Negative density of states: screening, Einstein relation, and negative diffusion
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Negative density of states: screening, Einstein relation, andnegative diffusion.
A. L. Efros ∗ Department of Physics, University of Utah, Salt Lake City UT, 84112 USA
Abstract
In strongly interacting electron systems with low density and at low temperature the thermo-dynamic density of states is negative. It creates difficulties with understanding of the Einsteinrelation between conductivity and diffusion coefficient. Using the expression for electrochemicalpotential that takes into account the long range part of the Coulomb interaction it is shown that atnegative density of states Einstein relation gives a negative sign of the diffusion coefficient D , butunder this condition there is no thermodynamic limitation on the sign of D . It happens because theunipolar relaxation of inhomogeneous electron density is not described by the diffusion equation.The relaxation goes much faster due to electric forces caused by electron density and by neutral-izing background. Diffusion coefficient is irrelevant in this case and it is not necessarily positivebecause process of diffusion does not contribute to the positive production of entropy. In the caseof bipolar diffusion negative D results in a global absolute instability that leads to formation ofneutral excitons. Graphene is considered as an example of a system, where the density relaxationis expected to be due to electric force rather than diffusion. It may also have a negative density ofstates. PACS numbers: 71.27. +a,73.50.-h . INTRODUCTION The idea of the Einstein relation was put forward by Einstein and Smoluchowski in1905-1906. Both scientists considered the Brownian motion in the presence of gravitationalforce. The result is the relation between mobility u in the field and diffusion coefficient D .In case of electric field and particles with the charge e it has a form eD = T u, (1)where T is the temperature in energy units. The main idea was equivalence of an exter-nal force and the density gradient. Of course, both Einstein and Smoluchowski did notcare about negligible mutual gravitational or any other small interactions of the Brownianparticles.The formulation of the Einstein relation for electrons is based upon electrochemical po-tential, the thermodynamic function that, like temperature and pressure, should be the sameat all points of the system in the equilibrium state. The usual arguments are as follows. If anexternal potential ψ is applied to the system, the condition of thermodynamic equilibriumreads E ec = µ ( n ) + eψ = Const, (2)where µ ( n ) is the chemical potential as a function of inhomogeneous electron density n .In the equilibrium both n and ψ are function of coordinates while E ec is constant. Thetemperature T should also be constant. Therefore, the electrical current density j at constant T can be written in a form j = − σe ∇ E ec = σ E − D ∇ en, (3)where σ is conductivity and E = −∇ ψ . Then one gets relation connecting σ and Dσe dµdn = D, (4)which is also called Einstein relation. For the Boltzman gas dµ/dn = T /n and one getsEq.(1) if σ = enu . It looks like derivation of Eq. (4) is independent of the properties of thesystem and this equation can be consider as general thermodynamic law.A simple observation shows however that in the case of non-ideal electron gas the Einsteinrelation needs some comments. We discuss an electron gas on the positive background at2ow temperatures and low densities when dimensionless parameter r s is not very small. Here r s = 3 / (4 πna B ) for 3-d case and r s = 1 /πn a B , where n and n are 3- and 2-dimensionalelectron densities respectively and a B = ~ κ/me is the Bohr radius, m is an effectiveelectronic mass, κ is an effective permittivity.The problems of dynamic screening and diffusion in slightly non-ideal electron gas ( r s <<
1) with electron-electron interaction were considered in details about 20 years ago (SeeRef.[ ])In this case the thermodynamic density of states is large and positive. I concentratehere on the strongly non-ideal case r s ≥ E of the order of − e n /d N/κ , where d = 2 , n is thedensity per area or volume respectively, N is total number of electrons. Then µ ∼ − e n /d /κ and E, µ , and dµ/dn are negative . The first experimental confirmation of this idea wasdone by Kravchenko et al , but direct quantitative study of this effect was performed byEisenstein et al .The derivative dµ/dn is proportional to the reciprocal compressibility of the electrongas. Note that compressibility has to be positive due to the thermodynamical condition ofstability. However, this principle cannot be applied to the charged systems, like electron gas,because part of their energy is outside the system in a form of the energy of electric field.On the other hand, in the case of a neutral electron-hole plasma, the situation of negativecompressibility can arise leading to collapse of the system. Such a situation is considered atthe end of Sec. III.It follows from Eq. 4 that if dµ/dn is negative, diffusion coefficient D and conductivity σ have opposite signs. This observation needs an explanation because near the thermodynamicequilibrium both of them have to be positive to provide positive entropy production due tothe Joule heat and due to the relaxation of inhomogeneous density. II. ELECTROCHEMICAL POTENTIAL AND STATIC SCREENING
