Measuring carrier density in parallel conduction layers of quantum Hall systems

Transcription

1 Measuring carrier density in parallel conduction layers of quantum Hall systems M. Grayson, F. Fiscer Walter Scottky Institut, Tecnisce Universität Müncen, D-8578 Garcing, Germany arxiv:cond-mat/5v [cond-mat.mes-all] 9 Nov (Dated: 7 Nov ) (Rsion 6) An experimental analysis for two parallel conducting layers determines te full resistivity tensor of te parallel layer, at magnetic fields were te oter layer is in te quantum Hall regime. In eterostructures wic exibit parallel conduction in te modulation-doped layer, tis analysis quantitatively determines te carge density in te doping layer and can be used to estimate te mobility. To illustrate one application, experimental data sow magnetic freeze-out of parallel conduction in a modulation doped eterojunction. As anoter example, te carrier density of a minimally populated second subband in a two-subband quantum well is determined. A simple formula is derived tat can estimate te carrier density in a igly resistive parallel layer from a single Hall measurement of te al system. Various ig-mobility eterostructures contain parallel conducting layers wose individual conductance parameters need to be caracterized. Suc systems include double-quantum well systems and quantum wells wit a second occupied subband. Anoter system of practical interest is a ig-mobility two-dimensional electron system (DES) wit a parallel conducting modulationdopant layer. In all cases, te well-known signature of parallel conduction is tat te minima in te longitudinal resistance R do not go to zero in te quantum Hall limit. One standard metod for separately caracterizing te layer densities is te Subnikov-de Haas (SdH) metod, wic requires tat bot systems sow oscillations in R in te same magnetic field range. Tese oscillations are ten Fourier transformed against / to give te densities of bot systems []. However, tis metod cannot be used if one of te systems as sufficiently low mobility tat SdH oscillations are indiscernable, and tis metod gives no separable measurement of R for eac layer. Oter metods ave been introduced wic can determine resistivities in non-quantized parallel cannels [,, ], but to date an explicit treatment of te quantized case is lacking. In tis paper, we present a new analysis of magneransport data wic determines bot components of te resistivity tensor in one layer, at values of te field were te oter layer is in te quantized Hall regime. Te density of carriers in te parallel layer ten follows directly n = / e, wit e te electron carge. We wis to first find an expression for te resistivity tensor of te combined quantum Hall and parallel conducting layers since resistivity is te experimentally measured quantity. We define separate resistivity tensors for te quantum Hall and parallel conduction layers as: ρ Q = ( νe νe ) (, = ) () Assuming tat te in-plane electric fields seen by te two layers are equal, te corresponding conductivities for eac layer, σ Q = ( νe νe ), σ = ( ) () can be summed to yield te seet conductivity of te combined system ( σ = σ Q +σ = νe + νe () were = det( ). Inverting σ gives te al resistivity tensor, = (σ ) wose components are directly measured in experiment: = ρ +( νe +ρ ), = ρ ( νe +ρ ) +( νe +ρ ) () Just as in single-layer Hall measurements of were expressions for te longitudinal and transverse resistance are solved for te two unknowns, density and Hall mobility, te above equations can be solved for te only two unknowns in tis problem, namely, te components of te resistivity tensor in te parallel layer. We will demonstrate tis analysis for data measured in te general Van der Pauw geometry, altoug te same analysis can be adapted to Hall bar and Corbino geometries. All -point resistances were measured in an Oxford He cryostat using standard lock-in tecniques wit a ig input impedance pre-amplifier (R in GΩ) to avoid measurement errors wic migt be mistaken for parallel conduction [5]. Te first sample of interest is a eterojunction quantum well wit parallel conduction in te modulation doping layer. We start experimentally by measuring te longitudinal resistance of te sample at finite -field in two )

