Giant Hall effect in the ballistic transport of two-dimensional electrons
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Giant Hall effect in the ballistic transport of two-dimensional electrons
Yu. O. Alekseev and A. P. Dmitriev Lyceum ”Physical-Technical High School”, Hlopina 8-3A, St. Petersburg, 194021, Russia Ioffe Institute, Politekhnicheskaya 26, 194021, St. Petersburg, Russia
We have studied magnetotransport of a degenerate two-dimensional electron gas in a Hall samplein the Knudsen regime, when the mean free paths of electrons with respect to their collisions witheach other and with impurities are much larger than the width of the sample. In contrast to theusually considered symmetric sample, whose both its edges reflect electrons diffusely, we consideredan asymmetric sample, one edge of which reflects them diffusely, while the other specularly. It isshown that in such structure in low magnetic fields the Hall coefficient is parametrically large incomparison with its standard value. Also the situation is discussed when all types of scattering canbe neglected except for scattering at the edges of the sample.
INTRODUCTION
In recent years, in connection with the impressiveprogress in the creation of two-dimensional systems witha record mobility of carriers, interest in the theoreticalstudy of the effect of interparticle interaction on trans-port phenomena has sharply increased [1-11], the roleof which in dirty systems with low mobility is insignif-icant. At the same time, research is being conductedin two directions. On the one hand, the hydrodynamicregime of electron transport is being intensively studied,which is realized, apparently, in experiments on gianttemperature-dependent magnetoresistance in ultrapuresemiconductor and graphene samples [12-24]. On theother hand, ballistic and intermediate between ballisticand hydrodynamic regimes are eximined extensively.This, second, direction of research is, first of all, of con-siderable theoretical interest, since the conditions for therealization of the hydrodynamic regime in a degenerateFermi gas differ from those in the case of an ordinary,non-degenerate gas and liquid. The effects caused by ex-ternal fields are often more pronounced in the ballisticregime than in the case of the local equilibrium hydrody-namic regime. Finally, in most experiments in small mag-netic fields, it is precisely the ballistic transport regimethat is realized, since the mean free path relative to in-terparticle collisions turns out to be on the order of, oreven larger than the characteristic spatial scales of theflow. As the magnetic field increases, the cyclotron ra-dius begins to play the role of the path length, and theconditions for the applicability of the hydrodynamic de-scription are satisfied.In papers [3] and [11], the Knudsen regime of currentflow in a long narrow two-dimensional sample with dif-fusely scattering boundaries was considered, where themean free path relative to interparticle collisions is muchlarger than the sample width. The electron gas was con-sidered to be degenerate. The limit of arbitrarily weakelectric and magnetic fields was studied, and the mag-netoresistance and Hall coefficient R H were found. Itturned out, in particular, that R H in this limit is half the value usual for Ohmic transport.Bearing in mind that, in experiments, the propertiesof the edges of the sample may differ, in this work westudied magnetotransport in ballistic regime in an asym-metric sample, one of the edges of which is smooth, i.e.reflects electrons specularly, while the other scatters themdiffusely. It is shown that in this case the Hall coeffi-cient is anomalously large as compared to its standardvalue. Finally, at a semi-quantitative level, the situationis discussed when all types of scattering can be neglectedexcept for scattering at the edges of the sample. Notethat the anomalously large value of the Hall coefficient inthe structure studied by us is, apparently, among the so-called ballistic anomalies in magnetotransport, discussedin the scientific literature in 1980 - 1990s (see, for exam-ple, a review [25]). PROBLEM STATEMENT AND BASICEQUATIONS
