Mobility versus quality in 2D semiconductor structures
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Mobility versus quality in 2D semiconductor structures
S. Das Sarma and E. H. Hwang , Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742-4111 SKKU Advanced Institute of Nanotechnology and Department of Physics,Sungkyunkwan University, Suwon, 440-746, Korea (Dated: October 8, 2018)We consider theoretically effects of random charged impurity disorder on the quality of high-mobility two dimensional (2D) semiconductor structures, explicitly demonstrating that the samplemobility is not necessarily a reliable or universal indicator of the sample quality in high-mobilitymodulation-doped 2D GaAs structures because, depending on the specific system property of inter-est, mobility and quality may be controlled by different aspects of the underlying disorder distri-bution, particularly since these systems are dominated by long-range Coulomb disorder from bothnear and far random quenched charged impurities. We show that in the presence of both channeland remote charged impurity scattering, which is a generic situation in modulation-doped high-mobility 2D carrier systems, it is quite possible for higher (lower) mobility structures to have lower(higher) quality as measured by the disorder-induced single-particle level broadening. In particular,we establish that there is no reason to expect a unique relationship between mobility and qualityin 2D semiconductor structures as both are independent functionals of the disorder distribution,and are therefore, in principle, independent of each other. Using a simple, but reasonably realistic,“2-impurity” minimal model of the disorder distribution, we provide concrete examples of situa-tions where higher (lower) mobilities correspond to lower (higher) sample qualities. We discussexperimental implications of our theoretical results and comment on possible strategies for futureimprovement of 2D sample quality.
I. INTRODUCTION
One of the most significant materials developmentsin the fundamental quantum condensed matter physics,which is not universally known outside the 2D commu-nity, has been the astonishing 3,000-fold increase in thelow temperature electron mobility of GaAs-based 2D con-fined quantum systems from ∼ cm /Vs in the firstmodulation-doped GaAs-AlGaAs 2D heterostructures in 1978 to the current world-record mobility of ∼ − × cm /Vs in the best available modulation dopedGaAs-AlGaAs quantum wells of today. This representsa truly remarkable more than three orders of magnitudeenhancement in the low temperature ( ∼ ∼
50 nm in 1978 to the essentially macroscopic lengthscale of ∼ . , the even-denominator FQHE , thebilayer half-filled FQHE , the anisotropic stripe and bub-ble phases , and many other well-known novel 2D phe-nomena. (As an aside we point out that, qualitatively similar to the situation in the Moore’s law, the exponen-tial enhancement in the 2D mobility of semiconductorheterostructures has slowed down considerably in the re-cent years with only a 30% increase in the mobility dur-ing the 2003 − ∼ × to ∼ × cm /Vs, after roughly a factor of 1,000 increases during1978 − T = 0 transport properties (or low-temperature transport properties) with the temperaturebeing much smaller than both the Fermi temperatureand the Bloch-Gr¨uneisen (BG) temperature of the 2Dsystem so that all thermal effects have saturated, and wedo not need to account for either phonon scattering orfinite temperature effects in the Fermi distribution func-tion. The high mobility of 2D semiconductor structuresapplies only to this low-temperature situation, and athigher temperatures ( > . Our theory is just restricted only to T = 0 impurity-scattering-limited 2D transport proper-ties, which limit the ultimate achievable mobility in thesesystems. It is also important to emphasize that we ignoreall weak localization aspects of 2D transport properties,restricting entirely to the semiclassical transport behav-ior where the concept of a mobility is valid. Thus, thetheory applies only at densities where weak-localizationbehavior does not manifest itself at the experimentaltemperatures ( ∼
25 mK − > mobility enhancement of 2D systems hasgenerally led to the improvement of sample quality on theaverage over the years as manifested in the observation ofnew phenomena, it has been known from the early daysthat the connection between mobility and quality is atbest a statistically averaged statement over many sam-ples and is not unique, i.e., a sample with higher mobilitythan another sample may not necessarily have a higherquality with respect to some specific property (e.g., theexistence or not of a particularly fragile fractional quan-tum Hall plateau). Thus, higher (lower) mobility doesnot necessarily always translate into higher (lower) qual-ity for specific electronic properties. Since 2D carriermobility is a function of carrier density , in fact, it is, inprinciple, possible for a sample to have a higher (lower)mobility than another sample at higher (lower) carrierdensity, implying that the measured mobility at somefixed high density is not (always) even a good indicatorof transport quality itself as a function of carrier den-sity, let alone being an indicator for the quality of otherelectronic properties!The reason for the above-mentioned mobility/qualitydichotomy is rather obvious to state, but not easy toquantify. Both mobility and quality (e.g., the measuredactivation gap for a specific FQHE state or some otherspecified electronic property) depend on the full disor-der distribution affecting the system which is in generalboth unknown and complex, and depends also on thesample carrier density in a complicated manner. Thedisorder distribution is characterized by many indepen-dent parameters, and therefore, all physical propertiesof the system, being unique functionals of the disorderdistribution, are independent of each other. In particu-lar, the dc conductivity σ , which determines the density-dependent mobility µ = σ/ne where n is the 2D carrierdensity (and e the magnitude of electron charge), is de-termined by essentially an integral over the second mo-ment of the disorder distribution whereas other physicalproperties (i.e., the quality, although there could, in prin-ciple, be many independent definitions of sample qualitydepending on independent experimental measurementsof interest) could be determined by other functionals ofthe disorder distribution. Thus, for any realistic disorderdistribution, we do not expect any unique relationshipbetween mobility and quality, and it should be possible,in principle, for samples of different mobility to have thesame quality or vice versa (i.e., samples of different qual-ity to have the same 2D mobility).There is still the vague qualitative expectation, how-ever, that if the sample mobility is enhanced by improv-ing the sample purity (i.e., suppressing disorder), thenthis should automatically also improve the sample qual-ity (perhaps not necessarily by the same quantitative fac-tor) since reduced disorder should enhance quality. Wewill show below that this may not always be the case since mobility and quality (for a specific property) maybe sensitive to completely different aspects of disorderand therefore enhancing mobility by itself may do noth-ing to improve quality in some situations. On the otherhand, when mobility and quality are both determined byexactly the same microscopic disorder in the sample, in-creasing (decreasing) one would necessarily improve (sup-press) the other although we will see later in this work[see Fig. 1(c)], for example) this is not necessarily trueif mobility and quality are both determined by remotedopant scattering arising from long-range Coulomb dis-order.It is important to emphasize a subtle aspect of themobility/quality dichotomy with respect to experimentalsamples. Theoretically, we can consider a hypotheticalsystem with continuously variable disorder (i.e., the pa-rameters characterizing the disorder distribution such asthe quenched charged impurity density and their strengthas well as their spatial locations including possible spa-tial correlations in the impurity distribution can all bevaried at will). Experimentally, however, the situation isqualitatively different. One does not typically change theimpurity distribution in a sample in a controlled mannerand make measurements as a function of disorder dis-tribution. Experimentally, measurements are made in different samples and compared, and in such a situationthere is no reason to expect two samples with identicalmobility at some specified carrier density to have identi-cal impurity distributions. The impurity distributions indifferent samples can be considered to be identical only ifthe full density-dependent mobility µ ( n ), or equivalentlythe density-dependent conductivity σ ( n ), are identical inall the samples. Such a situation of course never happensin practice, and typically when experimental mobilitiesin different samples are quoted to be similar in magni-tudes, one is referring to either the maximum mobility(occurring typically at different carrier densities in dif-ferent samples) or the mobility at some fixed high carrierdensity (and not over a whole range of carrier density). Iftwo samples happen to have the same maximum mobilityor the same mobility at one fixed density, there is no rea-son to expect them to have the same quality with respectto all experimental properties at arbitrary densities. Thisobvious aspect of mobility versus quality dichotomy hasnot much been emphasized in the literature.It should be clear from the discussion above that in the‘trivial’ (and experimentally unrealistic in 2D semicon-ductor structures) situation of the system having just onetype of impurities uniquely defining the applicable impu-rity distribution, both mobility and quality, by defini-tion, would be determined by exactly the same impurityconfiguration since there is just one set of impurities byconstruction. Such a situation is, in fact, common in 3Dsemiconductors where the applicable disorder is almostalways described by a random uniform background of un-correlated quenched charged impurity centers, which canbe uniquely characterized by a single 3D impurity den-sity n i . Obviously, in this situation both mobility andquality are uniquely defined by n i , and thus increasing(decreasing) n i would decrease (increase) both mobilityand quality (although not necessarily by the same quan-titative factor since mobility and quality