To resolve this contradiction one should include the long-range part of the Coulombpotential created by inhomogeneous electron gas into the function E ec in Eq. (2). Thiscontribution is a functional of n ( r ).To find E ec taking into account electron-electron interaction one should minimize the3elmholtz energy F with respect to electron density n ( r ) at a given value of T and N . Forlow T one gets F = e κ Z Z n ′ ( r ) n ′ ( r ′ ) d rd r ′ | r − r ′ | + Z f ( n + n ′ ) d r + Z en ′ ( r ) ψd r − E ec Z n ′ ( r ) d r, (5)where f is the Helmholtz energy density of a homogeneous electron system that results fromthe interaction in a neutral system, like the Wigner crystal or ”Wigner liquid”. Since thisinteraction comes mainly from the nearest neighbors and n ( r ) is a smooth function, onemay assume that both f and chemical potential µ = df /dn are local functions of n ( r ) Weassume also that n ( r ) = n + n ′ ( r ), where n is average density and n ′ ≪ n .Minimization of this expression with respect to n ′ gives the equation E ec = µ ( n ) + eψ + dµdn n ′ + e κ Z n ′ ( r ′ ) d r ′ | r − r ′ | . (6)It differs from Eq. (2) by the potential of electrons in the right hand side. Note thatthis potential is due to the violation of neutrality in a scale much larger than the averagedistance between electrons. To check this equation we consider thermodynamic equilibriumand find equations for the Thomas-Fermi static screening in 3- and 2-dimensional cases.Since E ec is independent of r in thermodynamic equilibrium one may take E ec − µ ( n ) as areference point for the total potential ϕ defined as ϕ = ψ + eκ Z n ′ ( r ′ ) d r ′ | r − r ′ | . (7)It follows from Eq. (6) that eϕ = − dµdn n ′ . (8)The Poisson equation has a form ∇ ϕ = − π ( en ′ − ρ ext ) κ , (9)where ρ ext is density of external charge. Using Eq. (8) one gets final equation for the 3-dlinear screening ∇ ϕ = − q ϕ − πρ ext κ . (10)Here q = 4 πe κ dndµ (11)4s the reciprocal 3-dimensional screening radius.Consider now a thin layer (x-y plane) with 2d electron gas separating two media withdielectric constants κ and κ . In this case one should substitute n ⇒ n δ ( z ) and κ ⇒ ¯ κ =( κ + κ ) /
2. The results is ∇ ϕ = − q ϕδ ( z ) − πρ ext ¯ κ , (12)where q = 2 πe ¯ κ dn dµ . (13)It is important that Eqs. (10), (12) are applicable only if the screening is linear ( n ′ ≪ n ) . There is another serious problem of applicability the Thomas-Fermi approximationin the case of the negative density of states. Indeed, the dielectric permittivity in thisapproximation has a form ǫ ( q ) = κ (1 − | q | q ) (14)in 3-d case and ǫ ( q ) = κ (1 − | q | q ) (15)in 2-d case. In both cases it has roots at q = | q | , | q | . The expression for the screenedpotential ϕ has a form ϕ ( r ) = Z ϕ ( q ) exp( i q · r ) d q ǫ ( q ) , (16)where ϕ is a bare potential. Thus, the roots of ǫ transform into the first order poles withoutany reasonable way of the detour. Such a detour follows from the casuality for the ω -planebut not for the q-plane. Moreover, the electrostatic potential should be real and one cannotadd a small imaginary part in the denominator. Therefore I think that the poles do nothave any physical sense.The reason is that negative sign of the density of states appears when q , q are of the orderof average distance between electrons ¯ r . At such distances the very concept of macroscopicfield does not have sense. However, if the bare potential has only harmonics with q ≪| q | , | q | , the Eqs.(10,12) have a sense. Consider, for example, the screening of the positivecharge Z at a distance z from the plane with 2-d gas(plane z = 0. The solution of Eq.(12)has a form ϕ ( ρ ) = Z ∞ Z exp( − qz ) κ ( q + q ) qJ ( qρ ) dq, (17)5here ρ is a polar radius in the plane z = 0. Suppose that | q | z ≫
1. Now the contri-bution to integral Eq.(17) from q ≃ | q | is exponentially small and one can ignore q in thedenominator. Then ϕ ( ρ ) = Zz κq ( z + ρ ) / . (18)Note that at q < negative potential in the plane withelectrons. That is what I call ”overscreening”.Extra electron density, as calculated from Eq. (8) is en ′ = − Zz π ( z + ρ ) / (19)It is negative and independent of the sign of q . One can see that the total charge Z ∞ en ′ πρdρ = − Z (20)Due to geometry of the problem electric field is zero below the plane with electrons. Asfollows from Eq. (8), the signs of charge density and potential are opposite if the density ofstates is negative.For the case of two such planes (double quantum well structure) Luryi has predicteda small penetration of electric field through the first plane. He has considered the case ofpositive density of states. Then the small penetrating field between two planes has the samedirection as the incident field.Eisenstein at al. studied this effect experimentally and found out that at negative densityof states the propagating field is opposite to the incident field and this is also a result of theoverscreening (see the quantitative theory in Ref. ).Negative density of states was also used for the explanation of magnetocapacitance databy Smith at al. . III. CONDUCTIVITY VERSUS DIFFUSION
