Zero-bias Anomaly of Tunneling into the Edge of a 2D Electron System. L. Shekhtman and L. I. Glazman cond-mat/9506138 29 Jun 1995 Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, 116 Church Str. SE, Minneapolis, MN 55455 (June 29, 1995 ; 14:49) Abstract We investigate the electron tunneling into the edge of a clean weakly interacting two-dimensional electron gas. It is shown that the corresponding di erential conductance G(V ) has a cusp at zero bias, and is characterized by a universal slope jdG=dV j at V = 0. This singularity originates from the electron scattering on the Friedel oscillation caused by the boundary of the system. PACS numbers:72.15.Lh, 71.25 Hc Typeset using REVTEX 1 It is well known that the electron-electron interaction in one dimension leads to a powerlaw singularity in the tunneling conductance at zero bias (see, e.g., 1]). When the interaction is weak, this anomaly may be attributed 2] to a singular backscattering of the incident electrons on the Hartree-Fock potential associated with the Friedel oscillation formed near the tunnel barrier. The Friedel oscillation a ects the electrons on the Fermi surface almost like a periodic potential a ects electrons with the wave vector on the boundary of the Brillouin zone. The tunneling density of states does not acquire a gap of nite width because unlike a strictly periodic potential, the oscillation decays with the distance from the barrier. However, this decay in 1D is su ciently slow (inversely proportional to the distance) to lead to a power-law singularity in the density of states at the Fermi level. It has been shown 3] that this power-law singularity survives also in a multichannel case { for tunneling into the edge of a long wire of a nite width. However, the corresponding exponent decreases with an increase in the number of the transverse modes. The question is, whether any reminiscence of this zero-bias anomaly persists in the limit of an in nite number of modes, i.e., in two dimensions. We show in this paper that the di erential conductance for tunneling into the edge of a clean, interacting two-dimensional electron gas (2DEG) is singular at zero bias. The backscattering from the Friedel oscillation in this case does not renormalize the zero-bias conductance to zero, but still leads to a non-analytic behavior of G(V ) at small applied voltages. The corresponding cusp in G(V ) is characterized by a nite slope jdG=dV j at V = 0, which is universal, i.e., does not depend on the interaction strength. To clarify the origin of the cusp in the di erential conductance, we investigate rst tunneling into the edge of 2DEG with a short-range electron-electron interaction. In this case, the lowest order perturbation theory in the interaction potential is applicable. Our calculation of conductance in a weakly interacting system is based on the Landauer 4] approach. Following Ref. 2] we will focus on the e ect of electron-electron interactions on the transmission coe cient for tunneling through a one-dimensional barrier separating two semiplanes. We assume that the transmission coe cient of the barrier is small, jt0j2 1, and that the barrier is homogeneous along the y-direction 5]. In the absence of interaction, the current I per unit length of the barrier may be obtained from a straightforward generalization of the Landauer formula: 1 d 1 dky I = 2e fl( eV ) fr ( )] jt0(ky ; )j2; (1) h 12 1 2 where fl( eV ) and fr ( ) are the Fermi distribution functions in the left and right semiplanes respectively, and is the kinetic energy of an electron. At small bias, Eq. (1) yields 2 (2) I (V ) = 2e h kF T0V; where kF is the Fermi wave vector in the 2DEG, and T0 = hjt0j2i is the bare transmission coe cient at the Fermi surface averaged over the directions of momenta of the incoming electrons. The bare transmission coe cient may depend on energy, which gives rise to the well-known eld e ect in tunneling. Therefore, the r.h.s. of Eq. (2) can be viewed as a linear term in expansion of the I (V ) function, the next term being proportional to V 3. Unlike this one-particle eld e ect, the electron-electron interaction leads to a stronger and non-analytical correction to the current (2). Z Z 2 The existence of the barrier breaks the translation invariance in the x direction, leading to the Friedel oscillation of the electron density. Due to this oscillation, the electron-electron ^ interaction produces an additional potential, Ve (x; k), which enhances the backscattering