are likely to bedifferent functions of system parameters in general). Insuch a simplistic situation, mobility and quality are likelyto monotonically connected by a unique relationship, andhence enhancing system mobility should always improvethe system quality since the same disorder determinesboth properties. In fact, this simple situation always ap-plies if the mobility/quality are both limited by purelyshort-range disorder in the system.By contrast, 2D semiconductor structures almost al-ways have several qualitatively distinct disorder mech-anisms affecting transport and other properties arisingfrom completely different physical origins. For exam-ple, it is well-known that Si-MOSFETs have at leastthree different operational disorder mechanisms: randomcharged impurities in the insulating SiO oxide layer, inthe bulk Si itself, and random short-range surface rough-ness at the Si-SiO interface. There may still be otherdistinct scattering mechanisms associated with still dif-ferent disorder sources in Si-MOSFETs such as neutraldefects or impurities, making the whole situation quitecomplex. In MOSFETs, low-density carrier transportis controlled by long-range charged impurity scatteringwhereas the high-density carrier transport is controlledby short-range surface roughness scattering, and this di-chotomy may very well lead to situations where a mea-sured electronic property (i.e., “quality”) does not nec-essarily correlate with the high-density maximum mobil-ity of the system. In high-mobility 2D GaAs-AlGaAs-based systems of interest in the current work, there areat least six distinct scattering mechanisms of varying im-portance arising from different physical sources of disor-der in the system. These are: Unintentional backgroundcharged impurities in the 2D GaAs conducting layer; re-mote dopant impurities in the insulating AlGaAs layer(which are necessary for introducing 2D carriers to formthe 2DEG); short-range interface roughness at the GaAs-AlGaAs interfaces; short-range disorder in the insulatorAlGaAs layer arising from alloy disorder (and neutraldefects); unintentional background charged impurities inthe insulating AlGaAs barrier regime; random chargedimpurities at the GaAs-AlGaAs interface. This is obvi-ously a highly complex situation where the complete dis-order distribution will have many independent parame-ters, and in general, there is no reason to expect a uniquerelationship between mobility and quality since mobil-ity could be dominated by one type of disorder (e.g.,background unintentional charged impurities) and qual-ity may be dominated by a different type of disorder (e.g.,remote dopants far from the 2D layer).It is clear from the above discussion that the min-imal disorder model capable of capturing the mobilityversus quality dichotomy in high-quality 2D semiconduc-tor structures is a “2-impurity” model with one type ofimpurity right in the 2D layer itself (arising, for exam- ple, from the unintentional background impurities in thesystem) and the other type of impurity being a remotelayer separated by a distance ‘ d R ’ from the 2D electronlayer. This minimal 2-impurity model is characterized bythree independent parameters: n R and d R , denoting re-spectively the 2D charged impurity density in the remotedopant layer separated by a distance d R from the 2D car-riers, and n B , the 2D impurity density corresponding tothe unintentional background impurities in the 2D layerwith d = 0. It is easy to go beyond this minimal modeland consider the remote impurities to be distributed overa finite distance (rather than simply being placed in a δ -function like layer located at a distance d R from the 2Dsystem) or assume the background impurities to be dis-tributed three-dimensionally (rather than in a 2D planeat d = 0), but such extensions do not modify any of ourqualitative conclusion in the current paper as we have ex-plicitly checked numerically. Also, we discuss our theoryassuming a strict 2D model (with zero thickness) for theelectron layer because the finite quasi-2D layer thicknesshas no qualitative effect on the physics of quality ver-sus mobility being discussed in this work. Many of ournumerical results are, however, obtained by incorporat-ing the appropriate quantitative effects of the quasi-2Dlayer thickness of the 2D carriers in calculating the mo-bility and the quality of the system within the 2-impuritymodel.One last issue we need to discuss in the Introductionis the question of how to define the sample quality sinceit is obviously not a unique property and depends onthe specific experiment being carried out. In order tokeep things both simple and universal, we have decidedto use the level broadening or the Dingle temperatureas a measure of the sample quality. The level broaden-ing Γ is defined as Γ = ~ / τ q , where τ q is the quantumscattering time (or the single particle relaxation time) incontrast to the mobility scattering time (or the trans-port relaxation time) τ t which defines the conductivity( σ ) or the carrier mobility ( µ ) through σ = ne τ t /m or µ = neτ t . In general, τ q ≤ τ t with the equality holdingfor purely short-range disorder scattering. Thus, for pure s -wave δ -function short-range scattering model, mobilityand quality are identical for obvious reasons. For a strict1-impurity model of underlying sample disorder, mobility( τ t ) and quality ( τ q ) are both affected by the same impu-rity density, and hence they must behave monotonically(but not necessarily identically) with changing disorder,i.e., if the impurity density is decreased (increased) ata fixed carrier density, both mobility and quality mustincrease (decrease) as well.As emphasized already, however, the 2-impurity model(near and far impurities or background and remote im-purities) introduces a new element of physics by allowingfor the possibility that mobility and quality could possi-bly be affected more strongly by different types of disor-der, for example, mobility (quality) could be dominatedby near (far) impurities in the 2-impurity disorder model,thus allowing, in principle, the possibility of mobility andquality being completely independent physical propertiesof realistic 2D semiconductor samples at least in somesituations. We find this situation to be quite prevalentin very high-mobility 2D semiconductor structures wherethe mobility (quality) seems to be predominately deter-mined by near (far) impurities. In low mobility samples,on the other hand, the situation is simpler and both mo-bility and quality are typically determined by the sameset of impurities (usually the charged impurities close tothe 2D layer itself).In section II we describe our model giving the theoret-ical formalism and equations for the 2-impurity model.In section III we provide detailed results for mobility andquality along with discussion. We conclude in section IVwith a summary of our findings along with a discussionof our approximations and of the open questions. II. MODEL AND THEORY
We assume a 2D electron (or hole) system at T = 0located at the z = 0 plane with the 2D layer being inthe x-y plane in our notation. The 2D system is charac-terized entirely by a carrier effective mass ( m ) definingthe single-particle kinetic energy ( E k = ~ k / m with k as the 2D wave vector), a background lattice dielec-tric constant ( κ ) defining the 2D Coulomb interaction[ V ( q ) = 2 πe /κq with q as the 2D wave vector), and a2D carrier density ( n ).As discussed in the Introduction, we use a minimal2-impurity model for the static disorder in the systemcharacterized by three independent parameters: n R , d R , n B . Here n R ( n B ) is the effective 2D charged impuritydensity for the remote (background) impurities with theremote (background) impurities being distributed ran-domly in the 2D x-y plane at a distance ‘ d ’ from the 2Delectron system in the z -direction with d = d R (0) for theremote (background) impurities. We assume the randomquenched charged impurities to all have unit strength(i.e., having an elementary charge of ± e each) with noloss of generality.Our model thus has four independent parameters withdimensions of length: n − / , n − / R , n − / B , d R . In ad-dition to these (experimentally variable) parameters, wealso have m and κ defining the material system which isfixed in all samples for a given material. In principle, twoadditional materials parameters should be added to de-scribe the most general situation, namely, the spin ( g s )and the valley ( g v ) degeneracy, but we assume g s = 2and g v = 1 throughout the current work (and for allour numerical results) since our interest here is entirelyfocussed on high-mobility n- and p-GaAs 2D systemswhere the mobility/quality dichotomy has mostly beendiscussed. Additional experimentally relevant (but, theo-retically non-essential) parameters, such as a finite widthof the 2D electron system (instead of the strict 2D limit)and/or a 3D distribution of the random impurities (in-stead of the 2D distribution assumed above), are straight- forward to include in the model and are not discussedfurther in details.The mobility (quality) is now simply defined by thecharacteristic scattering time τ t ( τ q ) as given by the fol-lowing equations in our leading-order transport theory(i.e., Boltzmann transport plus Born approximation forscattering):1 τ t ( k ) = 2 π ~ X k ′ X l Z ∞−∞ dzN ( l ) i ( z ) | u k − k ′ ( z ) | × (1 − cos θ ) δ ( E k − E k ′ ) , (1)and 1 τ q ( k ) = 2 π ~ X k ′ X l Z ∞−∞ dzN ( l ) i ( z ) | u k − k ′ ( z ) | × δ ( E k − E k ′ ) , (2)where N ( l ) i ( z ) is the 3D impurity distribution for the l -thkind of disorder in the system, and u q ( z ) is the screenedelectron-impurity Coulomb interaction given by: u q ( z ) = V q ( z ) ε ( q ) = 2 πe κq e − qz ε ( q ) , (3)with ε ( q ) being the static RPA dielectric function for the2D electron system. We note that V q ( z ) = V ( q ) e − qz = πe κq e − qz is simply the 2D Fourier transform of the 3D 1 /r Coulomb potential, which explicitly takes into accountthe fact that a spatial separation of ‘ z ’ may exist betweenthe 2D electron layer and the charged impurities. The 2Dstatic RPA dielectric function or the screening functionis given by ε ( q ) = 1 + 2 πe κq Π( q ) , (4)where the static 2D electronic polarizability functionΠ( q ) is given byΠ( q ) = N F h − θ ( q − k F ) p − (2 k F /q ) i , (5)where N F = m/π ~ and θ ( x ) = 0 (1) for x < x >