We will study the electrical transport of a degeneratetwo-dimensional electron gas in a long narrow samplewith a width W to which a time-independent uniformlongitudinal electric field E and a magnetic field B per-pendicular to the sample plane are applied (see Fig. 1).The sample will be assumed to be sufficiently clean, andthe temperature sufficiently low, so that the electron-phonon scattering can be neglected, and the mean freepath relative to the scattering of electrons by each otherand by impurities is much larger than its width. Oneedge of the sample is considered smooth, reflecting elec-trons ”specularly”, while the other is rough, scatteringthem diffusely. We direct the axis x along the field E and align it with the smooth edge of the sample, directthe axis y into the sample, and the magnetic field alongthe axis z (see Fig. 1). Bearing in mind to calculate thelinear response of the system to electric and magneticfields, we will consider them arbitrarily weak.We write the Boltzmann kinetic equation for the one-particle distribution function f ( r , v ) in the form v ∂f∂ r − e E ∂f∂ p + ω c ∂f∂ϕ = St ee [ f ] + St imp [ f ] , (1)where v and p = m v are the speed and momentum of theelectron, e > E is theelectric field equal to the sum of longitudinal and Hall E H fields, ϕ is the angle of the velocity vector measured fromthe ordinate axis, ω c = eB/mc is the cyclotron frequency, St imp [ f ] = − ( f − f ) /τ imp is the integral of collisionswith impurities, which we will assume to be short-range, f is the symmetric part of the distribution function,and St ee [ f ] is the integral of electron-electron collisions,as which we will use the model collision integral (see, forexample, [26] and [3]), which in the simplest way takesinto account the conservation of the number of particles,energy and momentum in electron-electron collisions: St ee [ f ] = − τ ee ( f − P [ f ]) , (2)where P is the operator of projecting the function ofthe angular variable ϕ onto the zero and first harmonics.Representing the distribution function in the form f = f F + ∂f F ∂ε g, (3)where f F is the equilibrium Fermi function, ε is the elec-tron energy, taking into account the smallness of the per-turbation, the degeneracy of the electron gas, and theindependence of the distribution function from x due tothe homogeneity of the system along the ordinate axis,from (1) we obtain:sin ϕ ∂g∂y − eE cos ϕ − eE H sin ϕ + 1 R c ∂g∂ϕ = − γ ee ( g − g − g s − g c ) − γ imp ( g − g ) , (4)where R c = v F /ω c is the cyclotron radius, γ ee =1 /l ee , γ imp = 1 /l imp , l ee/imp = v F τ ee/imp , g is thesymmetric part of the function g , g c is the projection g onto the cosine, and g s - onto the sine. g c ( y, ϕ ) = cos ϕπ π Z f ( y, ϕ ′ ) cos ϕ ′ dϕ ′ ,g s ( y, ϕ ) = sin ϕπ π Z f ( y, ϕ ′ ) sin ϕ ′ dϕ ′ , The function g ( y ) is responsible for the change in theconcentration of electrons at a given point, and throughthe functions g c and g s the densities of the longitudinaland transverse currents are expressed, respectively. Theboundary conditions for the function are written as: f + (0 , ϕ ) = f − (0 , − ϕ ) , ≤ ϕ ≤ π, (5) f − ( W ) = C − = 12 π Z f + ( W ) sin ϕdϕ, (6) FIG. 1: Ballistic sample with a rough and a specular edgesin external electric and magnetic fields. which means specular reflection from the bottom edge(Fig. 1) and diffuse - from the top [27]. Obviously, thefunction g ( y, ϕ ) also satisfies similar conditions. From (5)and (6) it can be seen, in particular, that the transversecurrent at the edges of the sample is zero, and due to thecontinuity equation it is zero everywhere in the sample,whence it follows that in the system under consideration g s = 0 . TRANSPORT IN THE ABSENCE OF AMAGNETIC FIELD
In the absence of a magnetic field, the contribution tothe symmetric part of the function g ( y, ϕ ) , which is lin-ear in perturbation, is also equal to zero, i.e. g = 0 . Thisfollows from the fact that for B = 0, the distribution ofelectrons across the sample cannot depend on the direc-tion of the electric field E applied to the sample (thisis no longer the case B = 0 because of the appearanceof the Lorentz force). Therefore, at B = 0 equation (4)takes the formsin ϕ ∂g ∂y − eE cos ϕ = − γ ee ( g − g c ) − γ imp g . (7)Finally, we will simplify it even further by omitting thefunction g c , which will be justified below.The solution of the resulting equation that satisfiesboundary conditions (5) and (6) has the form g ± ( y, ϕ ) = eE cos ϕγ " − exp − γ y ± W sin ϕ ! ,γ = γ ee + γ imp . (8)Note that g − ( W, ϕ ) = 0. For the function g c ( y, ϕ ) = π π R g ( y, ϕ ) cos ϕdϕ from this expression we obtain g c ( ϕ ) ≈ eE Wπ ln (cid:16) γW (cid:17) cos ϕ. (9)Bearing in mind the inequality γ | y ± W | ≪