of electrons. The corresponding correction to the transmission amplitude, t(k), can be found in the Born approximation. To do this, we notice that the component of the electron momentum parallel to the barrier is conserved, and, correspondingly, the electron wave function has the form k(x; y) = eiky y k (x). In the absence of the electron-electron interaction, the wave function k(x) kx (x) is the scattering state formed by the bare barrier potential. The correction found in the rst order in the interaction potential is k(x) = Z 1 ^ gk (x; x0)Ve 1 x (x0; k) kx (x0)dx0; (3) where gkx (x; x0) is the Green function of the non-interacting single-electron Hamliltonian in the presence of the barrier. It has the following asymptotic form at x0 < x, x ! 1: 8 < h m ikx (x x ) ikx (x+x ) 0) = ih2 kx e gkx (x; x : m ikx(x x ) + r0e ; ih2 kx t0 e 0 0 i 0 ; x0 > 0, x0 < 0, (4) where r0 is the re ection amplitude. For the wave incoming from x < 0, Eq. (3) gives the correction of the form 1 t(k)eikxx; x ! +1; (5) k (x) ' p 2 which de nes t(k). The non-analytic contribution to t(k) is determined by the Friedel oscillation present in the asymptotic behavior of the e ective potential at large x. This oscillation is caused by the re ection o the barrier of all the electrons forming the Fermi ^ sea, and is characteristic for both Hartree- and exchange-type terms in Ve (x; k): ^ ^ Ve (x; k) = VH (x) + Vex(x; k): (6) The Hartree term is local, Z VH (x) = Uee (r r1 )n(r1 ; r1)dr1 ; (7) and the exchange term is given by the following integral relation: Z ^ Vex(x; k) kx (x) = Uee (r r1)n(r; r1 ) kx (x1 )eiky (y y1 ) dr1 : (8) Here Uee (r r1) is the electron-electron interaction potential, and the electron density matrix is given by d (9) n(r; r1) = 2 q2 nF (q) qx (x1) qx (x)eiqy(y y1 ); where nF (q) = (kF q) is the zero-temperature Fermi distribution function, and kF is the Fermi wavevector. The discontinuity in the electron momentum occupation number Z 3 nF (q) results in the Friedel oscillation of the density n(r; r1), and in the e ective potential ^ Ve (x; k). The latter becomes local, 2 ~ VH (x) ' Uee (2kF ) 8kF 3=2 sin(2kF jxj + 3 ); (10) 4 (2kF jxj) 2 3 ~ (11) Vex(x; k) ' Uee (0) 8kF 3=2 sin(2kF jxj + 4 ); (2kF jxj) for distances x from the barrier exceeding both the Fermi wavelength F and the range d of the potential Uee (r). In derivation of (10) and (11) we assumed that the transmission coe cient is small, jt0j2 1, and the components of k satisfy the conditions: jkx kF j kF and ky kF . The latter two conditions allowed us to express the amplitudes of oscillation in ~ ~ (10), (11) in terms of Fourier harmonics Uee (2kF ), Uee (0) of the interaction potential Uee (r). Substituting now Ue (x; k) into (3), we can nd the correction t(kx; ky ). The oscillation in the potential (10), (11) leads to the non-analytical at kx = kF part of t(kx; ky ). The transmission coe cient jt(k)j2 for electrons with energies close to the Fermi level to the rst order in the strength of the electron-electron interaction is given by 8 < jt(k)j2 = jt0(k)j2 1 : ~ ~ + m 2 Uee(0) Uee (2kF ) 4 2h h i s 9 = kx kF (k k ) : x F kF ; (12) Here we presented explicitly only the non-analytical part of the dependence of the transmission coe cient on the incident wavevector. The bare transmission coe cient, jt0(k)j2, and ~ ~ the part of the correction m Uee(0) Uee (2kF )]=h2 are regular at k = kF functions. In order to calculate the corresponding contribution to the conductance, we have to substitute (12) instead of jt0j2 in Eq. (1). The calculation then amounts to integrating the transmission coe cient over the energies that are larger than F but smaller than F + eV , where F is the Fermi energy and eV is the applied bias. It follows from (12) that only the electrons incoming with the momentum kx larger than the Fermi wavevector contribute to the non-analytic correction. At eV F these are electrons moving in the small range of incident angles almost perpendicular to the barrier, and the corresponding bare transmission coe cient is jt0(ky = 0; F )j2. Averaging Eq. (12) over all the incident angles, we obtain the expression for the di erential conductance G(V ) dI=dV per unit length of the barrier at small biases: ~ ~ (13) G(V ) = G0 1 + m Uee (0) Uee (2kF ) jeV j : h2 F The non-analytical part in (13) provides a cusp in the di erential conductance at zero bias. The numerical coe cient jt0(ky = 0; F )j2=(2 )2T0 accounts for the di erence between the average transmission