0) isthe Heaviside step (or theta) function and the 2D Fermiwave vector k F is determined by the 2D carrier densitythrough the formula k F = (2 πn ) / . We note that the2D Fermi energy ( E F ) is given by E F = ~ k F / m = π ~ n/m . The RPA screening function ε ( q ) can be ex-pressed in the convenient form ε ( q ) = 1 + q s /q, (6)which is exactly equivalent to Eqs. (4) and (5) if we definethe 2D screening wave vector q s to be q s = q T F h − θ ( q − k F ) p − (2 k F /q ) i , (7)where the 2D Thomas-Fermi wave vector q T F is definedto be q T F = 2 me / ( κ ~ ) . (8)We note that (1) the Thomas-Fermi wave vector q T F isproportional to the 2D density of states at the Fermi en-ergy N F = m/π ~ (and is inversely proportional to theeffective background lattice dielectric constant κ ), and(2) screening is constant in 2D for 0 ≤ q ≤ k F [seeEq. (7)] because of the constant energy-independent 2Delectronic density of states. Since the δ -functions (neces-sary for energy conservation during the impurity-inducedelastic scattering of an electron from the momentum state | k i to the momentum state | k ′ i with a net wave vectortransfer of q = k − k ′ ) in Eqs. (1) and (2) restrict the wavevector transfer 0 ≤ q ≤ k F range (this is simply becausewe are at T = 0 so that the maximum possible scatter-ing wave vector is 2 k F corresponding to the pure back-scattering of an electron from + k F to − k F while obeyingenergy conservation), the relevant screening wave vectorfor our problem is purely the Thomas-Fermi wave vector q s = q T F as follows from Eq. (7) for q ≤ k F .We can, therefore, rewrite Eq. (3) as u q ( z ) = 2 πe κ ( q + q T F ) e − qz , (9)giving the effective screened Coulomb interaction (in the2D momentum space) between an electron in the 2D layerand a random quenched charged impurity located a dis-tance ‘ z ’ away.Finally, our 2-impurity disorder model is given by N i ( z ) = n R δ ( z − d R ) + n B δ ( z ) , (10)with three independent parameters n R , d R , and n B com-pletely defining the underlying disorder. We note thatwriting the second term in Eq. (10) as n B δ ( z − d B ), andthus introducing an additional length parameter into themodel, is completely unnecessary since, as we will seebelow, the physics of the mobility versus quality dual-ity in high-quality 2D structures is entirely dominatedby scattering from near impurities (controlling mobility)and far impurities (controlling quality), which allows usto put d B = 0 in the minimal model (thus making thenear impurities very near indeed). Putting Eqs. (3) –(10) in Eqs. (1) and (2), we can combine them into asingle 2D integral, obtaining1 τ t,q = 2 π ~ (cid:18) πe κ (cid:19) Z d k ′ (2 π ) δ ( E k − E k ′ )( q T F + | k − k ′ | ) f t,q ( θ ) × n n R e − | k − k ′ | d R + n B o , (11)where f t ( θ ) = 1 − cos θ kk ′ and f q ( θ ) = 1. For complete-ness, we mention that E k = ~ k / m .Before proceeding further, we emphasize that the“only” difference between mobility (i.e., τ t ) and qual-ity (i.e., τ q ) is the appearance (or not) of the angularfactor (1 − cos θ ) inside the double integral in Eq. (11)for τ t ( τ q ). This arises from the fact that the mobiltyor the conductivity is unaffected by forward scattering(i.e., θ ≈ θ ≈
1) whereas the single-particle level-broadening (Γ ∼ ~ /τ q ) is sensitive to scattering through all angles. Technically, the (1 − cos θ ) factor arises fromthe impurity scattering induced vertex correction in the2-particle current-current correlation function represent-ing the electrical conductivity whereas the single-particlescattering rate ( τ − q ) is given essentially by the imag-inary part of the impurity scattering induced 1-particleelectronic self-energy which does not have any vertex cor-rection in the leading-order impurity scattering strength.The absence (presence) of the vertex correction in τ − q ( τ − t ) makes the relevant scattering rate sensitive (insen-sitive) to forward scattering, leading to a situation where τ q and τ t could be very different from each other if for-ward (or small angle) scattering is particularly importantin a system as it would be for long range disorder poten-tial.This could happen in 3D systems if the scattering po-tential is strongly spatially asymmetric (or non-spherical)for some reason. It was pointed out a long time agoand later experimentally verified that such a stronglynon-spherically symmetric scattering potential arises nat-urally in 2D modulation-doped structures from randomcharged impurities placed very far ( k F d ≫
1) away fromthe 2D electron system due to the influence of the ex-ponential e − qd factor in the Coulomb potential whichrestricts much of the scattering to q ≪ /z , thus expo-nentially enhancing the importance of small-angle (i.e.,small scattering wave vector) scattering. This then leadsto τ t ≫ τ q for scattering dominated by remote dopants(the two scattering times could differ by more than twoorders of magnitude in high-mobility modulation-dopedstructures where k F d R ≫ k F d B ≪ d B ≈ d B canbe, but this has no qualitative significance for our con-sideration.Now, we immediately realize the crucial relevance ofthe 2-impurity model in distinguishing mobility (i.e., τ t )and quality (i.e., τ q ) in 2D systems since it now becomespossible for one type of disorder (e.g., remote impuri-ties) to control the quality (i.e., τ q ) and the other typeto control the mobility (i.e., τ t ). Of course, whether sucha distinction actually applies to a given situation or notdepends entirely on the details of the sample parameter(i.e., the specific values of n R , d R , n B ) as well as thecarrier density n , but the possibility certainly exists for d R to be large enough so that the remote impurity scat-tering is almost entirely small-angle scattering (thus onlyadversely affecting τ q in an appreciable way) whereas thebackground impurity scattering determines τ t . If thishappens, then mobility ( τ t ) and quality ( τ q ) could verywell be very different in 2D samples, and may have littleto do with each other.To bring out the above physical picture explicitly, wenow provide some analytical calculations for the integralsin Eq. (11) defining mobility ( τ t ) and quality ( τ q ). Werewrite Eq. (11) as τ − t,q = I ( R ) t,q + I ( B ) t,q , (12)where I ( R ) t,q = n R V π ~ Z d k ′ (2 π ) δ ( E k − E k ′ )( q T F + | k − k ′ | ) × f t,q ( θ ) e − | k − k ′ | d R , (13)and I ( B ) t,q = n B V π ~ Z d k ′ (2 π ) δ ( E k − E k ′ )( q T F + | k − k ′ | ) f t,q ( θ ) , (14)where V = (2 πe /κ ) . Eqs. (13) and (14) can be rewrit-ten in dimensionless forms as I ( R ) t,q = (cid:18) mn R V π ~ k F (cid:19) Z dx g t,q ( x ) e − xa R √ − x ( x + s ) (15)and I ( B ) t,q = (cid:18) mn B V π ~ k F (cid:19) Z dx g t,q ( x ) √ − x ( x + s ) (16)where a R ≡ k F d R and s = q T F / k F , and g t ( x ) = 2 x , g q ( x ) = 1.To proceed further analytically, we now make the as-sumption that, by definition, the remote dopants are farenough that the condition a R = 2 k F d R ≫ d R ,this “remote” impurity condition breaks down at a lowenough carrier density such that n . / πd R . Thus, thedistinction between ‘far’ and ‘near’ impurities in our 2-impurity model starts disappearing at very low carrierdensity. Putting a R ≫ s = q T F / k F ≫ I ( R ) t,q I ( R ) t = (cid:18) mn R π ~ k F (cid:19) (cid:18) πe κ (cid:19) (cid:18) s (cid:19) a R , (17) I ( R ) q = (cid:18) mn R π ~ k F (cid:19) (cid:18) πe κ (cid:19) a R s , (18) I ( B ) t = (cid:18) mn B π ~ k F (cid:19) (cid:18) πe κ (cid:19) πs , (19) I ( B ) q = (cid:18) mn B π ~ k F (cid:19) (cid:18) πe κ (cid:19) πs , (20)with τ − t,q = I ( R ) t,q + I ( B ) t,q . Equations (17) – (20) provideus with approximate analytical expressions for the con-tributions I ( R ) t,q and I ( B ) t,q by the remote and the back-ground scattering respectively to the transport and quan-tum scattering rates. We note that in these asymptotic limits ( a R ≫ s ≫ I ( B ) t I ( R ) t = 3 πa R n B n R , (21)and I ( R ) q I ( B ) q = 12 πa R n R n B . (22)In addition, I ( R ) q /I ( R ) t = 3 a R / , (23) I ( B ) q /I ( B ) t = 1 . (24)We note, therefore, that I ( R ) q ≫ I ( R ) t (since a R ≫ I ( B ) t > I ( R ) t if3 πa R > n R /n B , whereas I ( R ) q > I ( B ) q if n R /n B > πa R .Consistency demands that 3 πa R ≫ πa R , i.e., a R ≫ / a R ≫ B -scatterers to dominate τ − t ,i.e., I ( B ) t ≫ I ( R ) t , and R -scatterers to dominate τ − q , i.e., I ( R ) q ≫ I ( B ) q if the following conditions are both satisfied3 πa R ≫ n R /n B , πa R ≪ n R /n B . (25)The two inequalities defined by Eq. (25) are not mutuallyexclusive if n R ≫ n B and n R / πn B ≫ a R ≫ ( n R / πn B ) / . (26)It is easy to see the Eq. (26) is consistent as long as n R ≫ . n B , a perfectly reasonable scenario! In fact,we expect this condition to be extremely well-satisfiedin high-quality 2D systems where n B is very small, but n R ≈ n due to modulation doping and charge neutrality.The above analytic considerations lead to the conclu-sion that it is possible for τ − t to be dominated by back-ground impurities, and at the same time for τ − q to bedominated by the remote impurities provided the neces-sary conditions a = 2 k F d R ≫ n R ≫ . n B areobtained. We emphasize that these are only necessary conditions, and not sufficient conditions. Whether the2-impurity model indeed leads to mobility (i.e., τ t ) andquality (i.e., τ q ) being controlled by physically distinctdisorder mechanisms in real 2D semiconductor structurescan only be definitely established through explicit nu-merical calculations of τ − t,q for specific disorder configura-tions, which we do in the next section of this article. Thegeneral analytical theory developed above also explicitlyshows that there can be no mobility/quality dichotomy ifthere is only one type of disorder mechanism operationalin the sample since both quality and mobility will thenbe controlled by exactly the same disorder parameters(unlike the situation discussed above).It may be useful to write a full analytical expressionfor τ − t and τ − q (in the a R ≫ n R , d R , and n B enter the expressions for mobilityand quality: τ − t = I ( R ) t + I ( B ) t = A ( R ) t n R /a R + A ( B ) t n B , (27) τ − q = I ( R ) q + I ( B ) q = A ( R ) q n R /a R + A ( B ) q n B , (28)where A ( R ) t = (cid:16) m π ~ (cid:17) (cid:18) πe κ (cid:19) (cid:18) q T F (cid:19) A ( B ) t = (cid:16) m π ~ (cid:17) (cid:18) πe κ (cid:19) (cid:18) πq T F (cid:19) , (29) A ( R ) q = (cid:16) m π ~ (cid:17) (cid:18) πe κ (cid:19) (cid:18) q T F (cid:19) A ( B ) q = (cid:16) m π ~ (cid:17) (cid:18) πe κ (cid:19) (cid:18) πq T F (cid:19) . (30)Equations (27) and (28) immediately lead to the approx-imate sufficient conditions for mobility and quality to becontrolled by background and remote impurities, respec-tively: n B ≫ n R /a R , i.e., n R ≪ n B a R , (31)and n R /a R ≫ n B , i.e., n R ≫ n B a R . (32)Since a R ≫ a R ≫ n R /n B ≫ a R , (33)with a R ≫
