arXiv:cond-mat/0301112v3 [cond-mat.mes-hall] 4 Feb 2003 Typeset with jpsj2.cls Many-Body Effects on the Transmission Probability through a Tunnel Junction in a Strong Magnetic Field Toshihiro Kubo1 ,∗ and Arisato Kawabata2,† 1 Department of Physics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 2 Department of Physics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan (Received May 10, 2017) We investigate effects of electron-electron interaction on the transmission probability of electrons through a tunnel junction in a strong magnetic field. We start with the Hartree-Fock approximation, and we show that the coulomb interaction, which gives rise to the divergence of Fock correction, should be replaced by the dynamically screened interaction. We also show that we should use the bare coulomb interaction for Hartree term. We take into account higher order contributions using a simple renormalization group approach. The temperature dependence of the transmission probability is qualitatively similar to that of a one-dimensional system. KEYWORDS: Friedel oscillations, strong magnetic field, one-dimensional system, screened coulomb interaction, poor man’s scaling 1. Introduction Many-body effects on the electron transport in one-dimensional systems have been studied by several authors.1–4) The transport of interacting spinless Fermions through a single barrier was studied by Kane and Fisher1) and the case of spin-1/2 Fermions was investigated by Furusaki and Nagaosa.2) They found that the transmission probability vanishes with a power-law as T → 0. They treated the problem within the framework of the Tomonaga-Luttinger liquid theory,5, 6) i.e., they neglected the backward scatterings in the electron-electron interaction. After that, Matveev et al. developed a different treatment in which the effects of the backward scattering can be incorporated.3, 4) They started with the Hartree-Fock approximation and treated the logarithmic singularity by a simple renormalization group theory. They found that the temperature dependence of the transmission probability does not obey a simple power-law. According to them, Friedel oscillations of the electron density induced by the barrier potential give an essential effect on the electrical conduction in one-dimensional systems. Those theories give qualitatively the same results, as for the vanishing transmission probabilities at zero-temperature. Experimentally, this tendency was observed in quantum wires.7) In this experiment, however, electrons are probably scattered by the disorder such as the fluctuation of the width of the wire, and the situation is not as simple as the case of a single barrier.8, 9) In fact, in ∗ † E-mail: j1202701@ed.kagu.tus.ac.jp E-mail: arisato.kawabata@gakushuin.ac.jp 1 2 Toshihiro Kubo and Arisato Kawabata spite of the development of the technique, it is still hard to verify the theoretical predictions using artificial one-dimensional systems, because the localization effects also contribute to the decrease of the conductance at low temperatures. In this paper, we investigate the transport of electrons through a tunnel junction with a strong magnetic field perpendicular to it (see Fig. 1): The system is one-dimensional like when only the lowest Landau levels is occupied, and we can expect a similar effect. In fact, recently, Biagini et al. and Tsai et al. treated this problem and have shown that the transmission probability behaves like that of one-dimensional systems.10, 11) In this system, measurement of the transmission probability is much easier than in onedimensional systems. Moreover, one may be able to extract the interaction effects from the data by making use of the magnetic field dependence of the transmission probability. Fig. 1. Tunnel junctions in a strong magnetic field. Magnetic field B = (0, 0, B) is perpendicular to the insulator thin film. In order to do so, it is important to estimate the parameters which determine the temperature dependence of the transmission probability in approximations as good as possible. The important ˜ ˜ ˜ parameter are V (0), and V (2kF ), where V (q) is the Fourier transform of the interaction potential ˜ V (x) and kF is the Fermi wave number. For coulomb interaction, V (0) is divergent, and in Refs. ˜ ˜ 10 and 11 the authors used the static screened coulomb interaction for V (0), and V (2kF ). This replacement is not appropriate, however, as will be shown in the following parts of this paper. We ˜ will see that we should use dynamical screened interaction for V (0) and bare coulomb interaction ˜ for V (2kF ). In section 2 we present the calculations of Hartree-Fock correction to the transmission amplitude, and in section 3 we will show the way to take into account the higher order terms using renormalization group method. Many-Body Effects on Transmission through a Tunnel Junction 2. 