coe cient T0 that determines G0, and the relevant for the anomaly transmission coe cient jt0(ky = 0; F )j2. The value of depends on the detailed shape of the barrier, but typically ' 1. We are allowed to treat the potential energy of electron-electron interaction as a small perturbation only under the conditions i h 4 ! ~ mUee(2kF ) 1; (14) h2 ~ mUee(0) 1: (15) h2 Hence, the \bare" interaction must be weak and short-ranged . This latter condition can be implemented in experiments, if a metallic gate exists very close to the 2D electron gas. The interaction between electrons is described by the Coulomb potential only at distances r smaller than the separation d between the 2D electron gas and the gate. At r d the potential Uee (r) has a dipolar asymptotic behavior, and the condition (15) is met if d is much smaller than the e ective Bohr radius, aB = "h2=me2, where " is the dielectric constant of the semiconductor. Eq. (13) leads to the following estimate of the strength of the cusp: dG ' d G sign(eV ): (16) dV aB 0 In the most interesting case of a GaAs heterostructure, the thickness of a spacer separating the 2D electron gas from the gate normally exceeds aB . We will show that at d aB the proportional to the interaction strength factor d=aB in the estimate (16) is replaced by an interaction-independent universal constant. In the limit d aB , for a pure long-range Coulomb interaction, the requirement (14) is equivalent to the condition that the plasma parameter e2="hvF is small. This condition is satis ed in a su ciently dense interacting electron gas, and the above approach correctly gives the leading order Hartree-type contribution to the di erential conductance. However, ~ the condition (15) does not hold at any density since Uee (k ! 0) diverges. Therefore the slope of the cusp in G(V ) might be not small, and the exchange contribution to the Eq. (13) must be revised. Speci cally, we have to go beyond the perturbative single-electron picture presented above and to consider the many-body e ects leading to screening of the long-range part of the Coulomb interaction. To incorporate the screening e ects, we use the standard 6] relation of the tunneling current I with the product of the densities of states, which is valid at small jt0j2. This relation can be cast in the form (1), if one replaces the bare transmission coe cient jt0j2 by the renormalized value: 2h4 2 T (ky ; ) = jt0(ky ; F )j 4m2K 2 l (ky ; ) r(ky ; ): (17) x Here ! Kx = kx(ky ; ) q 2 (2m =h2) ky (18) is the x-component of the wavevector at given total energy and xed value of ky . The densities of electron states r(l)(ky ; ) are related 7] to the corresponding single-electron Green functions Gr(l)(x; x0; ky ; ) of the isolated right and left semiplanes respectively, r(l) (ky ; @2 ) = Im @x@x0 1 G (x; x0; k ; ) y r(l) 5 x=x =0 0 : (19) The latter relation for the densities of states accounts for the fact that tunneling occurs into the edge of 2DEG, and the electron wavefunctions should vanish at the edge of an isolated semiplane, Gr(l)(x; x0; ky ; ) x=x =0 = 0: (20) As we have shown above, the zero-bias anomaly in the di erential conductance originates from the scattering of the incoming electrons on the Friedel oscillation. Although this oscillation is induced by the barrier, only its tail at large distances ( h2kF =meV ) from the barrier is responsible for the anomaly. In the many-body formulation the potential of Friedel oscillation is represented by a speci c coordinate dependence of the electron self-energy . The latter contributes to the Green function, G(x; x0; ky ; ) = G(0)(x; x1; ky ; ) (x1; x2; ky ; )G(0)(x2; x0; ky ; )dx1dx2; (21) 0 Z and, therefore, to the density of states. It is essential, that the self-energy in Eq. (21) depends not only on the coordinate di erence x1 x2, but also on the distance from the barrier, i.e., on the sum of the coordinates x1 + x2. Our aim is to investigate the induced by the barrier contribution to , to show that it has the Friedel oscillation form, and to calculate the corresponding correction to the tunneling density of states. The outlined program based on the calculation of , enables us to generalize Eq. (11) so that the long-range nature of the bare Coulomb interaction between the electrons is accounted for. As may be anticipated from Eq. (11), the lowest order in electron charge e exchange ~ contribution to is proportional to Uee (k ! 