1. Equation (33) gives the sufficient conditionfor the existence of a mobility/quality dichotomy in 2Dsemiconductor structures. Since the unintentional back-ground charged impurity concentration is typically verylow in high-mobility 2D GaAs structures and since the re-mote charged dopant density is typically (at least) equalto the carrier density, the condition a R ≫ n R /n B ≫ a R ≫ x Ga − x Asstructure may have n ≈ n R ≈ × cm − ; d R ≈ n B ≈ cm − (corresponding roughly to a 3D bulkbackground charged impurity density of 3 × cm − fora 300 ˚A wide quantum well structure). These system pa-rameters satisfy the constraint defined by Eq. (33) with k F d R ≈
15, i.e., a R ≈ n R /n B ≈ k F d (and hence a R ), and even-tually at low enough carrier density, τ − t and τ − q aredetermined by the same disorder parameters! In concluding this theoretical section, let us considera concrete numerical example of the mobility/quality di-chotomy using two hypothetical samples (1 and 2) withthe following realistic sample parameters:Sample 1 : d (1) R = 500˚ A ; n (1) B = 10 cm − Sample 2 : d (1) R = 1000˚ A ; n (2) B = 10 cm − . (34)We assume the sample carrier density to be the samein both cases (so that “an apple-to-apple” comparisonin being made): n = 4 × cm − . For the purpose ofkeeping the number of parameters a minimum we assume n R = n = 4 × cm − also for both samples. Using theanalytical theory developed above (or by direct numericalcalculations), we find τ (1) t /τ (2) t ≈ τ (1) q /τ (2) q ≈ . (35)This means that sample 1 (with n (1) B < n (2) B ) has a fivetimes higher mobility than sample 2 whereas sample 2(with d (2) R > d (1) R ) has two times higher ‘quality’ thansample 1, i.e., sample 2 has a single-particle level broad-ening Γ (or “Dingle temperature”) which is half of that ofsample 1 although both samples have exactly the samecarrier density! This realistic example shows that it isgenerically possible in the 2-impurity model for µ > µ and Γ > Γ , with the conclusion that a higher mobilitydoes not necessarily ensure a higher quality. We em-phasize that (1) this would not be possible within the1-impurity model, and (2) this conclusion is density de-pendent – for much lower carrier density, where k F d ≫ τ t and τ q using the idealized 2-impurity model[i.e., Eq. (10)] and the strict 2D model for the semicon-ductor structure. These results presented in Figs. 1 – 4explicitly visually demonstrate that τ t and τ q cannot besingle-valued functions of each other as long as the under-lying disorder consists of (at least) two distinct scatteringmechanisms as operational within the 2-impurity model.The results shown in Figs. 1 – 4 also serve to establishthe validity of the analytical theory we provided abovein this section.In Fig. 1 we show the dependence of the calculated τ t and τ q on the individual scattering mechanism (i.e., thenear-impurity scattering strength defined by the back-ground impurity concentration n B or the far-impurityscattering strength defined by either n R or d R ) assumingthat the other mechanism is absent (i.e., just an effec-tive 1-impurity model applies). Results in Fig. 1 should -3 -2 -1 n (10 cm ) τ ( p s ) τ τ tq B -2 (a) -3 -2 -1 n (10 cm ) τ / τ t q B -2 (b) n (10 cm ) τ ( p s ) τ τ tq r 10 -2 (c) n (10 cm ) τ / τ t q r 10 -2 (d) d (nm) -2 τ ( p s ) τ τ t q R (e) d (nm) τ / τ t q r (f) FIG. 1. (a) The scattering times ( τ t and τ q ) and (b) the ratio( τ t /τ q ) as a function of the background impurity density n B for n = 3 × cm − and n R = 0. (c) The scattering timesand (d) the ratio ( τ t /τ q ) as a function of the remote impuritydensity n R for n = n R , n B = 0, and d R = 80nm. (e) Thescattering times and (f) the ratio ( τ t /τ q ) as a function of theimpurity location d r for n = n r = 3 × cm − and thebackground impurity density n B = 0. Here the δ -layer isconsidered (i.e., a = 0). be compared with the corresponding results in Figs. 2 –4 where both scattering mechanisms are operational toclearly see that τ t and τ q are manifestly not unique func-tions of each other by any means and a given value of τ t (or τ q ) could lead to distinct values of τ q (or τ t ) de-pending on the details of the disorder distribution. Thus,mobility ( τ t ) and quality ( τ q ) are not simply connected.In presenting the results for Figs. 1 – 4 we first notethat τ t,q = τ ( n, n B , n R , d R ) even within the simple 2-impurity model. Since the carrier density dependenceof τ is not the central subject matter of our interest inthe current work (and has been discussed elsewhere byus , we simplify the presentation by assuming n R = n inFigs. 1 – 4 which also assures a straightforward chargeneutrality. This, however, has important implicationssince the dependence on n R and n now become com- -3 -2 -1 n (10 cm ) τ ( p s ) τ τ t q B -2 (a) -3 -2 -1 n (10 cm ) τ / τ t q B -2 (b) n (10 cm ) τ ( p s ) τ τ tq r 10 -2 (c) n (10 cm ) τ / τ t q r 10 -2 (d) d (nm) -2 -1 τ ( p s ) τ τ t q R (e) d (nm) τ / τ t q (f) r FIG. 2. (a) The scattering times ( τ t and τ q ) and (b) the ratio( τ t /τ q ) as a function of the background impurity density n B for n = n R = 3 × cm − and d R = 80 nm. (c) Thescattering times and (d) the ratio ( τ t /τ q ) as a function of theremote impurity density n R for n = n R , n B = 10 cm − ,and d R = 80nm. (e) The scattering times and (f) the ratio( τ t /τ q ) as a function of the impurity location d R for n = n R =3 × cm − and the background impurity density n B = 0.Here the δ -layer is considered (i.e., a = 0). pounded rather than being independent. Thus the n R -dependence of τ shown in Figs. 1 – 4 is not the triv-ial τ t,q ∼ n − R behavior (as it is for the n B -dependence,where τ t,q ∼ n − B ) since the n R = n condition synergisti-cally combines both n R and n dependence. We mentionthat in ungated samples with fixed carrier density, thecondition n = n R is perfectly reasonable, and therefore,the results shown in Figs. 1 – 4 apply to modulation-doped samples with fixed carrier density n = n R .From Fig. 1, we immediately conclude the obvious:The functional relationship between τ t and τ q dependsentirely on which disorder parameter is being varied –in fact Figs. 1(a), (c), (e) give three completely distinctfunctional relationships between τ t and τ q depending onwhether n B , n R , or d R is being varied, respectively. Fromthe corresponding values of τ t /τ q , as shown in Figs. 1(b),(d), (f) respectively we clearly see that τ t and τ q varia- n (10 cm ) τ / τ t q r 10 -2 (b) d (nm) τ / τ t q r (d) n (10 cm ) τ ( p s ) τ τ tq r 10 -2 (a) d (nm) -2 -1 τ ( p s ) τ τ t q R (c) FIG. 3. (a) The scattering times and (b) the ratio ( τ t /τ q )as a function of the remote impurity density n R for n = n R , n B = 10 cm − , and d R = 80nm. (c) The scattering timesand (d) the ratio ( τ t /τ q ) as a function of the impurity location d R for n = n R = 3 × cm − and the background impuritydensity n B = 10 cm − . Here the δ -layer is considered (i.e., a = 0). tions with individual disorder parameters n B , n R , d R arevery different indeed. We point out an important qual-itative aspect of Fig. 1(c) which has not been explicitlydiscussed in the literature and which has important im-plications for the mobility/quality dichotomy. Here themobility (i.e., τ t ) increases with increasing n R = n , butthe quality (i.e., τ q ) decreases with increasing n = n R .This is a peculiar feature of long range Coulomb scatter-ing by remote dopants.Figs. 2 – 4 explicitly show how the 2-impurity modelcan very strongly modify the 1-impurity model func-tional dependence of τ t,q on the disorder parameters n B , n R (= n ), and d R . Clearly, depending on the specific2D samples, τ t and τ q could behave very differently asalready established in our analytical theoretical resultsgiven above. For example, in contrast to Fig. 1(a),where both τ t and τ q decrease monotonically (and triv-ially as n − B ) with increasing amount of unintentionalbackground impurity density n B , Fig. 2(a) shows that τ q is essentially a constant whereas τ t decreases with in-creasing n B , thus demonstrating a specific example ofhow the effective quality (i.e. τ q ) remains the same al-though the effective mobility decreases by more than anorder of magnitude due to the variation in the back-ground disorder. Similarly, in Fig. 1(c), increasing theremote dopant separation d R (keeping n = n R fixed, and n B = 0) increases both τ t and τ q monotonically (with τ t increasing as d in contrast to τ q increasing as d forlarge d ), but Fig. 2(e), 3(c), 4(c) show that, depending d (nm) -2 -1 τ ( p s ) τ τ t q R (c) d (nm) τ / τ t q r (d) n (10 cm ) -1 τ ( p s ) τ τ t q r 10 -2 (a) n (10 cm ) τ / τ t q r 10 -2 (b) FIG. 4. (a) The scattering times and (b) the ratio ( τ t /τ q )as a function of the remote impurity density n R for n = n r , n B = 10 cm − , and d R = 80nm. (c) The scattering timesand (d) the ratio ( τ t /τ q ) as a function of the impurity location d R for n = n r = 3 × cm − and the background impuritydensity n B = 10 cm − . Here the δ -layer is considered (i.e., a = 0). on the background impurity scattering strength, τ t basi-cally saturates for larger d since it is then dominated bythe unintentional background impurities rather than bythe remote dopants whereas τ q continues to be limitedby the remote dopants. This leads to an interesting non-monotonicity in τ t /τ q as a function of d R in Figs. 2 – 4 incontrast to Fig. 1(c) where τ t /τ q ∼ d R keeps on increas-ing forever in the absence of background scattering. Therealistic dependence of τ t,q on n R (with n = n R ) in thepresence of fixed n B and d R remains qualitatively similarin Figs. 1 – 4 although there could be large quantitativedifferences, indicating that increasing n R (= n ) wouldtypically by itself tend to enhance (suppress) τ t ( τ q ), butthe effect becomes whether for larger (smaller) values of n B ( d R ). This is an important result of our paper.The analytical and numerical results presented in thissection establish clearly that τ t and τ q can essentially beindependent functions of the disorder parameters in the2-impurity model, and thus, mobility and quality could,in principle, have little to do with each other in realistic2D semiconductor structures. We make this point evenmore explicit by carrying out calculations in experimen-tally realistic samples in the next section of this article.0 III. NUMERICAL RESULTS ANDDISCUSSIONS
