for the samedensity. The inset shows the Dingle plot for each case. (b) Quan-tum lifetime vs. n SL / N Si . Open circles (squares) are experimentalresults measured at 0.27 K with V FG set at 4.3 K (room tempera-ture). The error bar represents the uncertainty in n SL / N Si for the datataken at V FG = − .
70 V. The cyan and green lines are calculated life-times limited by BIs ( τ qBI ) and RIs ( τ qEES ). The dashed magenta lineshows the total quantum lifetime. with χ ( T ) = π k B T ¯ h ω c sinh 2 π k B T ¯ h ω c . (3)Here, ∆ R is the amplitude of the SdH oscillations, ω c is the cy-clotron frequency, R is the R xx at zero magnetic field, χ ( T ) is a thermal damping factor, and k B is the Boltzmann con-stant. Thus, the slope of ∆ R / R χ ( T ) vs. 1 / B , known as aDingle plot, shown in the inset of Fig. 5(a) gives the quan-tum lifetime τ q [30]. The obtained τ q is shown by opensymbols in Fig. 5(b). In contrast to µ , or transport lifetime( τ t = m QW µ / e ), τ q does not show a discernible change as afunction of n SL / N Si .We compared the measured τ q with the calculated quantumlifetimes limited by RIs and BIs ( τ qEES and τ qBI ), which areshown in Fig. 5(b) by the green and cyan lines, respectively.Here, τ qEES was calculated with the EES model [16], whereas τ qBI was calculated with the independent-scattering model us- ing the BI concentration obtained from µ BI ( =
224 m /Vs).The expected total quantum lifetime is shown by the dashedmagenta line. Although the measured τ q is close to the val-ues expected in the weak screening regime, it remains about2 . n SL / N Si increases, much lower than expected inthe intermediate and strong screening regimes. This discrep-ancy was not mitigated even when a more elaborate model wasemployed, such as one with different BI concentrations in theGaAs QW and AlGaAs barrier layers [17] and remote chargeson the sample surface and the back gate [31, 32]. Extrinsicmechanisms that might reduce the apparent quantum lifetime,such as the finite density gradient [33] [34] and the responsetime of the lock-in amplifier [35], did not explain the discrep-ancy, either. Ultra-high-quality samples with much longer τ q ,such as those in Refs. [33, 36], might be necessary to observethe predicted screening effect on τ q . Yet, it is interesting thatthe visibility of the spin gap varies with n SL / N Si even when τ q remains constant, as we observed. For n SL / N Si =
0, the den-sity inhomogeneity estimated from the analysis of the Dingleplot [33] is 1.8%, which may be partly responsible for thepoorly developed quantum Hall effects at even as well as oddinteger fillings. However, for n SL / N Si = .
18 and 0 .
93, theestimated density inhomogeneity is less than 0.2% with noclear difference, which cannot account for the difference inthe visibility of the spin gap. This suggests that the screeningof long-range disorder becomes more important for interac-tion phenomena such as FQHEs. The broad R xx minimumaround B = ν = / ν = / III. CONCLUSION
In summary, we investigated the screening effects of SLdoping on the mobility and quantum lifetime of a GaAs 2DESby controlling the excess electron density in the SL with a topgate. The dependence of mobility on excess electron densityis consistent with theory taking into account the screening ef-fect. On the other hand, the measured quantum mobility wasmuch lower than expected from theory and did not show adiscernible change with excess electron density. The excesselectrons also affected the depth of the spin-gap minima inthe Shubnikov-de Haas oscillations, which suggests the pos-sibility of controlling the visibility of FQHEs in-situ.
ACKNOWLEDGEMENTS
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