Now I come back to the problem of the negative diffusion. If the system is not inequilibrium the electric current can be written in the same form as Eq. (3) j = − σe ∇ E ec . (21)6sing Eq. (6) one gets j = σ E − D ∇ en ′ − σ eκ ∇ Z n ′ ( r ′ ) d r ′ | r − r ′ | . (22)Here D is connected to σ by the Einstein relation Eq. (4). Considering relaxation of thecharge density one can ignore external field E . The relaxation is described by the continuityequation ∂ ( en ) ∂t = −∇ · j (23)or ∂ ( en ) ∂t = σ (cid:18) e dµdn ∇ ( en ′ ) − πen ′ κ (cid:19) . (24)The ratio R of the first (diffusion) term in the right hand side to the second (field) term is R = ( q L ) − , where L − = ∇ n ′ /n ′ is the characteristic size of the extra charge and q isgiven by Eq. (11). If electron gas is non-ideal, q ∼ / ¯ r , where ¯ r is the average distancebetween electrons. However, the very concept of diffusion equation is valid at L ≫ ¯ r . Thismeans that for the non-ideal gas | R | ≪ n ′ ( r , t ) = n ′ ( r ,
0) exp − ( t/τ M ) , (25)where τ M = κ/ (4 πσ ) is well-known Maxwell’s time. Coefficient D does not enter in thiscase in the entropy production and it does not have a physical sense. Thus in 3-dimensionalnon-ideal electron gas negative dµ/dn does not create any contradiction with the Einsteinrelation.In the 3d gas of high density µ ∼ n / and R ∼ (¯ r/L ) /r s with r s <
1. In this case R might be large and diffusion is possible. However dµ/dn >
0, and
D > ∂ ( en ) ∂t = σ ( 1 e dµdn ∇ ( en ′ ) − e ¯ κ ∇ Z n ′ ( r ′ ) d r ′ | r − r ′ | ) . (26)Here n , σ and ∇ are 2-dimensional density, conductivity, and 2-dimensional gradient re-spectively. To consider the ratio R of the first (diffusion) term to the second (field) term it7s convenient to make the Fourier transformation. Then one gets ∂ ( n q ) ∂t = − σ ( 1 e dµdn q n q + 2 πq ¯ κ n q ) , (27)where n q is the Fourier transformation of n ′ .Now we find that the ratio of the first ( diffusion) term in the right hand side of Eq.(27) to the second (field) term R = q/q , where q is given by Eq. (13). Similar to the3d case in the non-ideal gas | q | ∼ / ¯ r and diffusion should be ignored. Then we get theDyakonov-Furman equation ∂ ( n q ) ∂t = − vqn q , (28)where velocity v = 2 πσ / ¯ κ . The physical meaning of this equation is that extra density ofelectrons localized initially at some spot propagates in all directions with velocity v conserv-ing the total amount of extra electrons. Of course, this way of relaxation is more efficientthan diffusion (random walk), because r ∼ vt while r ∼ √ Dt in the case of diffusion. Thus,diffusion coefficient D is irrelevant and negative dµ/dn does not create any contradictionwith the Einstein relation In a high density electron gas R = q ¯ r/r s and diffusion mechanismis possible. In this case dµ/dn > D > D appears in the term with the highest derivative that leadsto the absolute instability even if D is small . Consider, for example Eq. (24) for 3d case.After the Fourier transformation the solution for the charge density ρ = en ′ can be writtenin a form ρ q = ρ q exp (cid:18) − πσtκ − Dq t (cid:19) , (29)where D is given by the Einstein relation Eq. (4). One can see that at D < q ¯ r ≥ r . So they contain information that the charged liquid has a discreet electronicstructure. This information comes from the negative density of states which originatesfrom the interaction of the separate electrons. That is why macroscopic equations becomeunstable at small spacial harmonics. The message is that n ( r ) is rather a set of δ -functionsthan a continuous function. The instability is absent if D is positive.The instability of small spatial harmonics at small negative D does not affect largerharmonics because Eqs.(24,26) are linear. Due to the linearity different harmonics are in-8ependent and transformation of energy from small spacial harmonics to large harmonics isforbidden (cp. phenomenon of turbulence in non-linear hydrodynamics where the transfor-mation of energy is not forbidden, but the instability is initiated by large harmonics).Therefore, I think that at small D approximation D = 0 that gives Eqs.(25,28) is correct.One should note that the problem of the non-physical roots of electric permittivity dis-cussed in the previous section is of the same nature.Before we discussed the unipolar diffusion. Consider the simplest case of the ambipolardiffusion assuming that at t = 0 the densities of electrons and holes are equal in some finiteregion of space and are zero otherwise. Moreover we assume that the local macroscopiccharge density ρ ( r , t ) = 0 and a recombination of carriers is very slow. In this case Eq. (6)describes the electron-hole system in quasi-equilibrium. At large r s one gets E, µ, dµ/dn < r s , but at small density ( r s ≥