We begin presenting our realistic numerical results for2D transport properties (both τ t or mobility and τ q orquality) without any reference to the analytical asymp-totic theoretical results presented in the last section byshowing in Fig. 5 the calculated T = 0 mobility ( µ ),transport scattering time ( τ t ) and the single-particle (orquantum) scattering time ( τ q ) for a 2D GaAs-AlGaAssample at a fixed carrier density ( n = 3 × cm − ) inthe presence of two types of disorder: a remote chargedimpurity sheet ( n i = 1 . × cm − ) placed at a dis-tance ‘ d ’ from the edge of the quantum well (i.e. from theGaAs-AlGaAs interface) and a background 3D chargedimpurity density ( n iw ) which is distributed uniformlythroughout the inside of the GaAs quantum well (whichhas a well thickness of 300 ˚A). Results presented in Fig. 5involve no approximation other than assuming a uniformrandom distribution of the quenched charged impurities(2D distribution with a fixed 2D impurity density n i forthe remote impurities placed at a distance d from thequantum well and 3D distribution with a variable 3Dimpurity density n iw for the unintentional backgroundimpurities inside quantum well) as we include the fullquantitative effect of the quasi-2D nature of the quan-tum well width in the calculation and calculate all theintegrals [Eqs. (1) and (2)] for τ − t and τ − q numericallyexactly. Of course, our basic theory is a leading-ordertheory in the impurity scattering strength which shouldbe an excellent approximation at the high carrier densityof interest in the current work where our focus is on verylow-disorder and high-quality 2D semiconductor systems.The use of the realistic 3D background impurity distribu-tion is easily reconciled with our minimal model in sec-tion II by using: n i = n R , d R = d + a/ n B ≈ n iw a ,where a (= 300 ˚A in Fig. 5) is the quantum well-width.Results in Fig. 5 are quite revealing of the physicsdiscussed already in section II. First, we clearly see inFig. 5(a) the trend that for large (small) d , the mobility isdetermined by the background (remote) impurities, andhence for d = 800 (100) ˚A, the mobility depends strongly(weakly) on the background impurity density (until itbecomes very large, leading to µ < cm /Vs whichis no longer a high-quality situation). In particular, for d = 800 ˚A, µ − ( ∝ τ − t ) ∝ n iw approximately, implyingthat τ − t is dominated almost entirely (for d = 800 ˚A) bythe background impurities in the quantum well. By con-trast, for d = 100 ˚A, µ is almost independent of n iw for n iw . cm − (corresponds to n B ∼ × cm − ),indicating that τ − t is dominated almost entirely by the“remote” dopant scattering. In Fig. 5(a) we focus on theinteresting d = 800 ˚A situation where the 2-impuritymodel might apply – obviously, for d = 100 ˚A, the re-mote dopants dominate both τ − t and τ − q rendering the2-impurity model inapplicable since k F d < d ’.In Fig. 5(b) we show as a function of background dis- n (10 cm ) -1 µ ( c m / V s ) iw
15 -3 d=100A400A800A (a) n (10 cm ) τ ( p s ) iw 15 -3 τ q τ t τ /τ t q (b) FIG. 5. (a) Calculated mobilities as a function of back-ground 3D charged impurity density n iw for fixed remote im-purity density, n i = 1 . × cm − , and electron density, n = 3 × cm − . The red dashed line indicates µ ∼ n iw .The numbers indicate the location of remote impurities whichis measured from the interface of quantum well. (b) The cal-culated scattering times and their ratio τ t /τ q as a functionof background impurity density using the same parameters of(a) and d = 800 ˚A. The dashed line represents the ratio of τ t to τ q . A quantum well with the thickness of a = 300 ˚A isused in this calculation. -1 n (10 cm ) τ ( p s ) τ /τ
10 -2 τ dq τ iq dqdt τ /τ it iq τ Γ= E F (a) n (10 cm ) µ ( c m / V s )
10 -2 (b) di µµ µ FIG. 6. (a) Calculated scattering times and (b) mobilities asa function of carrier density. Here τ iq , µ i ( τ dq , µ d ) indicate thesingle particle relaxation time and mobility due to interfaceimpurities at d = 0 (remote impurities at finite d = 800 ˚A),respectively, and τ = ( τ − iq + τ − dq ) − , µ = ( µ − i + µ − d ) − . TheGreen line indicates τ q = 0 . /n in units of ps with n mea-sured by 10 cm − . The crossing point between green line andblue line represents Γ = E F . The following parameters areused: n i ( d = 0) = 10 cm − , n d ( d = 800˚ A ) = 10 cm − andquantum well width a = 300 ˚A. At a density n = 10 cm − we have µ = 35 . × cm /V s and τ q = 25 . n c = 0 . × cm − . order the calculated τ q (as well as τ t ) for d = 800 ˚A, andit is clear that for 5 × cm − < n iw < × cm − (i.e., over a factor of 40 increase in the background disor-der!) τ q (i.e., quality) remains almost a constant whereas τ t (i.e., mobility) decreases approximately by a factor of40 in this regime. Combining Figs. 5(a) and (b), we thenconclude that there could be an infinite series of samples,where the mobility decreases from ∼ − × cm /Vsto below 10 cm /Vs as n iw increases from 5 × cm − to 2 × cm − , with all of them having essentially thesame quality as characterized by the quantum scatteringtime τ q ∼ ∼ ~ / τ q ∼ . -1 n (10 cm ) τ ( p s ) τ /τ
10 -2 τ dq τ iq dqdt τ /τ it iq τ Γ= E F (a) n (10 cm ) µ ( c m / V s )
10 -2 (b) d i µµµ FIG. 7. The same as Fig. 6 with following parameters: n i ( d = 0) = 3 × cm − , n d ( d = 500˚ A ) = 3 × cm − , andquantum well width a = 300 ˚A. At a density n = 10 cm − we have µ = 14 . × cm /V s and τ q = 42 ps. The criticaldensity n c = 0 . × cm − . which are completely realistic, clearly bring out the factthat, when the underlying disorder has a basic 2-impuritymodel structure (one type of impurity with k F d ≫ k F d ≪ that the fragile 5/2 fractional quan-tum Hall effect (FQHE), which is traditionally only stud-ied in the highest quality samples with µ > cm /Vs,can actually be observed in much lower mobility sampleswith µ ∼ cm /Vs since, under suitable circumstances(as in the results of Fig. 5), it is possible for samples withorders of magnitude different mobilities (i.e. values of τ t )to have more or less the same “quality” (i.e., the samevalue of τ q ).In Figs. 6 – 9, we make the above issue very clear byshowing realistic transport calculation results (for both τ t and τ q ) in various situations within the 2-impurity model.In each case, the high carrier density mobility is deter-mined by the background impurity scattering whereasthe quality, i.e., the quantum lifetime τ q (or equivalentlythe single-particle level broadening Γ ∼ τ − q ) is deter-mined by remote impurity scattering, creating a cleardichotomy where mobility and quality are disconnectedand the high-density mobility by itself does not providea unique characterization of the sample quality.To make the physical implication of the mobil-ity/quality dichotomy very explicit, we have shown ineach figure the carrier density where the Ioffe-Regel cri-terion for strong localization, Γ = E F , is satisfied in eachof these samples. (We mention that Figs. 6 – 9 shouldbe thought of as representing five distinct 2D sampleswith fixed bare disorder each as described in each fig-ure caption, but with variable 2D carrier density, as forexample, can be implemented experimentally using anexternal back gate.) This Γ = E F Ioffe-Regel pointshould be thought of as the critical density below (above)which the system behaves insulating (metallic) as has re-cently been discussed by us in details elsewhere . Such -1 n (10 cm ) τ ( p s ) τ /τ
10 -2 τ dq τ iq dqdt τ /τ it iq τ Γ= E F (a) n (10 cm ) µ ( c m / V s )
10 -2 (b) di µµµ FIG. 8. (a) The same as Fig. 6 with following parameters: n i ( d = 0) = 3 × cm − , n d ( d = 500˚ A ) = 5 × cm − , andquantum well width a = 300 ˚A. At a density n = 10 cm − we have µ = 13 . × cm /V s and τ q = 28 . n c = 0 . × cm − . an apparent disorder driven effective 2D metal-insulatortransition (2D MIT) has been extensively studied in theliterature , and is usually discussed in terms of the max-imum mobility of the sample at high carrier density.One can think of the Ioffe-Regel criterion induced criti-cal density n c to be an approximate quantitative measureof the “sample quality” with n c decreasing (increasing)as the quality improves. The corresponding approximatemeasure of the sample mobility has traditionally beenthe so-called “maximum mobility ( µ m )”, or equivalentlyfor high-mobility GaAs-based modulation-doped struc-tures, the measured mobility at the highest possible car-rier density (since for modulation-doped high-mobilitystructures, as can be seen in Figs. 6 – 9 and as hasbeen extensively experimentally observed over the last20 years, the sample mobility decreases with decreasingcarrier density and the typically quoted sample mobilityis always the one measured at the highest carrier den-sity). The n¨aive expectation is that higher (lower) themaximum mobility, lower (higher) should be the criti-cal density for 2D MIT since the sample quality shouldimprove with sample mobility.Using µ m to be the mobility at n = 10 cm − inFigs. 6 – 9, we conclude that the sample quality, if itis indeed determined entirely by the maximum mobility(or the mobility at a very high carrier density), shoulddecrease monotonically as we go from the sample of Fig. 6( µ m = 35 . × cm − /Vs) to that of Fig. 9 ( µ m = 11 . × cm − /Vs). We list below in Table I the calculatedcritical density for each sample in Figs. 6 – 9 noting alsothe mobility µ m at n = 10 cm − . Figure µ m (10 cm /Vs) n c (10 cm − )Fig. 6 35.7 0.24Fig. 7 14.1 0.16Fig. 8 13.3 0.22Fig. 9 11.7 0.33TABLE I. µ m is the mobility calculated at n = 10 cm − and n c represents the critical density calculated from Γ = E F . -1 n (10 cm ) τ ( p s ) τ /τ
10 -2 τ dq τ iq dqdt τ /τ it iq τ Γ =E F (a) n (10 cm ) µ ( c m / V s )