0). For the long-range Coulomb interaction, this Fourier harmonic diverges. The standard way to deal with this di culty is to sum the ~ most singular at k ! 0 diagrams in each order of the self-energy expansion in Uee (k). In the translationally invariant system it is easy to identify these most singular diagrams, since for such a system the dielectric function of the electron gas (r1 r2; ) is diagonal in the planewave representation (see, e.g., Ref. 6]). It is well-known 8] that the described summation ~ is equivalent to the replacement of Uee (k) in the formula for the leading-order contribution to the exchange part of the self energy by the e ective screened interaction. The latter is ~ given by Uee (k)=~(k; ), where ~(k; ) is a Fourier transform of (r1 r2; ). In our problem, however, the translational invariance is destroyed by the barrier. The dielectric function (r1; r2 ; ) depends not only on r1 r2, but also on the distance from the barrier, i.e., on x1 + x2. Thus, the problem of diagonalization of the dielectric function becomes non-trivial, and the screened electron-electron interaction potential in the vicinity of the barrier can not be found in a closed form. Our task is simpli ed by the fact that we do not need to know this expression for the screened electron-electron interaction in the vicinity of the barrier. As we noted already, the zero-bias anomaly arises due to the scattering of the incident electrons by the Friedel oscillation far from the barrier , at the distances h2kF =meV . Thus, we are interested in the asymptotic form of the self energy (x1; x2; ky ; ) at x1; x2 aB ; F . There the dielectric function loses all the information about the barrier and becomes a function of j x1 x2 j only. Thus, while calculating the singular contribution to the electron self-energy, we may use the formulas for the dielectric function and, hence, for the screened electron-electron interaction in the translationally invariant case. In dense electron liquid (e2="hvF 1) this screened electron-electron interaction is well described by the expression found in RPA: 6 Z URPA (x1 x2; ky ; ) = dqxeiqx(x1 ~ x2 ) Uee (qx ; ky ) ; ~ ( q x ; ky ; ) (22) where ~(qx; ky ; ) is a two-dimensional dielectric function of the translation invariant electron gas, see, e. g., 9]. Below we concentrate on the leading order exchange-type term (x1; x2; ky ; )jx1 ;x2 Z aB ; F = i dqy d 0URPA (x1 x2; ky qy ; 0)G(0) (x ; x ; q ; 0) 1 2 y (23) since, as we already noted, in the considered dense electron liquid (e2="hvF 1) the Hartreetype contribution is correctly accounted for by the formula (13). It is easy to identify now the origin of the zero-bias anomaly. In Eq. (23) all the relevant for singularity information about the presence of the barrier is delivered by the Green function. Speci cally, the oscillating part of the dependence of on its coordinates originates in the oscillatory dependence of the Green function on the sum of the coordinates x1 + x2. It is easy to see, that far from the barrier G(0)(x1; x2; qy ; ) may be decomposed as G(0)(x1; x2; qy ; ) x1;x2 F G+ (x1 + x2; qy ; ) + G (x1 x2; qy ; ); and the singular contribution to the self-energy has the form: sing (x ; x ; k ; 1 2 y Z ) = i dqy d 0URPA(x1 x2; ky qy ; 0)G+ (x + x ; q ; 0): 1 2 y (24) Straightforward calculations based on Eqs. (19), (21) and (24) lead to the following expression for the anomalous contribution to the tunneling density of states: 2 2 (ky ; ) = Im 4m k4F 2h Z 1 e dxe2iKxxVex d ! (x; ky ; ) ; (25) where s 2 e 2ikF x ; (26) (2kF x)3=2 ~ ~ wave vector Kx kx (ky ; ) is given by (18), and URPA (q; ) = Uee (q)= (q; ). If the applied bias is small eV < (e2="hvF ) F ], we may neglect the energy and wavevec~ tor dependence of the screened interaction in Eqs. (25) and (26), and take URPA (q; ) ~ ~ URPA (0; 0). Now we note that URPA (0; 0) = h2=m is independent of the interaction strength. In complete analogy with the problem of tunneling into a disordered interacting 2DEG 10], this fact leads to universality of the correction to the electron density of states. Substituting Eq. (26) into (25) we nd for the correction to the density of states at energies close to F : e ~ Vex (x; ky ; ) = URPA (Kx kF ; ky ; F) 1=2 2 2 2 h2ky h2ky 1 2m 3 (ky ; ) = 2 2 (27) 2m F 2m F : h Thus, the anomalous exchange-type contribution of electron-electron interaction to the transmission coe cient is given by ! 