2.1 3 Hartree-Fock Correction to Transmission Probability Non-interacting electrons First we investigate the transmission of a noninteracting 3D electron through tunnel junctions under a strong magnetic field. We ignore the spin degrees of freedom, for we assume that the electrons are fully polarized. We choose the z-axis of the coordinate system along the magnetic field, and use Landau gauge A(x) = (0, Bx, 0) for the vector potential. Then the single-electron Hamiltonian is of the form H0 = 1 (p + eA(x))2 + U (z), 2m (2.1) where U (z) is the barrier potential of the insulator thin film, and we assume that the potential barrier is localized around z = 0 , i.e., U (z) = 0 for |z| > a. We consider the case when the electrons occupies only the lowest Landau level. It is realized if the magnetic field is strong enough so that B> e (2π 4 n2 )1/3 , e (2.2) where ne is the electron density. Then, we can label the eigenstates of the Hamiltonian H0 by a two-dimensional vector k = (ky , kz ), and the energy eigenvalues ǫ0 and the wave functions ϕ0 (x) k k are of the forms 2k 2 1 z ωc + , 2 2m (2.3a) ϕ0 (x) = φ0y (x, y)u0z (z) , k k k (2.3b) ǫ0 = k where φ0y (x, y) = k u0z (z) = k u0z (z) = k Here ωc = eB/m, ℓB = 1 π 1/4 ℓB 1/2 exp − (x + ky ℓB 2 )2 iky y e , 2ℓB 2 eikz z + r0 e−ikz z , t0 eikz z , t0 eikz z , eikz z + r0 e−ikz z , (2.4a) (z < −a), (kz > 0), (2.4b) (z < −a), (kz < 0). (2.4c) (z > a), (z > a), /eB, and t0 and r0 are the transmission and reflection amplitudes through the barrier, respectively. We assume an infinite system and neglect the effects of the boundaries. 4 Toshihiro Kubo and Arisato Kawabata 2.2 First Born approximation Next we calculate the correction to the transmission probability due to the electron-electron interaction. We calculate it to the lowest order in the interaction within the Hartree-Fock theory. Using the electron field operator ψ(x), we can write the many-body Hamiltonian H as H = H0 + H1 , (2.5) with dxψ † (x) H0 = 1 (p + eA(x))2 + U (z) ψ(x), 2m (2.6) and H1 = 1 2 dxdx′ ψ † (x)ψ † (x′ )V (x − x′ )ψ(x′ )ψ(x), (2.7) V (x) being the coulomb interaction potential, and if we do not write explicitly the ranges of the space integrations they mean integrations over the whole space region. We calculate the correction to the wave functions using Green’s function method.12) The single-electron wave functions ϕk(x, τ ) can be written in the form ϕk(x, τ ) = i dyGR (x, τ ; y, τ0 )ϕk (y, τ0 ), τ > τ0 (2.8) in terms of the retarded Green’s function GR (x, τ ; y, τ0 ). Assuming that the interaction is switched 0 0 on at time τ0 , we put ϕk (x, τ0 ) = ϕ0 (x)e−iξk τ0 / , ξk being the single-electron energy measured from k the Fermi level, and later we will let τ0 → −∞. The retarded Green’s function GR (x, y, ω) can be obtained from the Matsubara Green’s function G(x, y, ωn ) in terms of the analytic continuation GR (x, y, ω) = G(x, y, −iω + η) (2.9) where ωn = πkB T (2n+1)/ , n being an integer, and η → +0. The unperturbed Matsubara Green’s Fig. 2. Feynman diagram for the first Born approximation within Hartree-Fock theory: The thin solid lines indicate the Green’s function without the interaction correction, and the wavy lines indicate the interaction. Many-Body Effects on Transmission through a Tunnel Junction 5 function G 0 (x, y, ωn ) is defined by ∗ dk ϕ0 (x)ϕ0 (y) k k . 0 (2π)2 iωn − ξk/ G 0 (x, y, ωn ) = (2.10) Hereafter, if we do not write explicitly the ranges of the wave number integrations, they mean integrations over the whole wave number region. The Feynman diagrams for G(x, y, ωn ) to the lowest order in the interaction is shown in Fig. 2, and the expression corresponding to it is easily found to be G(x, y, ωn ) = G 0 (x, y, ωn ) + dx1 dx′ 1 × G 0 (x, x1 , ωn )ΣHF (x1 , x′ )G 0 (x′ , y, ωn ), 1 1 (2.11) where ΣHF (x1 , x′ ) is the Hartree-Fock self-energy which is independent of the frequency ωn and 1 is given by ΣHF (x1 , x′ ) =δ(x1 − x′ ) 1 1 × kB T n′ dx2 V (x1 − x2 ) G 0 (x2 , x2 , ωn′ ) − V (x1 − x′ )kB T 1 n′ G 0 (x1 , x′ , ωn′ ). 1 (2.12) Then we obtain the following form for single-electron wave functions (see Appendix A) ϕk(x) =ϕ0 (x) + k − dx1 Gk(x; x1 )VH (x1 )ϕ0 (x1 ) k dx1 dx′ Gk(x; x1 )VF (x1 , x′ )ϕ0 (x′ ), 1 1 k 1 where (2.13) ∗ Gk(x; x1 ) ≡ VH (x1 ) = dk′ ϕ0 ′ (x)ϕ0 ′ (x1 ) k k , 0 0 (2π)2 ξk − ξk′ dx2 V (x1 − x2 )ρ(x2 ), and VF (x1 , x′ ) = V (x1 − x′ ) 1 1 oc. dk 0 ∗ ϕ (x1 )ϕ0 (x′ ), 1 k (2π)2 k (2.14) (2.15) (2.16) ρ(x) being the electron density and the subscript oc. means the integration over the occupied states, i.e., |kz | ≤ kF . As for the treatment of the singularity in eq. (2.14), the reader is referred to Appendix B. 6 Toshihiro Kubo and Arisato Kawabata To calculate the correction to the transmission probability we need only the asymptotic form of the Gk(x, x1 ) at z → +∞. From eq. (2.14), we obtain (see Appendix B) 1 (x − x1 )2 + (y − y1 )2 exp − 2πi vF ℓB 2 4ℓB 2 i(x + x1 )(y − y1 ) × exp − 2ℓB 2 Gk (x; x1 ) = × t0 eikz (z−z1 ) , eikz (z−z1 ) + r0 eikz (z+z1 ) , z1 < 0, z1 > 0, (2.17) with vF ≡ kF /m. The electron density ρ(x) in eq. (2.15) is calculated from the wave functions (2.3b) with (2.4), and at large distances |z| it behaves like dk |ϕ0 (x)|2 (2π)2 k oc. |r0 | kF 1 ≃ 2 · 2π|z| sin(2kF |z| + arg r0 ) + π 2πℓB 1 · n(z). ≡ 2πℓB 2 ρ(x) ≡ (2.18) In the following we will neglect the space independent part in n(z) which gives a constant Hartree potential. Such oscillation of the electron density, i.e., the Friedel oscillation, gives an essential effect on the transmission probability, as in one-dimensional systems.3, 4) 2.3 Fock correction (1F ) First we calculate the Fock correction ϕk (x) to the wave function, i.e., the second term of the right side of eq. (2.13) (1F ) ϕk (x) = − =− dx1 dx′ Gk(x; x1 )VF (x1 ; x′ )ϕ0 (x′ ) 1 1 k 1 dx1 dx1 Gky (x, y; x1 , y1 )Gkz (z; z1 ) × V (x1 − x′ ) 1 × =− × × oc. dk′ 0 ∗ ϕ ′ (x1 )ϕ0 ′ (x′ ) 1 k (2π)2 k ′ ′ φ0y (x′ , y1 )u0z (z1 ) 1 k k ′ dx1 dy1 dx′ dy1 Gky (x, y; x1 , y1 ) 1 ′ dky 0 ∗ ′ ′ φky (x1 , y1 )φ0y (x′ , y1 ) φ0y (x′ , y1 ) ′ 1 1 k k′ 2π ′ ′ dz1 dz1 Gkz (z; z1 )u0z (z1 ) k × V (x1 − x′ ) 1 kF −kF ′ dkz 0 ∗ ′ u ′ (z1 )u0z (z1 ) , k′ 2π kz where Gky (x, y; x1 , y1 ) and Gkz (z; z1 ) are defined in Appendix B. (2.19) Many-Body Effects on Transmission through a Tunnel Junction 7 ′ ′ As is shown in the Appendix C, the integration over z1 , z1 , and kz gives ′ dz1 dz1 Gkz (z; z1 ) × V (x1 − x′ ) 1 = kF −kF ′ dkz 0 ∗ ′ ′ ukz (z1 )u0z (z1 ) u0z (z1 ) ′ k k′ 2π |2 t0 |r0 ˜ ′ V (x1 − x′ , y1 − y1 ; 0) ln 1 2π vF 1 |kz − kF |d · eikz z , (2.20) ′ ˜ where V (x1 −x′ , y1 −y1 ; qz ) is the Fourier transform of V (x1 −x′ ) with respect only to z component 1 1 of x1 − x′ and is defined by, 1 ′ ˜ V (x1 − x′ , y1 − y1 ; kz ) ≡ 1 ∞ −∞ ′ ′ ′ ′ dz2 V (x1 − x′ , y1 − y1 , z2 )e−ikz z2 . 1 (2.21) and d is the cut-off length. From Eqs. (2.19) and (2.20), we have (1F ) ϕk ′ dx1 dy1 dx′ dy1 Gky (x, y; x1 , y1 ) 1 (x) = − ′ dky 0 ∗ ′ ′ φ ′ (x1 , y1 )φ0y (x′ , y1 ) φ0y (x′ , y1 ) 1 1 k k′ 2π ky t0 |r0 |2 ˜ ′ Vs (x1 − x′ , y1 − y1 ; 0) × 1 2π vF 1 × ln · eikz z . |kz − kF |d × (1F ) Thus the contribution from the Fock term ϕk (1F ) ϕk (x) is written as (see Appendix C) (x) = −α2 (B)t0 (1 − |t0 |2 ) ln × φ0y (x, y)eikz z , k where α2 (B) ≡ ∞ 1 2π 2 vF (2.22) 1 |kz − kF |d (2.23) 2 q⊥ dq⊥ e−q⊥ ℓB 0 2 /2 ˆ V (q ⊥ ) , (2.24) ˆ V (q ⊥ ) being the Fourier transform of interaction V (x) with q⊥ ≡ (qx , qy , 0), and we obtain the following 1st-order correction to the transmission amplitude by the Fock term t(1F ) = −α2 (B)t0 (1 − |t0 |2 ) ln 1 |kz − kF |d . (2.25) It is to be noted that if V (x) is coulomb interaction we have e2 ˆ V (q) = 2 , ǫq (2.26) ǫ being the dielectric permittivity of the matter, and that the right hand side of eq. (2.24) is divergent. In this context, one of the authors (A.K.) reinvestigated the problem from a point of 8 Toshihiro Kubo and Arisato Kawabata view different from those of the theories mentioned above:13) Using the parquet diagram method developed by Bychkov et al,14) he has shown that, to be consistent, this singularity should be renormalized, too, and that the interaction potential should be replaced by the screened interaction within Random Phase Approximation (RPA).15) As the screened interaction is frequency dependent, the second term of the right hand side of eq. (2.12) should be replaced by − n′ V (x1 − x′ , iωn − iωn′ )G 0 (x′ , x1 ; ωn′ ) . 1 1 (2.27) According to the arguments in Ref. 13, relevant energy is vF qz ≈ ωn ≈ kB T and we have to ˆ replace V (q ⊥ ) in eq. (2.24) with ˆ lim Vs (q, ωn = vF qz ) . (2.28) qz →0 where frequency dependent screened interaction is given by eq. (D.15). Then we easily find that ∞ 2 α2 (B) ≡ 2(κℓB ) 0 2 2 qe−q ℓB /2 dq , 2 2 q 2 + κ2 e−q ℓB /2 (2.29) κ being given by eq. (D.18). We easily find that this κ is smaller than that in Ref. 11 by a factor √ 1/ 2. 2.4 Hartree correction Next we calculate the correction to the transmission amplitude by the Hartree term. We can (1H) write the contribution of the Hartree term ϕk (1H) ϕk (x) as dx1 Gk(x; x1 )VH (x1 )ϕ0 (x1 ) k (x) = dx1 Gky (x, y; x1 , y1 )Gkz (z; z1 )φ0y (x1 , y1 ) k = × dx2 V (x1 − x2 )ρ(x2 ) u0z (z1 ) k dx1 dy1 Gky (x, y; x1 , y1 )φ0y (x1 , y1 ) k = × dx2 dy2 2πℓB 2 × dz2 V (x1 − x2 )n(z2 ) . dz1 Gkz (z; z1 )u0z (z1 ) k (2.30) Here we should not replace V (x) with the screened interaction according to the following reason. The above mentioned replacement can be expressed by Feynman diagrams in Fig. 3 as the renormalization of the interaction. On the other hand, this term can be interpreted as the correction to the one-electron Green’s function as is shown in Fig. 4. In the section 3 we take into account the higher order correction, which can be interpreted as the corrections to the one-electron states. Many-Body Effects on Transmission through a Tunnel Junction 9 Fig. 3. The Hartree correction in terms of the screened coulomb interaction Fig. 4. The Hartree correction in terms of the one-electron Green’s function Therefore, if we replace the coulomb interaction with the screened interaction, it gives rise to the overcounting of the corrections. In