10 -2 (b) di µµ µ FIG. 9. (a) The same as Fig. 6 with following parameters: n i ( d = 0) = 3 × cm − , n d ( d = 500˚ A ) = 10 cm − , andquantum well width a = 300 ˚A. At a density n = 10 cm − we have µ = 11 . × cm /V s and τ q = 16 . n c = 0 . × cm − . We conclude from Table I that there is simply noone-to-one relationship between mobility and quality inthese numerical transport results based on the 2-impuritymodel. For example, although the “lowest mobility” sam-ple (Fig. 9, µ m = 11 . × cm /Vs) does indeed havethe “lowest quality” as reflected in the highest value of n c ( ∼ . × cm − ), the highest mobility sample (withalmost three times the mobility of all the other samples)has the second highest value of n c ( ∼ . × cm − )instead of having the lowest n c as it would if quality isdetermined exclusively by mobility. The other two in-termediate mobility samples with mobilities 14 . × cm − /Vs and 13 . × cm − /Vs also have their n c val-ues “reversed” (0 . × cm − and 0 . × cm − , re-spectively) compared with what they should be if the mo-bility really determined quality. We note that the sam-ples of Figs. 6 and 8 have almost identical quality (i.e.,essentially the same values of n c ) although the sample ofFig. 6 has almost three times the high-density mobilityas that of Fig. 8!We do mention that the values of n c ( ∼ × cm − )we obtain in our Figs. 6 – 9 are consistent with the ob-served 2D MIT critical density in ultra-high-mobility 2DGaAs structures where n c ∼ × cm − has been re-ported for µ m ∼ cm /Vs for n . cm − . The last set of numerical results we show for the 2-impurity model is based on HIGFET (heterojunction-insulator-gated-field-effect-transistor) structures (in con-trast to MODFET structures or modulation-doped-field-effect-transistors, which we have discussed so far in thispaper) and is motivated by the recent experimental workby Pan and his collaborators on the effect of disorder onthe observation, existence, and stability of the 5/2 FQHEin high-mobility GaAs-AlGaAs HIGFET structures .This work of Pan et al . is closely related to similarwork by Gamez and Muraki and by Samkharadze etal. who also studied disorder effects on the stability ofthe 5/2 FQHE in modulation doped GaAs-AlGaAs 2Dsystems. All three of these experimental studies con-clude, using different phenomenology and methodology,that the quality of the observed 5/2 FQHE in 2D systems n (10 cm ) µ ( c m / V s ) B 10 -2 FIG. 10. (a) The calculated mobility as a function of back-ground impurity density n B in a GaAs HIGFET structure.Here n R = 10 cm − , d R = 630 nm, and the carrier density n = 4 . × cm − are used. is not directly connected in an one-to-one manner withthe sample mobility, and it is possible to find robust 5/2FQHE in samples with mobility in the & cm /Vsrange whereas much of the earlier work had to useultra-high mobility ( > cm /Vs) for the observation ofstable 5/2 FQHE. This observation by these three exper-imental groups of the mobility/quality dichotomy is verysimilar to the theory being developed in the current workwith the only difference being that our work specificallyfocuses on the quality being associated with the single-particle quantum scattering rate τ − q or the collisionallevel-broadening Γ ∼ τ − q rather than the 5/2 FQHE gapsince we do not know of any quantitative microscopictheory which directly connects FQHE gap values withdisorder. We comment further on this feature below inour discussion after presenting our HIGFET numericalresults.A HIGFET system is different from modulation-dopedquantum well structures we considered so far in ourwork with the important qualitative difference being thatHIGFETs are undoped (except, of course, for uninten-tional background charged impurities as represented by n B , which are unavoidable in a semiconductor) with noremote modulation doping layer present in the system.Instead, the 2D carriers are induced in the GaAs surfacelayer at the AlGaAs-GaAs interface by a remote heav-ily doped gate placed very far from the GaAs-AlGaAsinterface. Thus, HIGFETs are basically the GaAs ver-sion of Si-MOSFETs (metal-oxide-semiconductor-field-effect transistors) with the insulator being the AlGaAslayer instead of SiO . An additional difference be-tween HIGFETs and modulation-doped quantum wellsis that the quasi-2D carrier confinement in the HIGFETis in an asymmetric triangular potential well (similar toMOSFETs ) in contrast to the symmetric square wellconfinement in the AlGaAs-GaAs quantum well system.Given that HIGFETs have no intentional modulation3 n (10 cm ) µ ( c m / V s )
10 -2 (a) n (10 cm ) Γ ( m e V )
10 -2 (b) n (10 cm ) τ ( p s ) τ
10 -2 t tRtot τ tB τ (c) n (10 cm ) -1 τ ( p s ) τ
10 -2 q (d) qR tot τ qB τ FIG. 11. The calculated (a) mobility, µ , (b) level broadening,Γ, (c) transport lifetime, τ t and (d) quantum lifetime, τ q , asa function of carrier density in a GaAs HIGFET structurewith parameters n R = 10 cm − , d R = 630 nm, and n B =1 . × cm − . doping, it may appear that the 2-impurity model is sim-ply inapplicable here since the background unintentionalcharged impurities seem to be the only possible typeof Coulomb disorder in the system so that the systemshould belong to a 1-impurity disorder description (i.e.,just the unintentional background random charged impu-rities). This is, however, incorrect because the presenceof the far-away gate, which induces the 2D carriers inthe HIGFET, introduces remote charged disorder (albeitat a very large value of d ) arising from the gate chargeswhich must be present due to the requirement of chargeneutrality. We, therefore, use exactly the same minimal2-impurity model for the HIGFETs that we have usedfor the modulation doped systems assuming n R to bethe charged impurity density on the far away gate at avery large distance d R away from the induced 2D electronlayer on the GaAs side of the GaAs-ALGaAs interface.(Later in this section we will present results for a realis-tic 3-impurity model in order to provide a quantitativecomparison with the HIGFET data of Pan et al . .)In Figs. 10 – 12 we show our full numerical results of an-GaAs HIGFET structure using the 2-impurity model.The specific HIGFET structure used for our numericalcalculations is motivated by the sample used in Ref. [16],but we do not attempt any quantitative comparison withthe experimental transport results, which necessitates a3-impurity model to be described later. At this stage,i.e., for Figs. 10 – 12, our goal is to establish the mobil-ity/quality dichotomy for HIGFET 2D systems based onour minimal 2-impurity model.In Fig. 10 we show the calculated mobility as a function n (10 cm ) µ ( c m / V s )
10 -2 R (a) n (10 cm ) -3 -2 -1 Γ ( m e V )
10 -2 (b) R FIG. 12. The calculated (a) mobility and (b) level broad-ening as a function of remote impurity density in a GaAsHIGFET structure with parameters n = 1 . × cm − , d R = 630 nm, and n B = 1 . × cm − . of n B , the background impurity density and in Fig. 11we show the mobility ( µ ), the level broadening (Γ), thetransport lifetime ( τ t ), and the quantum lifetime ( τ q ) as afunction of the 2D carrier density n in a GaAs HIGFETstructure using the 2-impurity model with n R = 10 cm − ; d R = 630 nm; n B = 1 . × cm − . We notethat µ = neτ t and Γ = ~ / τ q are simple measures of mo-bility and quality which are directly linearly connectedto τ t and τ − q . Our choice of d R = 630 nm is specifi-cally aimed at the sample of Ref. [16] where the gate islocated 630 nm away from the GaAs-AlGaAs interface.Our choice of n R = 10 cm − and n B = 1 . × cm − is arbitrary at this stage (and the precise choicehere is irrelevant with respect to our qualitative conclu-sions) except that this combination of a large (small) n R ( n B ) is the appropriate physical situation in high-qualityHIGFETs. Our choice of n R , n B , and d R (which we getfrom the actual experimental system) gives the correct2D “maximum” mobility of µ = 14 × cm /Vs at a2D carrier density of n = 4 . × cm − consistent withthe experimental sample in Ref. [16] as shown in Fig. 10.The calculated mobility in Fig. 10 decreases monoton-ically with increasing n B , and we choose n B = 1 . × cm − to get the correct maximum mobility of 14 × cm /Vs reported in Ref. [16] with the correspondingvalue of the level broadening being 0.638 meV at thesame density. We emphasize that the level broadening(or equivalently, τ q ) here is determined entirely by theremote scattering from the gate in spite of the gate beingan almost macroscopic distance ( ∼ . µm ) away from the2D electrons – changing n B by even a factor of 100 doesnot change the the value of Γ or τ − q (but does changemobility µ or τ − t by a factor of 100) whereas the mobil-ity is determined entirely by the background scattering(and therefore changing n R does not affect the mobility).In Fig. 11 we show the calculated µ , Γ, τ t , and τ q (remembering µ = neτ t and Γ = ~ / τ q ) as a func-tion of 2D carrier density n for fixed n R , d R , n B asshown. These results clearly show the mobility/qualitydichotomy operational within the 2-impurity model inthis particular HIGFET structure (for the chosen realis-4 n (10 cm ) µ ( c m / V s ) B 10 -2 FIG. 13. (a) The calculated mobility as a function of back-ground impurity density n B in a GaAs MODFET structurewith a well width a = 300˚A. Here n = n R = 1 . × cm − and d R = 2000 nm are used. tic disorder parameters n R , d R , n B ). For n & × cm − , the mobility is determined essentially by the back-ground impurity scattering (i.e. n B ) whereas the levelbroadening or the quantum scattering rate is determinedentirely by the remote scattering for the entire densityrange (10 cm − < n < cm − ) shown in Fig. 11. Atlow carrier density ( n . cm − ) k F d R ( .
10) is nolonger very large, and given the rather large value of n R (= 10 cm − ) corresponding to the remote gate charges,the scattering by n R starts affecting the mobility. But,the high density mobility (determined by n B ) and thequality at all density (determined by n R ) are still com-pletely independent quantities, and therefore it is possi-ble for the quality (e.g., the FQHE gap at high density)to be completely independent of the mobility as foundexperimentally in Ref. [16].This is demonstrated explicitly in Fig. 12 where weshow that the variation in the mobility is essentiallynon-existent for four orders of magnitude changes in n R whereas Γ changes essentially by four orders of magni-tude. Similarly, Fig. 11 indicates that (since τ − qB ∝ Γ ∝ n B ), a 2 orders of magnitude change in n B will hardlychange Γ, but µ will change by 2 orders of magnitude(due to a 2-orders of magnitude change in n B ) at highcarrier density.We believe that our Figs. 10 – 12 provide a com-plete explanation for the puzzling observation in Ref. [16]where a drop in the mobility of the sample at high carrierdensity hardly affected its quality as reflected in the mea-sured 5/2 FQHE energy gap. This is because the highcarrier density mobility is determined by background im-purity density n B which does not affect the quality at allwhereas the quality is affected by remote scattering whichdoes not much affect the mobility at high carrier density.For the sake of completeness, and to make connectionwith the interesting recent works of Refs. [12 and 17],who also independently conclude in agreement with Pan n (10 cm ) µ ( c m / V s )