7 ! T (ky ; ) / jt0(ky ; F )j2 2 h2ky 2m (ky ; ) = jt0(ky ; F )j2 0(ky ; ) 1=2 ! F 2 h2ky 2m ! F ; (28) where 0(ky ; ) = 2mkx (ky ; )= h2 is the density of states (19) for tunneling into an edge of non-interacting electron system. Substituting (28) in (1) we nd the expression for the di erential conductance to the rst order on the screened electron-electron interaction: (29) G(V ) = G 1 ~ + jeV j ; ! 0 F where ~ e2="hvF is small and ' 1. We neglected in Eq. (29) the Hartree-type contribution see Eq. (13)], since its ratio to the exchange-type contribution is of the order ~ ~ of URPA(2kF )=URPA (0) e2="hvF , and therefore small. Clearly, at a larger bias, eV > 2="hvF ) F , one can not approximate URPA (q; ) in (26) by URPA (0; 0), since the screening ~ ~ (e is not e ective at jqj > 1=aB . At these biases the exchange contribution to di erential conductance becomes comparable with the Hartree contribution, the slope of the curve G(V ) decreases, and the interaction-induced correction to the conductance eventually vanishes. The qualitative dependence of the tunneling di erential conductance on the applied bias is shown in Fig. 1. For simplicity we considered above the case of the uniform barrier, where the parallel to the barrier component of the tunneling electron momentum is conserved. It is clear, however, that our result does not depend on this barrier uniformity. Indeed, the zero-bias anomaly is not sensitive to the changes in the bare transmittion coe cient, and originates exclusively from the singularity in the electron density of states (ky ; ) at the Fermi surface. It is easy to check, that even if the barrier is not uniform, the tunneling current to leading order in electron-electron interaction is still given by the formula i i 2e 1 d f ( eV ) f ( )] 1 dky F (ki ; ) 1 + i(ky ; ) ; I= (30) l r y i (k i ; ) 1 2 0 y i=r;l h 1 2 X Z Z ! where the superscript i = r; l refers to the left or right semiplane respectively. Clearly, the i i singular behavior of the density of states i(ky ; ) at the Fermi surface (at ky = 0) does not depend on details of tunneling. This behavior is adequately accounted for by Eq. (27). All the information about the shape of the barrier, i.e., about the expression for the bare i transmission coe cient jt0(kr; kl; )j2, is absorbed in the factor F (ky ; ). It is obvious, that i i F (ky ; ) is a regular function of energy and momentum at = F and ky = 0. This directly leads to Eq. (29). In conclusion, we demonstrated that a weak interaction in the clean two-dimensional electron gas leads to a singular contribution to tunneling into the edge of electron system. This singularity is characterized by a non-analytical behavior of the corresponding density of states near the Fermi level, and leads to a characteristic universal cusp in the di erential conductance at zero bias. The authors are grateful to I. L. Aleiner for critical reading of the manuscript and useful comments. This work was supported by NSF Grant DMR-9423244. 8 REFERENCES 1] C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 (1992); L. I. Glazman, I. M. Ruzin, and B. I. Shklovskii, Phys. Rev. B45, 8454 (1992); A. Furusaki and N. Nagaosa, Phys. Rev. B47, 4631 (1993). 2] K. A. Matveev, Dongxiao Yue, and L. I. Glazman, Phys. Rev. Lett. 71, 3351 (1993). 3] K. A. Matveev and L. I. Glazman, Phys. Rev. Lett. 70, 990 (1993). 4] R. Landauer, IBM J. Res. Dev. 1, 223 (1957). 5] The latter assumption makes the presented calculations simpler, but does not a ect the main conclusion about the existence of the cusp in G(V ). The magnitude of the non-analytical part of the conductance is barely a ected by a particular geometry of the barrier. 6] G. D. Mahan, Many-Particle Physics, (Plenum Press, New York, 1990). 7] T. E. Feuchtwang, Phys. Rev. B13, 517 (1976). 8] J. J. Quinn and R. F. Ferrell, Phys. Rev. 112, 812 (1958). 9] F. Stern, Phys.Rev. Lett. 18, 546 (1967). 10] B. L. Altshuler and A. G. Aronov, in Electron-electron interaction in Disordered Systems, ed. by A. L. Efros and M. Pollak, (Elsevier Science Publishers, Amsterdam 1985). 9 G/G 1 1- ~ -~ ~ eV/ FIG. 1. The schematic dependence of the di erential conductance G(V ) = dI=dV on the applied bias V for tunneling into the edge of interacting two-dimensional electron gas. The amplitude of the interaction-induced correction is proportional to the interaction strength and small, ~ ' e2 ="hvF , and the correction vanishes at larger bias. However the cusp in G(V ) at V = 0 does not depend on the interaction strength. A smooth variation of G(V ) due to the eld e ect on the shape of the tunneling barrier is not shown. 10 F