other words, the effects of the Hartree terms are essentially the screening of the barrier potential, and the screening should not be counted twice. Thus, in the Hartree term, we should keep the bare coulomb potential in eq. (2.12). As is shown in the Appendix C, the integration over z1 and z2 in eq. (2.30) gives dz1 Gkz (z; z1 )u0z (z1 ) k dz2 V (x1 − x2 )n(z2 ) × = t0 |r0 |2 ˜ V (x1 − x2 , y1 − y2 ; 2kF ) 2π vF 1 × ln · eikz z . |kz − kF |d From Eqs. (2.30) and (2.31), we have (1H) ϕk dx1 dy1 Gky (x, y; x1 , y1 ) (x) = × × dx2 dy2 1 φ0y (x1 , y1 ) k 2πℓB 2 t0 |r0 |2 ˜ V (x1 − x2 , y1 − y2 ; 2kF ) 2π vF (2.31) 10 Toshihiro Kubo and Arisato Kawabata × ln 1 |kz − kF |d (1H) hence the contribution from the Hartree term ϕk (1H) ϕk · eikz z , (2.32) (x) is written as (see Appendix C) 1 |kz − kF |d (x) = α1 (B)t0 (1 − |t0 |2 ) ln × φ0y (x, y)eikz z , k (2.33) where κ2 . (2.34) 4kF 2 Thus we obtain the following 1st-order correction to the transmission amplitude by the Hartree α1 (B) ≡ term. t(1H) = α1 (B)t0 (1 − |t0 |2 ) ln 1 |kz − kF |d . 1 |kz − kF |d , (2.35) Eqs. (2.25) and (2.35) give the 1st-order correction to the transmission amplitude within Hartree-Fock theory t(1) = −α(B)t0 (1 − |t0 |2 ) ln (2.36) where dimensionless parameter α(B) of the electron-electron interaction is α(B) = α2 (B) − α1 (B), (2.37) and the 1st-order correction to the transmission probability is given by T (1) = −2α(B)T0 (1 − T0 ) ln 1 |kz − kF |d . (2.38) Since this is the result obtained by perturbation theory, it is applicable as long as α(B) ln 1 |kz − kF |d ≪ 1. (2.39) However eq. (2.39) is no longer valid at low temperature since only electrons of kz ≃ kF contribute electric conduction at low temperatures. Thus we have to take into account the higher order contributions in the interaction. 3. Higher order contributions In order to include the higher order corrections, we use a simple renormalization group (RG) approach called the poor man’s scaling developed by Anderson for the Kondo problem.16) We assume that only electrons in the strip of the wave number kz of halfwidth Λ0 = 1/d near the Fermi wave number kF contribute to the correction (2.38) (see Fig. 5). In this strip we linearize the dispersion relation of electron energy: 0 ξk = vF (|kz | − kF ). (3.1) Many-Body Effects on Transmission through a Tunnel Junction 11 ˜ Fig. 5. The reduction of the cutoff Λ0 . kz is the wave number measured from kF . Note that d is the cutoff of integrations not only due to the validity of eq. (2.18) but also due the validity of the linearized dispersion. Next we reduce the cutoff Λ0 to sΛ0 with 1 − s ≪ 1 (see Fig. 5). If we simultaneously renormalize the transmission probability T0 so that the effects of the states excluded by this RG transformation are taken into account, the new problem by this RG transformation is equivalent to the original problem. The change in T0 found in the first Born approximation is (see Appendix E) δT = −2α(B)T0 (1 − T0 ) ln E0 E , (3.2) where E ≡ vF sΛ0 , E0 ≡ vF Λ0 are respectively the electron energy measured from Fermi level. We apply the RG transformation again, reducing the bandwidth E → sE step by step. Dur- ing each step of rescaling the cutoff, transmission probability T is renormalized according to eq. (3.2) with T0 being substituted by the modified T from the previous step. The effect of these renormalizations may be found as a solution of the differential equation (RG equation) dT (E) = −2α(B)T (E)(1 − T (E)). d ln(E0 /E) (3.3) If we integrate this RG equation from E = E0 to E = E with the initial condition T |E=E0 = T0 , the transmission probability becomes T (E) = T0 (E/E0 )2α(B) , R0 + T0 (E/E0 )2α(B) (3.4) where R0 = 1 − T0 . At low temperature E in eq. (3.4) should be replaced by kB T and the temperature dependence of the transmission probability is found to be T (T ) = T0 (kB T /E0 )2α(B) . R0 + T0 (kB T /E0 )2α(B) (3.5) Our result is of the same form as that of one-dimensional electron systems except for the parameter of the electron-electron interaction. 3.1 Temperature dependence of the transmission probability From eq. (3.5) we find that the transmission probability vanishes as T → 0 if α(B) > 0, as in the case of one-dimensional systems. On the other hand, in Fig. 6 and Fig. 7, we see that α(B) can be negative for large magnetic fields. In this case, according to eq. (3.5), the transmission probability 12 Toshihiro Kubo and Arisato Kawabata should become 1 at 0K. Since the calculation of α(B) is based on random phase approximation, it is not clear whether the values of α(B) are quantitatively reliable. Nevertheless, the tendency that α(B) decreases as the magnetic field increases should be reliable. As can be seem from eq. (2.37) the Fock term suppresses the transmission while the Hartree term enhances it. The role of the Hartree term can be interpreted as the screening of the barrier potential, and it is not surprising if the barrier becomes transparent because of the strong screening due to the one-dimensional nature of the system. 