10 -2 (a) n (10 cm ) -2 -1 Γ ( m e V )
10 -2 (b) n (10 cm ) τ ( p s ) τ
10 -2 t (c) tR tot τ tB τ n (10 cm ) τ ( p s ) τ
10 -2 q (d) qRtot τ qB τ FIG. 14. The calculated (a) mobility, µ , (b) level broadening,Γ, (c) transport lifetime, τ t and (d) quantum lifetime, τ q , asa function of carrier density in a GaAs MODFET structurewith a well width a = 300 ˚A. Here the parameters n R = n , d R = 200 nm, and n B = 6 . × cm − are used. et al . that very high mobility ( > cm /Vs) is notnecessarily required for the experimental observation ofa robust 5/2 FQHE in standard modulation-doped GaAsquantum wells (in contrast to Pan’s usage of undopedHIGFETs), we show in Figs. 13 – 15 ( which correspondto the HIGFET results shown in Figs. 10 – 12 respec-tively) our calculated transport results for a modulation-doped quantum well structure with a high-density mo-bility identical (i.e., 14 × cm /Vs) to the HIGFETstructure considered in Figs. 10 – 12.The main differences between the 2D systems forFigs. 10 – 12 (HIGFET) and Figs. 13 – 15 (MODFET)are the following: (1) the HIGFET has a triangular quasi-2D confinement potential (determined self-consistentlyby the carrier density) and the MODFET has a square-well confinement imposed by the MBE-grown AlGaAs-GaAs-AlGaAs structure with a given confinement width( a = 30 nm in Figs. 13 – 15); (2) the 2D carriers are in-duced by a very far away gate in the HIGFET whereas itis induced by the remote dopants (we choose n R = n inFigs. 13 – 15) in the modulation doping layer (we choose d R = 200 nm in Figs. 13 – 15); (3) the specific necessaryvalues of n B are somewhat different in the two systems inorder to produce the same high-density maximum mobil-ity. The quantitative differences described in items (1)- (3) above are sufficient to produce substantial differ-ences between the numerical results in the HIGFET andthe MODFET system as can easily be seen by comparingthe results of Figs. 10 – 12 with those of Figs. 13 – 15,respectively, although we ensured that both have exactlythe same high-density mobility ( µ m = 1 . × cm /Vs).5 n (10 cm ) µ ( c m / V s )
10 -2 (a) R n (10 cm ) -2 -1 Γ ( m e V )
10 -2 (b) R FIG. 15. The calculated (a) mobility and (b) level broaden-ing as a function of remote impurity density in a GaAs MOD-FET structure with a well width a = 300 ˚A. The parameters n = 1 . × cm − , d R = 200 nm, and n B = 6 . × cm − are used. However, qualitatively the two sets of results shown inFigs. 10 – 12 and 13 – 15 are similar in that the mobility(quality) at high carrier density ( > cm − ) is invari-ably determined by the background (remote) scatteringrespectively, leading to the possibility that a substantialchange in mobility (quality) by changing n B ( n R ) respec-tively may not at all affect quality (mobility), and thusit is possible at high carrier density for a system to havea modest mobility ( ∼ cm /Vs) by having a large n B with little adverse effect on quality (i.e., Γ). Thus, theexperimental observations in Refs. [12, 16, and 17] are allconsistent with our theoretical results.Finally, we show in Figs. 16 and 17 the numerical trans-port results for the HIGFET structure (of Figs. 10 –12) using a more realistic 3-impurity model going be-yond the 2-impurity model mostly used in our currentwork. The 3-impurity model is necessary for obtainingagreement between experiment and theory since exper-imentally the measured mobility, µ ( n ), as a function ofcarrier density manifests non-monotonicity with a max-imum in the mobility around n ∼ × cm − . Sucha non-monotonicity, where µ increases (decreases) withincreasing n at low (high) carrier density, is common inSi-MOSFETs , and is known to arise from short-range in-terface scattering which becomes stronger with increasingcarrier density as the self-consistent confinement of the2D carriers becomes stronger and narrower pushing theelectrons close to the interface and thus increasing theshort-range interface roughness scattering as well as thealloy disorder scattering in AlGaAs as the confining wavefunction tail of the 2D electrons on the GaAs side pushesinto the Al x Ga − x As side of the barrier. We include thisrealistic short-range scattering effect, which becomes im-portant at higher carrier density leading to a decrease ofthe mobility at high density (as can be seen in Fig. 16(a)).Importantly, however, this higher-density suppression ofmobility (by a factor of 3 in Fig. 16(a) consistent withthe observation of Pan et al . ) has absolutely no effecton the quality (see Fig. 16(b)) with the level broaden-ing Γ decreasing monotonically with increasing carrierdensity (since Γ is determined essentially entirely by the n (10 cm ) µ ( c m / V s )
11 -2 (a) n (10 cm ) Γ ( m e V )
10 -2 (b) n (10 cm ) τ ( p s ) τ
10 -2 t (c) ts tR τ tRtot τ tB τ n (10 cm ) -1 τ ( p s ) τ
10 -2 q (d) τ qsqR tot τ qB τ FIG. 16. The calculated (a) mobility and (b) level broad-ening with long range remote impurity at d R , short rangeimpurities at the interface, and background short range im-purities. We assume that the density dependence of scatter-ing time with interface short range impurities is τ − qs ∝ n α .In (a) the blue line is calculated with n B = 0 . × cm − and τ − qs = 5 × n . The green line is calculated with n B =0 . × cm − and τ − qs = 1 . × n . . The black line iscalculated with n B = 1 . × cm − and τ − qs = 0 . × n .The red dots are experimental data from Pan et al. In (c)and (d) we show the total scattering times as well as the in-dividual scattering time corresponding to the each scatteringsource. Here n B = 0 . × cm − and τ − qs = 5 × n areused. remote scattering – see Fig. 16(d)). Thus, we see anapparent paradoxical situation (compare Figs. 16(a) and(b)) where the mobility decreases at higher carrier den-sity, but the quality keeps on improving with increasingcarrier density! This is precisely the phenomenon ob-served by Pan et al . who found that, although the mo-bility itself decreased in their sample by a factor of 3 athigher density, the sample quality, as measured by the5/2 FQHE gap, improved with increasing density pre-cisely as we predict in our work. In Fig. 17 we show thatthe 3-impurity model, except for allowing the mobilityto decrease at high carrier density due to the increas-ing dominance of short-range scattering (thus bringingexperiment and theory into agreement at high densityin contrast to the 2-impurity results), has no effect onthe basic quality/mobility dichotomy being discussed inthis work – for example, Fig. 17 shows that while qualitydecreases (i.e., Γ increases) monotonically with increas-ing remote scattering, nothing basically happens to themobility!Before concluding this section, we provide a criticaland quantitative theoretical discussion of two distinct ex-periments (one from 1993 and the other from 2011 ),6separated by almost 20 years in time, involving high-mobility 2D semiconductor structures in the context ofthe mobility versus quality question being addressed inthe current work. Our reason for focusing on these twopapers is because both report τ t and τ q for the sam-ples used in these experimental studies, thus enabling usto apply our theoretical analyses quantitatively to thesesamples.In ref. [21], two GaAs-AlGaAs heterojunctionswere used (samples A and B) with the followingcharacteristics : Sample A: n = 1 . × cm − ; µ = 6 . × cm /Vs; τ t = 270 ps; τ q = 9 ps, and SampleB: n = 2 . × cm − ; µ = 12 × cm /Vs; τ t = 480ps; τ q = 4 . d R = 80 nm for the remote dopants,but we should consider d A > d B &
80 nm since sampleA has a lower carrier density and therefore the quasi-2Dlayer thickness for sample A must be slightly higher sincethe self-consistent confinement potential must be weakerin A than in B due to its lower density. We note thatsample A and B indeed manifest the mobility/quality di-chotomy in that A (B) has higher (lower) quality (i.e. τ q ), but lower (higher) mobility!We start by assuming the absence of any backgroundimpurity scattering ( n B = 0), then the asymptotic for-mula for k F d R ≫ τ t ∼ ( k F d R ) /n R ; τ q ∼ ( k F d R ) /n R . Making the usualassumption n R = n , since no independent informa-tion is available for n R , we conclude that the theorypredicts τ Bt /τ At = p n B /n A ( d B /d A ) ≈ . d B ≈ d A , and τ Bq /τ Aq = p n A /n B d B /d A ≈ . d B ≈ d A . Experimentally, A and B samples satisfy: τ Bt /τ At ≈ . τ Bq /τ Aq = 0 .
5. Thus, just the considera-tion of only remote dopant scattering which must alwaysbe present is all modulation-doped samples already givessemi-quantitative agreement between theory and experi-ment including an explanation of the apparent paradoxi-cal finding that the sample B with higher mobility has alower quality! The key here is that the higher density ofsample B leads to a higher mobility, but also leads to ahigher values of τ − q (and hence lower quality) by virtueof higher carrier density necessitating a higher value of n R leading to a lower value of τ q [see, for example, Fig. 1(c)where increasing n = n R leads to increasing (decreasing) τ t ( τ q )].We can actually get essentially precise agreement be-tween theory and experiment for the dichotomy in sam-ples A and B of Ref. [21], with τ t higher (lower) in sampleB (A) and τ q higher (lower) in sample A (B) by incor-porating the fact that a higher (by a factor of 2) car-rier density in sample B compared with sample A makes d A > d B due to self-consistent confinement effect in het-erostructures and hence the theoretical ratios of τ A and τ B change from the values given above to τ Bt /τ At < . τ Bq /τ Aq ≈ . < . τ Bq /τ Aq of samples A and B can be un-derstood quantitatively on the basis of remote scatter-ing (which determines the quality almost exclusively in n (10 cm ) µ ( c m / V s )
10 -2 (a) R n (10 cm ) -3 -2 -1 Γ ( m e V )
10 -2 (b) R FIG. 17. The calculated (a) mobility and (b) level broaden-ing as a function of the remote impurity density. The sameparameters as Fig. 16(a) are used. high-mobility modulation-doped structures), the mobil-ity ratio τ Bt /τ At is not determined exclusively by remotedopant scattering. Inclusion of somewhat stronger back-ground disorder scattering in sample A compared withsample B immediately gives τ Bt /τ At = 1 . a = 56 nm; d R = 320 nm; n = 8 . × cm − ; µ = 12 × cm /Vs; Γ = 0 .