2 complementarygeometriesr A andr (seefig. ), and determine te al -dependent seet resistance () according to te Van der Pauw equations [6]: = π ln() f(x)ra +R f(x) = ln ( x x+ ) [ (ln) (ln) (5) ]( ) x x+ Here x = R/R A for R A > R, and f(x) is a weak function of x of order unity. Te Hall resistance = R can be measured directly (Fig. ). We now solve Eq. for in terms of te measured and ρ. Te al resistivity tensor ρ inverted to give σ = ( ), and since σ = σ Q + σ, σ = σ = ρ, σ = σ + νe = ρ (6) Knowing te filling factor index ν of a given quantum Hall minimum, we solve for σ = σ and σ = σ νe. Inverting σ to find = (σ ) yields: = σ = σ σ = (7) ( ) +( ρ νe ) ρ νe ) (8) ( ) +( ρ νe ) σ = ρ ( e wic is noting oter tan Eq. () solved for and. Te result is plotted in Fig. for values of were te DES Hall resistance is quantized. We can estimate te carrier density in te parallel layer wit eiter of two metods. First, te slope of (bot witin eac quantum Hall effect domain and across all filling factors) indicates a density of n = =. cm. Secondly, as a useful backof-te-envelope sortand, we note tat in te limit of < ρ as in Fig., Eq. (8) can be simplified to, Solving for n gives = n e νe ( ) n n Q νe (9) () were n Q = νe(ν) is te density of te QH system, and (ν) is te magnetic field value in te center of te ν-t plateau in. In Fig., te flat lines above eac ρ plateau sow te resistance quantum νe demonstrating tat te first term in parentesis in Eq. () is always greater tan. Te remarkable utility of Eq. () is tat te parallel carrier density can be estimated from a single measurement. For example in ρ from Fig., te position of te ν = plateau at (ν = ) = 8.7 T determines n Q =.95 cm and te plateau resistance of =.76 kω gives n =.6 cm witin 5 % of te more exact grapical analysis of Eq. (8) and te resulting Fig.. We can separately caracterize te conductance of te parallel delta-doping layer as a result of tis analysis. Looking at te two components of te resistivity tensor in Fig., one observes first te sarp rise in resulting from so-called magnetic freeze-out in tis igly disordered delta-doping layer, expected in systems wit opping conduction. Te al number of mobile carriers in te delta layer, owever, stays fixed as confirmed by te constant slope of. We can also make a roug estimate of te mobility of tis layer, by extrapolating te parabolic fit sown in te figure to =. Wit an intercept resistivity of around = 7kΩ and te known density, te = mobility of te parallel conduction layer is about µ =. cm /Vs. In te temperature dependence sown in Fig. 5, we see increase wit decreasingtemperature, wile remains fixed, indicating a temperature independent density. ot beaviors are consistent wit opping conduction in te delta-layer. Te same analysis can be applied to a two-subband quantum well sample. In Fig. 6 we sow te measured and ρ wic are analized to give ρ as plotted in Fig. 7. Te measured parallel carrier density of n =. cm is in good agreement wit selfconsistent Poisson calculations wic predict a second subband occupied to n calc =.5 cm. We note tat te SdH signal from tis low density second subband was too weak to give and estimable density using alternate te Fourier transform metod. Comparing te resistivity components in Fig. 7 wit Fig., one seesin tatonceagainte densityofcarriersisrater constant overte wole -field range. Te measured is of order5 Ω orless, implying a raterig mobility of over 6 cm /Vs for tis second subband. Altoug tis value is not quantitatively conclusive, qualitatively it indicates tat te low density subband wic would normally ave a correspondingly low mobility, benefits from te Tomas-Fermi screening of te densely populated subband to screen most of te disorder. As a final note, we remind te reader tat measurementerrorsariseifteinputimpedanceoftelock-inr in is too low, and tese patological signals look misleadingly like parallel conduction. Wit a low R in, finite R minima of order R min = (/νe ) /R in are observed wit te astonising property tat te false parallel signal disappears upon reversing te current contact polarity.

3 Suc subtleties in quantum Hall measurement penomena are discussed in detail elsewere [5]. In conclusion, we ave demonstrated a new analysis for te resistivity of parallel D systems tat determines te resistivity tensor of one layer at magnetic fields were te oter layer is in te quantum Hall regime. We demonstrate te validity of tis analysis for te case of a quantum well wit two occupied subbands, as well as for double-quantum wells. We also demonstrate te analysis of carrier density in parallel conducting modulation doped layers. In te latter case, tis tecnique will be particularly useful for crystal growers in caracterizing doping efficiency, because it elimates te expensive and time-consuming trial-and-error metod currently used for doping calibration of parallel conduction layers. Acknowledgement Tis work was supported in part by te DFG (SF 8). M.G. tanks te A.v. Humboldt Foundation for support. (a) mgrayson@alumni.princeton.edu [] J. Jo, Y. W. Sen, L. W. Engel, M.. Santos and M. Sayegan, Pys. Rev. 6, 9776 (99). [] M. van der urgt, V. C. Karavolas, F. M. Peeters, J. Singleton, R. J. Nicolas, F. Herlac, J. J. Harris, M. Van Hove and G. orgs, Pys. Rev. 5, 8 (995). [] C. M. Hurd, S. P. McAlister, W. R. McKinnon,. R. Stewart, D. J. Day, P. Mandeville, and A. J. SpringTorpe, J. Appl. Pys. 6, 76 (988). [] M. J. Kane, N. Apsley, D. A. Anderson, L. L. Taylor and T. Kerr, J. Pys. C: Solid State Pys. 8, 569 (985). [5] F. Fiscer and M. Grayson, submitted to J. Appl. Pys. in tandem wit tis paper. [6] L. J. van der Pauw, Pilips Res. Rep., (958).

4 a) parallel dopant layer b) parallel quantum well c) second subband E F Q E F E F Q Q Fig. FIG. : Conduction band scematic for te tree kinds of parallel conduction systems of interest: a) te dopant layer of a modulation-doped eterostructure, b) a double-quantum well system, or c) a quantum well wit a second occupied subband.

5 5 FIG. : Te tree measurements necessary to completely caracterize te resistivity tensor at any magnetic field. Fig. Sample --. n Q =.9 x / cm µ Q =.9 x 5 cm /Vs. K ρ ρ ρ ρ R A R (T) FIG. : Plot of measured, and for a sample wit parallel conduction in te modulation doped layer. is calculated from te Van der Pauw relation in Eq. 5 using R A and R as measured in te inset.

6 6 5 Sample --. n =. x / cm µ ~. x cm /Vs. K Fig. ρ 5 ρ ρ 5 ρ (T) FIG. : Plot of for te parallel conducting modulation doped layer calculated from te data in Fig. using Eqs. 7 and 8. Te slope of yields te carrier density in te parallel conductor n = e. 5 Sample --. n =. x / cm Fig. 5 ρ 5 5 mk ρ (T). K. K (T) FIG. 5: Plot of for te parallel conducting modulation doped layer at various temperatures. Te opping conduction in tis system is strongly temperature and magnetic field dependent. Inset: Plot of for te parallel conducting layer. Note tat te density of opping carriers is independent of and temperature, except above T at 5 mk under extreme magnetic freezeout.

7 7 ( ) 5 Sample --. n Q =.5 x / cm Q =.57 x 6 cm /Vs 5 mk Fig (k ) R A (T) R FIG. 6: Plot of,, R A and R for a quantum well wit a ligtly populated second subband. (Ω) ρ Sample --. n =. x / cm 5 mk Fig. 7 8 ρ ρ ρ 5 (T) FIG. 7: Plot of calculated using te data in Fig. 6 for a ligtly populated second subband.