15 20 25 30 -2 -4 -6 -8 Fig. 6. In the case of n = 1023 m−3 , parameter α(B) of interaction vs. magnetic field B. (i) ǫ = 10ǫ0 , m = 0.5me , (ii) ǫ = 11.9ǫ0 , m = 0.26me , (iii) ǫ = 13.1ǫ0 , m = 0.067me (typical values of GaAs), and (iv) ǫ = 15ǫ0 , m = 0.05me , where ǫ0 is the permittivity of vacuum and me is the rest mass of electron. 0.5 60 80 100 120 -0.5 -1 Fig. 7. In the case of n = 1024 m−3 , parameter α(B) of interaction vs. magnetic field B. (i) ǫ = 10ǫ0 , m = 0.5me , (ii) ǫ = 11.9ǫ0 , m = 0.26me , (iii) ǫ = 13.1ǫ0 , m = 0.067me (typical values of GaAs), and (iv) ǫ = 15ǫ0 , m = 0.05me . 4. Summaries We investigated the effects of electron-electron interaction on the transmission of electrons through a tunnel barrier in a strong magnetic field. We have found that the temperature dependence of the transmission probability is at large the same as that of one-dimensional systems, as in Refs. 10 and 11. In fact, such behavior can be expected because in both cases the Friedel oscillations of the electron density play an essential role. In estimating the parameter α(B), we have shown that we should use dynamical screened Many-Body Effects on Transmission through a Tunnel Junction 13 coulomb interaction and bare coulomb interaction for Fock correction and Hartree correction, respectively: It is important to use approximations as good as possible, in order to verify the interaction effect from the experimental data. It is interesting to investigate the magnetic field dependence of α(B). To do so, one need not go to very low temperature. In fact, at moderate temperatures from eq. (3.2) we can expect that δT = 2α(B)T0 (1 − T0 ) ln(kB T ) . (4.1) If α(B) decreases with the increase of the magnetic field, we can expect it to become negative for a larger magnetic field, as in Figs. 6 and 7. Some line in these figures are for the values of effective mass etc. of typical semiconductor. In the case of doped semiconductors, the scatterings of electrons by impurities will suppress the long tail of the Friedel oscillations. In order to observe clear interaction effects, the mean free path of the electrons has to be much longer than the Fermi wave length. Very pure semimetals are good candidates for the observation of the interaction effects. In semimetals, however, the band structures are generally very complex, and the theory needs some modifications to be applied to them. Acknowledgments This work is partly supported by the ”High Technology Research Center Project” of Ministry of Education, Culture, Sports, Sciences and Technology. Appendix A: The derivation of eq. (2.13) by Green’s function method First we consider only the Hartree term. From Eqs. (2.11) and (2.12), the corresponding part of the Matsubara Green’s function G H (x, y, ωn ) is given by G H (x, y, ωn ) = 1 (A.1) dx1 G 0 (x, x1 , ωn )VH (x1 )G 0 (x1 , y, ωn ), where VH (x1 ) is the Hartree-potential VH (x1 ) = (A.2) dx2 V (x1 − x2 )ρ(x2 ) , ρ(x2 ) being the electron density. Thus the corresponding retarded Green’s function can be written as GH (x, y, ω) R = 1 dk′ (2π)2 dk′′ (2π)2 ∗ ∗ ϕ0 ′′ (x1 )ϕ0 ′′ (y) ϕ0 ′ (x)ϕ0 ′ (x1 ) VH (x1 ) k 0 k . dx1 k 0 k ω − ξk′ / + iη ω − ξk′′ / + iη (A.3) 14 Toshihiro Kubo and Arisato Kawabata The Fourier transform of eq. (A.3) is given by GH (x, y, τ − τ0 ) = R dk′ (2π)2 1 dk′′ (2π)2 ∗ ∗ dx1 ϕ0 ′ (x)ϕ0 ′ (x1 )VH (x1 )ϕ0 ′′ (x1 )ϕ0 ′′ (y) k k k k e−iω(τ −τ0 ) dω . 0 0 2π (ω − ξk′ / + iη)(ω − ξk′′ / + iη) × (A.4) The correction to the wave functions ϕH (x, τ ) by the Hartree term is given by k (1H) ϕk dyGH (x, y, τ − τ0 )ϕ0 (y, τ0 ) , R k (x, τ ) = i (A.5) 0 where ϕk(y, τ0 ) = ϕ0 (y)e−iξk τ0 / and we will let τ0 → −∞. Putting eq. (A.4) into the above k expression and performing the integral over y, k′′ , and ω, we obtain (1H) ϕk (x, τ ) = dk′ (2π)2 1 ∗ dx1 ϕ0 ′ (x)ϕ0 ′ (x1 )VH (x1 )ϕ0 (x1 , τ0 ) k k k 0 0 × e−iξk ′ (τ −τ0 )/ e−iξk (τ −τ0 )/ + 0 0 0 0 ξk/ − ξk′ / ξk′ / − ξk/ . (A.6) In the limit τ0 → −∞, the second term in the curly bracket of the above equation oscillates very rapidly when the integral over k′ is done, hence its contribution can be neglected. Therefore we have ∗ (1H) ϕk 0 (x, τ ) = e−iξk τ / dx1 dk′ ϕ0 ′ (x)ϕ0 ′ (x1 ) k k VH (x1 )ϕ0 (x1 ). k 0 0 (2π)2 ξk − ξk′ (A.7) Hereafter we omit the time-dependence of the wave functions, and it can be written in the form (1H) ϕk (x) = dx1 Gk (x; x1 )VH (x1 )ϕ0 (x1 ), k (A.8) where Gk(x; x1 ) is the single-electron Green’s function for noninteracting electrons and is defined as ∗ Gk(x; x1 ) ≡ dk′ ϕ0 ′ (x)ϕ0 ′ (x1 ) k k . 0 0 (2π)2 ξk − ξk′ (A.9) As for the treatment of the singularity of the integrand, the reader is referred to the Appendix B. Next we consider the Fock term. Using Eqs. (2.11) and (2.12), we can write the corresponding part of the Matsubara Green’s function G F (x, y, ωn ) as G F (x, y, ωn ) = − 1 dx1 dx′ G 0 (x, x1 , ωn )VF (x1 , x′ )G 0 (x′ , y, ωn ) 1 1 1 (A.10) where VF (x1 , x′ ) is the Fock-potential 1 VF (x1 , x′ ) = V (x1 − x′ ) 1 1 dk′ 0 ∗ ϕ (x1 )ϕ0 ′ (x′ ). 