1, from (8)and (9) it is easy to see what g c is greater g for all ϕ , ex-cept for narrow regions around the directions ϕ = 0 and ϕ = π , where, on the contrary, g ≫ g c . In this regard,it may seem that the rejection g c in (7) was unjustified,however, firstly, it is these narrow regions that made themain contribution to (9) and, secondly, outside these re-gions, the entire right-hand side of (7) is small in param-eter γW and can be omitted. The correction h ( y, ϕ ) tofunction (8), caused by taking into account g c in equation(7), can be found by the perturbation method, writing g ( y, ϕ ) in the form g = g + h and substituting in (7) as g c expression (9). From the resulting equation, we find, h ± ( y, ϕ ) = 4 eE W cos ϕπ " − exp − γ y ± W sin ϕ ! , which, due to the inequality γW ≪ I = en πp F W Z dy π Z g ( y, ϕ ) cos ϕdϕ and resistivity ρ , from (8) we obtain I ≈ e n E W πp F ln (cid:16) γW (cid:17) , ρ ≈ πp F e n W ln (cid:16) γW (cid:17) . (10)In the expression for the current density, we neglected theterms that depend on y and do not contain a large loga-rithm ln[1 / ( γW )]. Note that expression (10) for ρ is halfthat obtained in [3], which is not surprising, since therewas considered a problem with two diffusely reflectingedges.Result (10) has a simple physical meaning. In thesystem under consideration, the electron gas momentumcan relax either upon collisions of electrons with a dif-fusely scattering edge of the sample, or upon their col-lisions with impurities. Due to the condition γW ≪ W/v F ≪ τ ee,imp andtheir contribution to the conductivity is relatively small.The main contribution to the conductivity is made byelectrons moving at small angles to the axis x .If l imp ≪ l ee , then the relaxation length of the momen-tum of such electrons is of the order of l imp or less, and they make a proportional contribution Z γ imp W dϕ/ϕ ∼ ln γ imp W ! to the conductivity. In the opposite limiting case l ee ≪ l imp , an electron moving at a small angle, hav-ing passed a length of the order of l ee or less, is scatteredat a rough edge or collides with another electron, afterwhich it becomes typical and after a short time of theorder W/v f is diffusely scattered at the rough edge ofthe sample. The corresponding contribution to the con-ductivity is proportional ln[1 / ( γ ee W )].In the general case, formula (10) is obtained. It is alsoclear from the last reasoning why, at l imp ≫ l ee , the out-flow processes described by the integral of interparticlecollisions St ee [ f ] play the main role in the formation ofthe current, while the role of the incoming processes as-sociated with the function g c ( y, ϕ )is small.In sufficiently narrow samples, a different situation ispossible. Due to the uncertainty principle, the minimumtransverse momentum of electrons is of the order ~ /W ,so that the maximum time of motion without scatteringbefore collision with the edge is of the order mW / ~ , andthe minimum angle between the electron velocity vectorand the ordinate axis is of the order of the diffraction an-gle ϕ ∼ ~ /W p F ∼ λ F /W . As a result, if the inequality issatisfied 1 /k F W ≫ γW , i.e. W ≪ √ lλ F , where l = 1 /γ ,we again get formulas (10), which will enter ln ( k F W )instead ln [1 / ( γW )]. HALL EFFECT