24 K, and Sample B: a = 30nm; d R = 78 nm; n = 2 . × cm − ; µ = 11 × cm /Vs; Γ = 2 .
04 K. Thus, in this case , although thetwo samples have almost identical mobilities, the lower-density sample A has almost 8 times higher quality withΓ B / Γ A = τ Aq /τ Bq ≈
8. We note that the lower qual-ity sample has three times the carrier density, and goingback to our Figs. 1 – 4, we see that a higher carrier den-sity n (= n R ) always leads to higher mobility and lowerquality since the quality (i.e., τ q ) is determined mostlyby long-range remote scattering whereas the mobility isdetermined by a combination of both remote and back-ground scattering with the background scattering oftendominating the mobility. The fact that d AR ≫ d BR con-siderably improves the quality of sample A with respectto sample B without much affecting the mobility sincethe quality (mobility) is limited by remote (background)scattering.Using the asymptotic formula (for k F d ≫ τ q ∝ k F d/n R and n R ≈ n , we conclude for the compara-tive quality of the two samples: τ Aq /τ Bq = Γ B / Γ A ≈ p n B /n A d A /d B ≈ d A = 348 nm and d B = 93 nm by taking into account their differences inboth the set back distances and the well thickness. Ex-perimentally, Γ B / Γ A ≈ . τ q (or equivalently Γ ∝ τ − q ) as a mea-sure of the quality because it is well-defined and theo-retically calculable. Experimentally, the quality can bedefined in a number of alternative ways as , for exam-ple, done in the recent experiments where the 5/2FQHE gap is used as a measure of the quality. There is noprecise microscopic theory for calculating disorder effectson the FQHE gap, but there are strong indications that the FQHE gap ∆ Γ in the presence of finite disorderscales approximately as∆ Γ ≈ ∆ − Γ , (36)where Γ is indeed the quantum level broadening we usein our current work as the measure of quality and ∆ isthe FQHE gap in the absence of any disorder. If this iseven approximately true (as it seems to be on empiricalgrounds), then our current theoretical work shows com-plete consistency with the recent experimental resultsconcerning the dichotomy between mobility and FQHEgap values in the presence of disorder. In this context,it may be worthwhile to emphasize an often overlookedfact: the mobility itself (i.e., τ − t and not τ − q ) can beconverted into an energy scale by writing (for GaAs)Γ µ = ~ τ t ≈ (10 − / ˜ µ ) meV ≈ ( . / ˜ µ ) K , (37)where ˜ µ = µ/ (10 cm /V s ). Thus, a mobility of10 cm /Vs corresponds only to a broadening of 10mK which is miniscule compared with the theoreticallycalculated − cm /Vs corresponds to a mobility broadeningof only 100 mK, which is much less than the expected5/2 FQHE gap. Thus, the quality of the 5/2 FQHE can-not possibly be determined directly by the mobility value(unless the mobility is well below 10 cm /Vs) and theremust be some other factor controlling the quality, whichwe take to be the quantum level broadening in this work.It must be emphasized here that the mobility/qualitydichotomy obviously arises from the underlying disor-der in high-mobility semiconductor structures being long-ranged. If both mobility and quality are dominated byshort-range disorder, then τ q ≈ τ t , and a mobility of 10 cm /Vs with Γ ≈
100 mK will be a high-quality sample!In concluding this section, we should mention that thevery first experimental work we know of where the mobil-ity/quality dichotomy was demonstrated and noted ex-plicitly in the context of FQHE physics is a paper bySajoto et al. from the Princeton group which appearedin print an astonishing 24 years ago! In this work, (see the “Note added in proof” in Ref. [25]), it was specificallystated that the samples used by Sajoto et al. manifestedas strong FQHE states as those observed in other samplesfrom other groups with roughly 5 −
10 times the mobil-ity of the Sajoto et al. samples, thus providing a clearand remarkable early example of the mobility/quality di-chotomy much discussed during the last couple of yearsin the experimental 2D literature. We note that the sam-ples used by Sajoto et al. had unusually large set-backdistances ( d R ∼
270 nm), leading to rather small val-ues of τ − q and Γ corresponding to our theory althoughthe mobility itself, being limited by background impurityscattering (i.e. by n B ), was rather poor ( ∼ cm /Vs).We believe that the reason the samples of Sajoto et al.had such high quality in spite of having rather modestmobility is the mobility/quality dichotomy studied in ourwork where the mobility determined by background scat-tering is disconnected from the quality determined by theremote dopant scattering. IV. SUMMARY AND CONCLUSION
In summary, we have theoretically discussed the impor-tant issue of mobility versus quality in high-mobility 2Dsemiconductor systems such as modulation-doped GaAs-AlGaAs quantum wells and GaAs undoped HIGFETstructures. We have established, both analytically (sec-tion II) and numerically (section III), that modulation-doped (or gated) 2D systems should generically manifesta mobility/quality dichotomy, as often observed experi-mentally, due to the simple fact that mobility and qual-ity are often determined by different underlying disordermechanisms in 2D semiconductor structures – in particu-lar, we show definitively that in many typical situations,the mobility (quality) is controlled by near (far) quenchedcharged impurities, particularly at higher carrier densityand higher mobility samples. We show that often the2D mobility (or equivalently, the 2D transport scatter-ing time) is controlled by the unintentional backgroundcharged impurities in the 2D layer whereas the quality,which we have parameterized throughout our work bythe quantum single-particle scattering time (or equiva-lently, the quantum level broadening), is controlled bythe remote charged impurities in the modulation dopinglayer whose presence is necessary for inducing carriers inthe 2D layer. Somewhat surprisingly, we show that thesame mobility/quality dichotomy could actually apply toundoped HIGFET structures where the charges on thefar-away gate play the role of remote scattering mecha-nism. Quite unexpectedly, we show that a very far awaygate (located almost 10 − cm away from the 2D layer)can still completely dominate the quantum level broad-ening, while at the same time having no effect on themobility. We develop a minimal 2-impurity model (nearand far or background and remote) which is sufficient toexplain all the observed experimental features of the mo-bility/quality dichotomy. The key physical point here is8that the dimensionless parameter ‘ k F d ’, where k F ∝ √ n is the Fermi wave number of the 2D electron system and‘ d ’ is the distance of the relevant charged impurities fromthe 2D system completely controls the mobility/qualitydichotomy. Impurities with k F d ≫ ≪
1) could to-tally dominate quality (mobility) without affecting theother property at all. We give several examples of situ-ations where identical or very similar sample mobilitiesat high carrier density could lead to very different sam-ple qualities (i.e., quantum level broadening differing bylarge factors) and vice versa. The mobility/quality di-chotomy in our minimal 2-impurity model arises fromthe exponential suppression of the large angle scatteringby remote charged impurities which leads to the interest-ing situation that remote scattering contributes little tothe resistivity, but a lot to the level broadening throughthe accumulation of substantial small angle scattering.We emphasize that the mobility/quality dichotomy arisesentirely from the long-range nature of the underlying dis-order, and would disappear completely if the dominantdisorder in the system is short-ranged.It is important to realize that τ t (mobility) and τ q (quality) both depend not only on the disorderstrength, but also on the carrier density, i.e., τ t,q ≡ τ t,q ( n, n R , d R , n B ). Thus, even within the 2-impuritymodel (parameterized by disorder parameters n R , d R , n B ), τ t,q are both functions of carrier density. For verylow carrier density, the dimensionless parameter k F d maybe small for all relevant impurities in the system, andeventually our 2-impurity model will then fail since atsuch a low carrier density, all impurities are essentiallynear impurities with the distinction between R-impuritiesand B-impurities being merely a semantic distinctionwith no real difference. Mobility and quality at suchlow densities then will behave similarly. The same situa-tion may also apply as a matter of principle at very highcarrier densities (i.e., very large k F ) where all impuritiesmay satisfy k F d ≫ n ∼ − × cm − ),and (2) the very low and high density regimes where the2-impurity model is no longer operational are completelyout of the experimentally relevant density range of in-terest in high-mobility 2D semiconductor structures forthe physics (e.g., FQHE) studied in this context. As-suming a high-mobility modulation doped GaAs quan-tum well of thickness 200 −
400 ˚A and a set-back dis-tance of 600 − × cm − . n . × cm − , which is the ap-plicable experimental regime of interest. Thus, the ap-plicability of the 2-impurity model for considering themobility/quality dichotomy is not a serious issue of con- cern.A second concern could be the validity (or not) ofour theoretical approximation scheme for calculating τ t,q ,where we have used the zero-temperature Boltzmann the-ory and the leading-order Born approximation for ob-taining the scattering rates. The T = 0 approxima-tion is excellent as long as T ≪ T F = E F /k B , whichis valid in all systems of interest in this context. Forhigh-mobility 2D semiconductor structures of interest inthe current work, where the issue of the dichotomy ofmobility/quality is relevant (since in low-mobility sam-ples, typically τ t ≈ τ q ), the leading order theory (inthe disorder strength) employed in our approximationscheme should, however, be excellent since the conditions n ≫ n B and n ≫ n R e − k F d R are both satisfied makingBorn approximation essentially an exact theory in thismanifestly very weak disorder situation (consistent withthe high carrier mobility under consideration). An equiv-alent way of asserting the validity of Born approximationin our theory is to note that the conditions E F ≫ Γ and k F l ≫ l being the quantum level broadeningand the transport mean free path respectively). A re-lated issue, which is theoretically somewhat untractable,is the possible effect of impurity correlation effects onthe mobility versus quality question in 2D semiconductorstructures. It is straightforward to include impurity cor-relation effects among the dopant ions in our transporttheory, but unfortunately no sample-dependent experi-mental information is available on impurity correlationsfor carrying out meaningful theoretical calculations. Wehave carried out some representative numerical calcula-tions assuming model inter-impurity correlations amongthe remote dopants, finding that such correlations en-hance both τ t and τ q , as expected (with τ q being en-hanced more than τ t in general), compared with the com-pletely random impurity configuration results presentedin the current article, but our qualitative conclusionsabout the mobility/quality dichotomy remain unaffectedsince the fact that τ t and τ q are controlled respectivelyby background and remote scattering in high-mobilitymodulation-doped structures continues to apply in thepresence of impurity correlation effects. We thereforebelieve that our current theory involving Born approxi-mation assuming weak leading order disorder scatteringfrom random uncorrelated quenched charged impuritiesin the environment (both near and far) is valid in theparameter regime of our interest.Finally, we comment on the possibility of future ex-perimental work to directly verify (or falsify) our the-ory. Throughout this paper, we have, of course, madeextensive contact with the existing experimental resultswhich, in fact, have motivated our current theoreticalwork on the mobility versus quality dichotomy. For adirect future experimental test of the theory, it will benecessary to produce a large number of high-mobility 2Dsemiconductor structures with different fixed carrier den-sities and with varying values of the remote dopant set-9back distance, and then measure the values of τ t,q ina large set of samples which are all characterized bytheir high-density mobility. The measurement of thetransport relaxation time τ t is simple since it is directlyconnected to the carrier mobility µ (or conductivity σ ): τ t = mσ/ne = mµ/e . The measurement of the single-particle relaxation time (or the quantum scattering time) τ q is, however, not necessarily trivial although its theo-retical definition is very simple. In particular, the Dingletemperature or equivalently the Dingle level broadeningΓ D obtained from the measured temperature dependenceof the amplitude of the 2D SdH oscillations may not nec-essarily give the zero-field quantum scattering time τ q defining the sample quality in our theoretical considera-tions (i.e. Γ D = ~ / τ q may not necessarily apply to the2D SdH measurements) because of complications arisingfrom the quantum Hall effect and inherent spatial densityinhomogeneities (associated with MBE growth) in the 2Dsample. Since our theory is explicitly a zero-magneticfield theory, it is more appropriate to obtain τ q simplyby carefully monitoring low-field magneto-resistance os-cillations finding the minimum magnetic field B wherethe oscillations disappear. The corresponding cyclotronenergy ω = eB /mc then defines the single particlelevel-broadening Γ ∼ ~ ω , providing τ q = 1 / ω . Anadvantage of this method of obtaining τ q is that one isnecessarily restricted to the low magnetic field regime inhigh-mobility systems (i.e., E F ≫ ~ ω ), where our the-ory should be applicable. A much stronger advantagebe applicable. A much stronger advantage of using thisproposed definition (i.e., the disappearance of magneto-resistance oscillations at the lowest experimental temper-ature) for the experimental determination of τ q is thatthis is much easier to implement in the laboratory thanthe full measurement of the Dingle temperature whichrequires accurate measurements of the temperature de-pendent SdH amplitude oscillations. We therefore sug- gest low-temperature measurements of µ and ω to obtain τ t and τ q respectively in a large number of modulationdoped samples with varying n , n R , d R , and n B in orderto carry out a quantitative test of our theory. A largesystematic data base of both τ t and τ q in many differ-ent samples should manifest poor correlations betweenthese two scattering times (i.e., the mobility/quality di-chotomy) provided the samples are high-mobility samplesdominated by long-range charged impurity disorder. Asemphasized (and as is well-known) throughout this work,if one type of disorder completely dominates both τ t and τ q , then they will obviously be correlated, but this shouldbe more an exception than the rule in high-mobilitymodulation-doped 2D semiconductor structures, whereboth near (“ n B ”) and far (“ n R ”) impurities should, ingeneral, play important roles in manifesting the mobil-ity/quality dichotomy.One important open question is whether the mobil-ity/quality dichotomy we establish in the current workcan be extended to other definitions of quality beyondour definition of quality in terms of the single-particlescattering rate or quantum level-broadening. The advan-tage of using the quantum scattering rate as the measureof sample quality is that this definition is generic, uni-versal, and simple to calculate (and to measure). 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