1 k 2 k′ (2π) (A.11) Many-Body Effects on Transmission through a Tunnel Junction 15 Thus the corresponding retarded Green’s function can be written as dk′′ dk′ dx1 dx′ 1 (2π)2 (2π)2 ∗ 0 0 ∗ ϕ0 ′ (x)ϕ0 ′ (x1 ) ′ ϕk′′ (x1 )ϕk′′ (y) k k × VF (x1 , x1 ) . 0 0 ω − ξk′ / + iη ω − ξk′′ / + iη GF (x, y, ω) = − R 1 (A.12) The Fourier transform of eq. (A.12) is given by GF (x, y, τ − τ0 ) = − R dk′ (2π)2 1 dk′′ (2π)2 ∗ × ϕ0 ′′ (x′ )ϕ0 ′′ (y) 1 k k ∗ dx1 dx′ ϕ0 ′ (x)ϕ0 ′ (x1 )VF (x1 , x′ ) 1 k 1 k e−iω(τ −τ0 ) dω 0 / + iη)(ω − ξ 0 / + iη) . 2π (ω − ξk′ k′′ (A.13) Thus, as in the case of Hartree term, we can calculate the correction to ϕF (x, τ ) by the Fock term k from (1F ) ϕk dyGF (x, y, τ − τ0 )ϕk (y, τ0 ) , R (A.14) dx1 dx′ Gk(x; x1 )VF (x1 , x′ )ϕ0 (x′ ), 1 1 k 1 (A.15) (x, τ ) = i i.e., (1F ) ϕk (x) = − where we omit the time-dependence of the wave functions and Gk(x; x1 ) is defined by eq. (A.9). From Eqs. (A.8) and (A.15), we obtain eq. (2.13). Appendix B: The calculation of the single-electron Green’s function To calculate the correction to the transmission probability we need only the asymptotic form of the Gk(x, x1 ) at z → +∞. From eqs. (2.14), (2.3b) and (2.4) we have Gk(x, x1 ) = ′ dky 0 ∗ φ ′ (x, y)φ0y (x1 , y1 ) · k′ 2π ky ∗ 0 ′ 0′ ′ dkz ukz (z)ukz (z1 ) 0 − ξ0 2π ξk k′ ≡ Gky (x, y; x1 , y1 ) · Gkz (z; z1 ), (B.1) 0 Note that the energy eigenvalues ξk depend only on kz and first we calculate Gky (x, y; x1 , y1 ): ′ dky 0 ∗ φ ′ (x, y)φ0y (x1 , y1 ) k′ 2π ky Lx /2ℓB 2 dk ′ ′ 1 y iky (y−y1 ) e = 1/2 π ℓB −Lx /2ℓB 2 2π Gky (x, y; x1 , y1 ) ≡ × exp − We let Lx → ∞, then we obtain ′ ′ (x + ky ℓB 2 )2 + (x1 + ky ℓB 2 )2 2ℓB 2 (x − x1 )2 + (y − y1 )2 1 exp − 2πℓB 2 4ℓB 2 i(x + x1 )(y − y1 ) × exp − . 2ℓB 2 . (B.2) Gky (x, y; x1 , y1 ) = (B.3) 16 Toshihiro Kubo and Arisato Kawabata Next we calculate Gkz (z; z1 ) for kz > 0: ∗ 0 ′ 0′ ′ dkz ukz (z)ukz (z1 ) . 0 0 2π ξk − ξk′ Gkz (z; z1 ) ≡ (B.4) In treating the singularities of the integral, we replace kz with kz + iη (η → +0) so that we extract only the out going waves at z → +∞. Then, for z1 > 0, from eqs. (2.4b) and (2.4c) we have ∞ Gkz (z; z1 ) = 0 0 + −∞ ′ ′ dkz |t0 |2 eikz (z−z1 ) ′ 2π vF (kz + iη − kz ) ′ ′ ′ ′ ′ dkz (eikz z + r0 e−ikz z )(e−ikz z1 + r0 ∗ eikz z1 ) , ′ 2π vF (kz + iη + kz ) (B.5) where we used the linearized energy dispersion relation eq. (3.1). Since the integrands of the above ′ integrals oscillate very rapidly for large z, the main contributions come from only such kz that the denominators vanish. Therefore we may extend the regions of the integrals to (−∞, ∞) and, after adding a contour of an infinitely large semi-circle, we obtain (see Fig. 8) 1 Gkz (z; z1 ) = eikz (z−z1 ) + r0 eikz (z+z1 ) . i vF (B.6) Fig. 8. The contour of the first integral in eq. (B.5) (R → ∞). Similarly, for z1 < 0, we obtain Gkz (z; z1 ) = 1 t0 eikz (z−z1 ) , i vF (B.7) where we used the relation t0 r0 ∗ + t0 ∗ r0 = 0 , (B.8) which can be derived from a time reversal symmetry argument. Thus from eqs. (B.1), (B.3), (B.6) and (B.7) we obtain eq. (2.17). Appendix C: The calculations of the correction to the transmission amplitude by the perturbation theory C.1 The derivation of eq. (2.20) In eq. (2.20), because of V (x1 − x′ ), main contribution comes from the regions where z1 and 1 ′ ′ z1 are of the same sign. First we consider only the integration over kz in the Fock potential. When Many-Body Effects on Transmission through a Tunnel Junction 17 ′ both z1 and z1 are positive, it is easily found that kF −kF ′ dkz 0 ∗ ′ u ′ (z1 )u0z (z1 ) k′ 2π kz = |t0 |2 0 kF 0 ′ dkz ikz (z1 −z1 ) ′ ′ e + |r0 |2 2π ′ dkz ikz (z1 −z1 ) ′ ′ 0 0 −kF ′ dkz ′ dkz −ikz (z1 −z1 ) ′ ′ e 2π ′ ′ r0 ∗ eikz (z1 +z1 ) + c.c. e + 2π 2π −kF −kF ′ )] sin[kF (z1 − z1 = ′ π(z1 − z1 ) ′ sin[kF (z1 + z1 ) + arg r0 ] − sin(arg r0 ) . + |r0 | ′ π(z1 + z1 ) + (C.1) ′ Similarly, when both z1 and z1 are negative, it is easily found that kF ′ dkz 0 ∗ ′ ukz (z1 )u0z (z1 ) ′ k′ −kF 2π ′ ′ sin[kF (z1 − z1 )] sin[kF (z1 + z1 ) − arg r0 ] + sin(arg r0 ) = + |r0 | . ′) ′ π(z1 − z1 π(z1 + z1 ) (C.2) ′ ′ ′ ′ Next we transform z1 and z1 into z2 = z1 + z1 and z2 = z1 − z1 , respectively. Then the left side of eq. (2.20) becomes t0 r0 |r0 | ikz z e 2πi vF ∞ × 0 ∞ −∞ ′ ′ ′ dz2 V (x1 − x′ , y1 − y1 , z2 ) 1 0 eikz z2 sin(kF z2 + arg r0 ) + dz2 z2 t0 r0 |r0 | ikz z ˜ ′ e = V (x1 − x′ , y1 − y1 ; 0) 1 πi vF dz2 −∞ ∞ dz2 0 e−ikz z2 sin(kF z2 − arg r0 ) z2 eikz z2 sin(kF z2 + arg r0 ), z2 (C.3) ′ ˜ where V (x1 −x′ , y1 −y1 ; kz ) is the Fourier transform for only z component and defined by eq. (2.21) 1 Since the expression for the electron density, eq. (2.18), is only valid at large distances |z|, we restrict the integration in eq. (C.3) in the regions |z| ≥ d. At low temperatures, the electronic conduction is determined entirely by electrons with energy close to the Fermi energy. Therefore in the integral in eq. (C.3), we leave only the terms which are divergent for kz → kF . Then, we find that the integral becomes i −i arg r0 ∞ cos[(kz − kF )z2 ] e dz2 , 2 z2 d and that for |kz − kF |d ≪ 1 eq. (C.3) reduces to t0 |r0 |2 ˜ ′ V (x1 − x′ , y1 − y1 ; 0)eikz z ln 1 2π vF 1 |kz − kF |d (C.4) . (C.5) 18 C.2 Toshihiro Kubo and Arisato Kawabata The derivation of eq. (2.23) ′ The integral over ky in eq. (2.22) is easily found to be ′ dky 0 ∗ ′ φ ′ (x1 , y1 )φ0y (x′ , y1 ) 1 k′ 2π ky ′ ′ 1 (x1 − x′ )2 + (y1 − y1 )2 i(x1 + x′ )(y1 − y1 ) 1 1 = exp − exp − . 