In this section of the article, we will find the Hall co-efficient for our system in an arbitrarily weak magneticfield. For this, it is necessary to take into account in thekinetic equation (4) the terms with the Lorentz force andthe Hall electric field. The function g ( y ) is now nonzero,since the action of the Lorentz force leads to a redistri-bution of electrons across the sample. Let us write it inthe form g + g , where g is the correction caused bythe magnetic field, which will be considered arbitrarilysmall and taken into account as a disturbance.We are in-terested in the contribution to g , linear in the magneticfield; therefore, in the term R − c ∂g/∂ϕ in equation (4),we can use function (8) as a function g . In addition,since the influence of the magnetic field on the currentappears only in the second order in B , we will not take itinto account by setting g c = 0. Then from (4) we obtainthe equationsin ϕ ∂g ∂y − sin ϕeE H + γ ( g − g ) = − R c ∂g ∂ϕ , or sin ϕ ∂ ˜ g ∂y + γ (˜ g − ˜ g ) = − R c ∂ ˜ g ∂ϕ , (11)where the function ˜ g = g + e Φ is introduced, Φ is thepotential of the electric field.The procedure for solving this equation is completelysimilar to the procedure for solving equation (7): havingomitted the function ˜ g ( y ) in (11), we obtain the follow-ing expressions:˜ g +1 ≈ eE R c ( sin ϕγ − exp − γ y + W sin ϕ ! × " sin ϕγ γW sin ϕ ! − ! + y − Wγ − cos ϕ ϕ (cid:0) y + 2 yW − W (cid:1) − πγ and ˜ g − ≈ eE R c ( sin ϕγ − exp − γ y − W sin ϕ ! × " sin ϕγ + y − Wγ −− cos ϕ ϕ (cid:0) y − W (cid:1) − πγ . (12)Then we show that the correction arising from the ac-count ˜ g ( y ) is parametrically small (see Appendix A).Substituting these expressions in2 π ˜ g ( y ) = Z π g ( y, ϕ ) dϕ and keeping only the main contribution, we find˜ g ≈ − eE πR c " W γ ( y + W ) − γ . (13)The function ˜ g /e is the electrochemical potential Ψ( y ),its minus derivative is equal to the Hall field, and thedifference in values at the edges of the sample is measuredwith a voltmeter and is equal to the Hall voltage, U H =Ψ( W ) − Ψ(0). Therefore, we have E H ≈ − E W πR c γ ( y + W ) , U H ≈ − E πR C γ . (14)From here and from (10) for the Hall coefficient we obtain R H = U H BI = 316 γ W ln(1 /γW ) 1 en c ≫ en c . (15)Near the points ϕ = 0 and ϕ = π functions (12) havesingularities of the form 1 /ϕ and 1 / ( ϕ − π ) , making the main contribution (13) to the function ˜ g ( y ), the diver-gences arising in this case are ”cut off” by exponentialfactors exp[ − γ ( y ± W ) / sin ϕ ]. Note that in the case of asymmetric structure, which was studied in papers [3] and[11], the main contributions to ˜ g ( y ) from functions g +1 and g − cancel each other, and the Hall coefficient turnsout to be equal 1 / en c . In narrow samples, W ≪ √ lλ F ,the divergences are ”cut off” due to the principle of un-certainty at angles | ϕ | and | ϕ − π | order 1 /W k F , and asa result, the following expression is obtained for the Hallcoefficient R H = A ( k F W ) ln( k F W ) · en c , (16)where A is a numerical coefficient of the order of unity,which remains undefined within the framework of ourconsideration.In conclusion of this section, we will explain the anoma-lously large value of the Hall coefficient in the asymmet-ric structure we studied. To do this, let’s approach theproblem from a slightly different point of view. We rep-resent the distribution function in the form f ( ε, r , ϕ ) = f ( ε, r ) + ˜ f ( ε, r , ϕ ) , R π ˜ f dϕ = 0, multiply equation (1)by p y and integrate over 2 d p / (2 π ~ ) . It will turn out ∂ε∂y + enE y + ∂ Π yx ∂x + ∂ Π yy ∂y − mω c nV x + mnτ imp V y , (17)where the notation n ( r ) = Z f d p (2 π ~ ) , ε = Z f ε d p (2 π ~ ) , V ( r ) = 1 n ( r ) Z ˜ f v d p (2 π ~ ) , Π ik ≡ m Z ˜ f v i v k d p (2 π ~ ) (18)is introduced. The equation ∂ε∂x + enE x + ∂ Π xx ∂x + ∂ Π xy ∂y + mω c nV y + mnτ imp V x (19)is obtained similarly. Equations (17) and (19) are exact.It is convenient to write them in the form enE x + ∂ε∂x + ∇ · Π x + mω c nV y + mnτ imp V x = 0 ,enE y + ∂ε∂y + ∇ · Π y − mω c nV x + mnτ imp V y = 0 , Π i = Z ˜ f v i v d p (2 π ~ ) . (20)In equilibrium, the electric field and average velocity V ( r ) are equal to zero; therefore, in the linear approxi-mation, these equations take the form en E x + ∂ε∂x + ∇ · Π x + mω c n V y + mn τ imp V x = 0 ,en E y + ∂ε∂y + ∇ · Π y − mω c n V x + mn τ imp V y = 0 . (21)Let us introduce the electric Φ and electrochemical po-tentials Ψ = Φ − ε/en and rewrite the equations (21)through them en ∂ Ψ ∂x − n eBc V y + mn τ imp V x = ∇ · Π x ,en ∂ Ψ ∂y + n eBc V x + mn τ imp V y = ∇ · Π y . (22)The function Ψ = Φ − ε/en is the same as the functionΨ = ˜ g /e = Φ + g /e we introduced above. Really: ε ≡ Z f νεdε = Z ∂f F ∂ε g νεdε = − νε F g = − n g , (23)where ν = m/π ~ is the density of states.In the left-hand sides of equations (22), there are forcesacting on a unit volume of the electron gas in x and y out of directions, and on the right, divergence of themomentum flow in the same directions. We are interestedin the second equation. Let us rewrite it, taking intoaccount that in our problem there is no dependence onthe coordinate x and the average velocity along the axis y is zero en ∂ Ψ ∂y + n eBc V x = ∂ Π yy ∂y , and integrate across the sample. It turns out: U H = 1 en (cid:2) Π yy ( W ) − Π yy (0) (cid:3) + BIen c . (24)Substituting now the function (12) multiplied ∂f f /∂ε into the definition Π yy and performing the integration,we again obtain expression (14) for the Hall voltage (notethat the contribution to the integral of the symmetricpart of the function ˜ g is zero). It can be seen from theforegoing that the Hall voltage arising in our asymmetricsample is primarily intended to compensate for the gradi-ent of the transverse-pulse flux density across the sample,and not the Lorentz force, as is usually the case in sys-tems with an ohmic current flow. A nonzero gradient ofthe momentum flux density in the transverse directionalso arises in a symmetric structure, but it is of the sameorder of magnitude as the Lorentz force and leads to ahalving of the Hall coefficient compared to its standardvalue. CONCLUSION
In this work, we studied the Knudsen regime of a de-generate electron gas flow in a Hall sample, one edgeof which reflects electrons diffusely, and the other spec-ularly. The collisions of electrons with each other andtheir collisions with impurities were taken into account. It is shown that, in such an asymmetric sample, the Hallcoefficient is parametrically large in comparison with itsstandard value R H = 1 /en c : R H /R H ∼ l /W ln( l/W ), where l is the effective momentum relaxation length l ≫ W . In addition, the situation is discussed at a semi-quantitative level when all types of scattering can be ne-glected except for scattering at the edges of the sample.Finally, we note that, when deriving formula (15) forthe Hall resistance, the role of “skipping orbits” near therough edge was not thoroughly analyzed, which can leadto a change in the numerical factor in the expression forthe Hall resistance [28].We thank P. S. Alekseev for valuable discusions. Yu.A.also thanks to his teachers M. E. Kompan, N. M. Khimin,M. G. Ivanov. This work was supported by the RussianScience Foundation (grant No. 17-12-01182-c). APPENDIX A
Having performed the calculations described in thetext, we obtain for corrections to functions (12) h +1 ( y, ϕ ) = γ sin ϕ " y Z exp − γ y − y ′ sin ϕ ! ˜ g ( y ′ ) dy ′ + W Z exp − γ y + y ′ sin ϕ ! ˜ g ( y ′ ) dy ′ ,h +1 ( y, ϕ ) = γ sin ϕ y Z W exp − γ y − y ′ sin ϕ ! ˜ g dy ′ , (25)where ˜ g = ˜ g ( y ′ ) is the symmetric part of function (12).Having integrated it over and over, we obtain for thecorrection to the Hall coefficient δR H ≈ − πγW · en c , which is parametrically less than expression (15). [1] M. Hruska and B. Spivak, Phys. Rev. B 65, 033315(2002).[2] P. S. Alekseev, Phys. Rev. Lett. 117, 166601 (2016).[3] P. S. Alekseev and M. A. Semina, Phys. Rev. B 98,165412 (2018).[4] L. Levitov and G. Falkovich, Nat. Phys. 12, 672 (2016).[5] H. Guo, E. Ilseven, G. Falkovich, and L. Levitov, Proc.Natl. Acad. Sci. USA 114, 3068 (2017).[6] P. S. Alekseev, Phys. Rev. B 98, 165440 (2018).[7] T. Scaffidi, N. Nandi, B. Schmidt, A. P. Mackenzie, andJ. E. Moore, Phys. Rev. Lett. 118, 226601 (2017).[8] E. I. Kiselev and J. Schmalian, Phys. Rev. B 99, 035430(2019).[1] M. Hruska and B. Spivak, Phys. Rev. B 65, 033315(2002).[2] P. S. Alekseev, Phys. Rev. Lett. 117, 166601 (2016).[3] P. S. Alekseev and M. A. Semina, Phys. Rev. B 98,165412 (2018).[4] L. Levitov and G. Falkovich, Nat. Phys. 12, 672 (2016).[5] H. Guo, E. Ilseven, G. Falkovich, and L. Levitov, Proc.Natl. Acad. Sci. USA 114, 3068 (2017).[6] P. S. Alekseev, Phys. Rev. B 98, 165440 (2018).[7] T. Scaffidi, N. Nandi, B. Schmidt, A. P. Mackenzie, andJ. E. Moore, Phys. Rev. Lett. 118, 226601 (2017).[8] E. I. Kiselev and J. Schmalian, Phys. Rev. B 99, 035430(2019).