1) coefficient
D <
0. Thenthe absolute instability takes place for all harmonics that means a collapse of the system.Thus the electron-hole ”Wigner liquid” and crystal are unstable.This result is very transparent. It happens because negative µ just means that the energyof the system decreases with increasing density. In bipolar case neutrality is provided bythe particles and we do not consider any background. Thus the instability is a result of thenegative compressibility in a neutral system. At large enough r s these particles are classical,and the absence of the mechanical equilibrium follows also from the Earnshaw theorem. Inreality quantum mechanics becomes more important with increasing density. As a resultthe excitons are formed. These neutral particles have a positive diffusion coefficient D a andtheir density smears with time through all available space. This process is described by aregular diffusion equation. In the case of optical excitation the carriers may appear in theform of the excitons from the very beginningFor the coefficient of the ambipolar diffusion D a a textbook equation D a = 2 D e D h D e + D h (30)is often used, where D e,h are diffusion coefficients of electron and holes in unipolar case.As follows from the previous discussion, one should be careful with this equation becausefor the non-ideal electron (or hole) gas these unipolar coefficients might be negative andmeaningless. It happens because in unipolar case there is a deviation from neutrality that9reates electric field, while in bipolar case the system is neutral. In this case Eq. (30) doesnot work and one should calculate D a in a different way as a diffusion of the exciton.In the recent paper by Zhao the experimental results for the ambipolar diffusion insilicon-on-insulator system are compared with Eq. (30). At high temperatures a goodagreement is found while at low temperatures the observed values of D a are 6-7 times less.The previously reported values show similar temperature dependence.The author’s explanation is that coefficients D e,h are taken for the bulk silicon usingEinstein relation and they might be larger than in the film at low temperatures. However,the reason discussed above cannot be excluded. IV. GRAPHENE AS A POSSIBLE EXAMPLE OF A NON-IDEAL ELECTRONSYSTEM
It is interesting to discuss the single layer graphene as an example of the system withnon-ideal electron gas. Graphene is a gapless material with the linear spectrum of electronsand holes near the Dirac point. Due to some reasons, that are not quite clear now, thevelocity v of electrons and holes in equation ǫ = ± pv is of the order of e / ~ . It followsthat at any Fermi energy inside this linear spectrum electron gas in graphene is non-idealin a sense mentioned above: the absolute value of the chemical potential is of the order ofinteraction energy e n / . It means that unipolar density relaxation in this system shouldbe described by the Dyakonov-Furman equation rather than by diffusion equation.However,without magnetic field the electron gas in graphene is marginally non-ideal. Itcannot be classical, like an electron gas of a low density with quadratic spectrum. Themarginal situation makes theoretical calculations very difficult. Nevertheless, it is acceptedthat the Wigner crystal in single layer graphene is absent without magnetic field . Thesign of dµ/dn is also an interesting question but very difficult for theoretical study. Recentlytunneling microscopy experiment has been done by Martin et al. . They claim that theirmeasurement give the thermodynamic density of states and that it is positive. The laststatement might be a result of disorder. 10 . CONCLUSION Finally I argue that the negative sign of diffusion coefficient that follows from the Einsteinrelation at negative density of states does not lead to any contradiction because diffusioncoefficient is irrelevant for the unipolar transport under this condition. The sign of thediffusion coefficient in this case should not be definitely positive because the diffusion is notthe main source of the entropy production. In bipolar situation negative diffusion meansthe collapse of the system and formation of neutral excitons.I am grateful to Boris Shklovskii and Yoseph Imry for important discussion. I am espe-cially indebted to David Khmelnitskii and Emmanuel Rashba for multiple discussions andcriticism. ∗ Electronic address: [email protected] A. Einstein, Annalen der Physik , 549 (1905). M. von Smoluchowsky, Annalen der Physik , 756 (1906). L. D. Landau and E. Lifshitz,
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