2πℓB 2 4ℓB 2 2ℓB 2 (C.6) Then we find that eq. (2.22) is given by (1F ) ϕk ∞ 1 t0 |r0 |2 ′ ln · eikz z dx1 dy1 dx′ dy1 1 2π vF |kz − kF |d −∞ 1 (x1 − x1 )2 + (y1 − y1 )2 i(x1 + x1 )(y1 − y1 ) × exp − 2 exp − 2 2πℓB 4ℓB 2ℓB 2 ′ ′ 1 (x1 − x′ )2 + (y1 − y1 )2 i(x1 + x′ )(y1 − y1 ) 1 1 × exp − exp − 2πℓB 2 4ℓB 2 2ℓB 2 (x′ + ky ℓB 2 )2 iky y′ ˜ 1 ′ e 1 Vs (x1 − x′ , y1 − y1 ; 0) , exp − 1 × 1 2ℓB 2 π 1/4 ℓB 1/2 (x) = − (C.7) and after performing the integrals, we obtain eq. (2.23). C.3 The derivation of eq. (2.31) From eq. (B.6) and (B.7), we have ∞ dz1 Gkz (z; z1 ) |z2 |≥d −∞ = + t0 ikz z e i vF t0 r0 ikz z e i vF dz2 V (x1 − x2 )n(z2 ) u0z (z1 ) k ∞ dz1 |z2 |≥d −∞ 0 dz2 V (x1 − x2 )n(z2 ) ∞ dz1 e−2ikz z1 + dz1 e2ikz z1 |z2 |≥d 0 −∞ dz2 V (x1 − x2 )n(z2 ) , (C.8) where the integrations over z2 are done for |z2 | ≥ d because the expression for the electron density n(z2 ) is valid only at large distances |z2 |. In eq. (C.8), we neglect the first term of the right hand side because its integrand oscillates and its contribution to the integral is very small. Then the right hand side of eq. (C.8) is easily found to be t0 r0 ikz z e i vF ∞ ∞ dz1 sin(2kz z1 ) dz1 cos(2kz z1 ) + 2i |z2 |≥d 0 −∞ dz2 V (x1 − x2 )n(z2 ) . (C.9) Here we consider the integral with cos(2kz z1 ): ∞ I1 ≡ dz1 cos(2kz z1 ) −∞ |z2 |≥d dz2 V (x1 − x2 )n(z2 ) . (C.10) ′ Changing the integration over z1 to that over z1 ≡ z1 − z2 and making use of that V (x1 − x2 ) is an even function of the arguments, we easily find that ˜ I1 = V (x1 − x2 , y1 − y2 ; 2kz ) dz2 cos(2kz z2 )n(z2 ) , |z2 |≥d (C.11) Many-Body Effects on Transmission through a Tunnel Junction 19 where we used eq. (2.21). In the integral over z2 , we leave only the term which is divergent at kz → kF . Then from eq. (2.18) we easily find that ∞ |r0 | ˜ dz2 V (x1 − x2 , y1 − y2 ; 2kz ) sin[2(kz − kF )z2 − arg r0 ] 2π z2 d |r0 | ˜ 1 = V (x1 − x2 , y1 − y2 ; 2kz ) sin(arg r0 ) ln . 2π |kz − kF |d I1 = − (C.12) In the same way, we can calculate the other term in eq. (C.9) and we find that the right side of eq. (C.8) becomes t0 |r0 |2 ikz z ˜ 1 e V (x1 − x2 , y1 − y2 ; 2kF ) ln , (C.13) 2π vF |kz − kF |d where we have replaced 2kz with 2kF in the potential. C.4 The derivation of eq. (2.33) ˜ We easily find that V (x1 − x2 , y1 − y2 ; 2kF ) in eq. (2.31) is given by e2 ˜ K0 V (x1 − x2 , y1 − y2 ; 2kF ) = 2πǫ (x1 − x2 )2 + (y1 − y2 )2 1/2 · 2kF , (C.14) where K0 (r) is the modified Bessel function. Then from eqs. (B.3) and (C.13), we find that eq. (2.31) becomes (1H) ϕk (x) = t0 |r0 |2 e2 ln 2π vF 2πǫ 1 |kz − kF |d ∞ · eikz z 1 1 2 2 exp − 2 (x − x1 ) 2πℓB 4ℓB 1 (x + x1 )(y − y1 ) exp − 2ℓB 2 dx1 dy1 dx2 dy2 × −∞ 1 (y − y1 )2 4ℓB 2 1 1 2 2 × eiky y1 exp − 2 (x1 + ky ℓB ) 1/4 ℓ 1/2 2ℓB π B 1 1/2 (x1 − x2 )2 + (y1 − y2 )2 · 2kF . × 2 K0 2πℓB × exp − (C.15) We first do the integrations over x2 and y2 , and using the formula ∞ (C.16) rK0 (r)dr = 1 , 0 we have (1H) ϕk (x) = t0 |r0 |2 e2 ln 2π vF 2πǫ 1 |kz − kF |d = α1 (B)t0 (1 − |t0 |2 ) ln φ0y (x, y) k 1 |kz − kF |d 1 4kF 2 ℓB 2 eikz z φ0y (x, y)eikz z , k (C.17) 20 Toshihiro Kubo and Arisato Kawabata where 1 e2 1 2π vF 2πǫ 4kF 2 ℓB 2 κ2 = . 4kF 2 α1 (B) ≡ Appendix D: (C.18) The Screened Coulomb Interaction by RPA Below we will consider the case of zero-temperature, and we will not take into account the effects of the barrier on the screening, for simplicity. Suppose we put an external charge Q(x, t) = Qq ei(q·x−ωt) (D.1) in the system, where q is a three-dimensional vector (below k will be a two-dimensional vector in y-z plane, and we define k ± q ≡ (ky ± qy , kz ± qz )). Then the electron density induced by this external charge is given by Q′ (x, t) = χ(q, ω)Q(x, t). (D.2) Here χ(q, ω) is the density response function defined by17) χ(q, ω) = ∞ 1 ˆ V (q) i dt [ρ−q , ρq (t)] eiωt−ηt , (D.3) 0 ˆ where · · · indicates the thermal average, V (q) is the Fourier component of the coulomb interaction potential e2 ˆ V (q) = 2 , ǫq ǫ being the dielectric permittivity of the matter. and ρq is the electron density operator (D.4) ˆ ˆ dxψ † (x)ψ(x)e−iq·x ρq = ak−q † ak exp i = k ℓB 2 2 ℓB 2 qx (2ky − qy ) − (qx + qy 2 ) . 2 4 (D.5) We define the Matsubara Green’s function D(q, ωn ), β D(q, ωn ) = dτ ρq (τ )ρ−q eiωn τ , (D.6) 0 where ωn = πkB T (2n + 1)/ , n being an integer. Then χ(q, ω) can be expressed in terms of D(q, ωn ) by analytic continuation χ(q, ω) = ˆ V (q) D(q, −iω + η) , (D.7) and the dielectric function is defined as ε(q, ω) = ǫQ(x, t) ǫ = . Q′ (x, t) + Q(x, t) 1 + χ(q, ω) (D.8) Many-Body Effects on Transmission through a Tunnel Junction 21 From eq. (D.5), we find D(q, ωn ) = exp − ℓB 2 2 (qx + qy 2 ) 2 β × k1 ,k2 0 dτ ak1 −q † (τ )ak1 (τ )ak2 +q † ak2 eiωn τ exp iℓB 2 qx (k1y − k2y − qy ) . (D.9) Fig. 9. Feynman diagram for D(q, ωn ). In calculating D(q, ωn ) using Feynman diagram, among the various terms we retain only those corresponding to the diagrams shown in Fig. 9. Then we obtain D(q, ωn ) = D 0 (q, ωn ) , ˆ 1 − V (q)D 0 (q, ωn ) (D.10) where D 0 (q, ωn ) = exp − × k 1 β ℓB 2 2 (qx + qy 2 ) 2 νn F 0 (k, νn )F 0 (k − q, ωn + νn ), (D.11) with νn = πkB T (2n + 1)/ , n being an integer and F 0 (k, νn ) is the Matsubara Green’s function for free electrons F 0 (k, νn ) = It is easy to calculate D 0 (q, ωn ): 2 2 D 0 (q, ωn ) = e−lB q⊥ /2 k 1 . iνn − ξk/ fF (ξk) − fF (ξk−q ) , (ξk − ξk−q )/ + iωn (D.12) (D.13) 2 2 2 where fF (ξp) is the Fermi distribution function and q⊥ = qx + qy . From eqs. (D.7), (D.8) and (D.10), the analytic continuation of the dielectric function is given by ˆ ε(q, ωn ) = ǫ 1 − V (q)D 0 (q, ωn ) . (D.14) The screened interaction is given by ˆ Vs (q, ωn ) = e2 q 2 ε(q, ω n) . (D.15) 22 Toshihiro Kubo and Arisato Kawabata As is mentioned in the last part of subsection 2.3, the relevant parameter to the Fock correction is given by eq. (2.28). For small qz , from eq. (D.13) we have 2 D 0 (q, ωn ) = − hence we find that ˆ lim Vs (q, vF qz ) = qz →0 (D.16) e2 2 ǫ q⊥ + κ2 exp − 1 (ℓB q⊥ )2 2 where q ⊥ ≡ (qx , qy , 0) and κ≡ Appendix E: 2 e−lB q⊥ /2 2(vF qz )2 , 2 2 4π 2 lB vF (vF qz )2 + ωn , e2 . 4π 2 ǫ vF ℓB 2 (D.17) (D.18) The derivation of eq. (3.2) First we consider only the Hartree term. The 1st-order Hartree correction to the transmitted wave with cutoff Λ0 can be written as (1H) ϕk (x, Λ0 ) = = dx1 Gk(x; x1 )VH (x1 )ϕ0 (x1 ) k dx2 dy2 1H ′ uk (x , Λ0 ), 2πℓB 2 z dx1 dy1 Gky (x, y; x1 , y1 )φ0y (x1 , y1 ) k (E.1) where x′ = (x1 − x2 , y1 − y2 , z) and u1H (x′ , Λ0 ) ≡ kz dz2 Vs (x1 − x2 )n(z2 ) u0z (z1 ). k dz1 Gkz (z; z1 ) (E.2) Here only u1H (x′ , Λ0 ) is dependent on the cutoff. Within the strip shown in Fig. 5, using linearized kz energy dispersion, the single-electron Green’s function for noninteracting electrons can be written as (see eq. (B.4)) kF +Λ0 Gkz (z; z1 ) = = + + ≡ kF −Λ0 1 ∗ 0 ′ 0′ ′ dkz ukz (z)ukz (z1 ) 0 − ξ0 2π ξk k′ −sΛ0 2π vF −Λ0 1 sΛ0 2π vF −sΛ0 1 Λ0 2π vF sΛ0 (A) Gkz (z; z1 ) ˜′ dkz ˜′ dkz ˜′ dkz u0 ′ +k (z)u0 ′ +k ˜ ˜ k k F z z ∗ F (z1 ) ˜′ kz − kz − kF ∗ u0 ′ +k (z)u0 ′ +k (z1 ) ˜ ˜ k k F z F z ˜′ kz − kz − kF ∗ u0 ′ +k (z)u0 ′ +k (z1 ) ˜ ˜ k k z (B) F z F ˜′ kz − kz − kF (C) + Gkz (z; z1 ) + Gkz (z; z1 ), (E.3) Many-Body Effects on Transmission through a Tunnel Junction 23 ˜′ ˜′ where (A), (B) and (C) denote the regions of −Λ0 ≤ kz ≤ −sΛ0 , −sΛ0 ≤ kz ≤ sΛ0 and sΛ0 ≤ ˜′ kz ≤ Λ0 , respectively. Similarly, the electron density n(z2 ) can be written as kF dkz 0 |u (z2 )|2 2π kz kF −Λ0 −sΛ0 ˜ dkz 0 = |u˜ (z2 )|2 + 2π kz +kF −Λ0 n(z2 ) = ≡n (1H) Thus ukz (A) (z2 ) + n (B) 0 −sΛ0 ˜ dkz 0 |u˜ (z2 )|2 2π kz +kF (E.4) (z2 ). (x′ , Λ0 ) is of the form (1H) ukz (A) (x′ , Λ0 ) = dz1 Gkz (z; z1 ) dz2 V (x1 − x2 )n(A) (z2 ) u0z (z1 ) k (A) dz2 V (x1 − x2 )n(B) (z2 ) u0z (z1 ) k (B) dz2 V (x1 − x2 )n(A) (z2 ) u0z (z1 ) k (B) dz2 V (x1 − x2 )n(B) (z2 ) u0z (z1 ) k (C) dz2 V (x1 − x2 )n(A) (z2 ) u0z (z1 ) k (C) dz2 V (x1 − x2 )n(B) (z2 ) u0z (z1 ). k + dz1 Gkz (z; z1 ) + dz1 Gkz (z; z1 ) + dz1 Gkz (z; z1 ) + dz1 Gkz (z; z1 ) + dz1 Gkz (z; z1 ) (E.5) (1H) We find that the 4th term of the right hand side in eq. (E.5) can be written as ukz (x0 , sΛ0 ), hence (1H) uk z (1H) (1H) (x′ , Λ0 ) = ukz (x′ , sΛ0 ) + δukz (E.6) (x′ , Λ0 ), where (1H) δukz (A) (x′ , Λ0 ) ≡ dz1 Gkz (z; z1 ) dz2 Vs (x1 − x2 )n(A) (z2 ) u0z (z1 ) k (A) dz2 V (x1 − x2 )n(B) (z2 ) u0z (z1 ) k (B) dz2 V (x1 − x2 )n(A) (z2 ) u0z (z1 ) k (C) dz2 V (x1 − x2 )n(A) (z2 ) u0z (z1 ) k (C) dz2 V (x1 − x2 )n(B) (z2 ) u0z (z1 ), k + dz1 Gkz (z; z1 ) + dz1 Gkz (z; z1 ) + dz1 Gkz (z; z1 ) + dz1 Gkz (z; z1 ) (E.7) The 1st and 4th terms of the right hand side in eq. (E.7) are the order of (1 − s)2 , and we neglect them. Here we consider only most divergent terms at low temperature which we used in the calculations of the perturbation theory. Then we have (1H) δukz (x′ , Λ0 ) = t0 (1 − |t0 |2 ) ˜ V (x1 − x2 , y1 − y2 ; 2kF ) ln 2π vF 1 s · eikz z . (E.8) 24 Toshihiro Kubo and Arisato Kawabata We put it into eq. (E.1) and perform the integrations of x and y directions. Then the transmitted wave within 1st Born approximation by Hartree term is written as (1H) (H) ϕk (x) =t0 eikz z φ0y (x, y) + ϕk k (x, Λ0 ) =t0 eikz z φ0y (x, y) k (1H) + ϕk 1 s (x, sΛ0 ) + α1 (B)t0 (1 − |t0 |2 ) ln (1H) ≡ t0 + δt(1H) eikz z φ0y (x, y) + ϕk k · eikz z φ0y (x, y) k (E.9) (x, sΛ0 ), where 1 s δt(1H) ≡ α1 (B)t0 (1 − |t0 |2 ) ln (E.10) . Similarly, the transmitted wave within 1st Born approximation by Fock term is written as (1F ) (F ) ϕk (x) =t0 eikz z φ0y (x, y) + ϕk k (x, Λ0 ) =t0 eikz z φ0y (x, y) k (1F ) + ϕk 1 s (x, sΛ0 ) − α2 (B)t0 (1 − |t0 |2 ) ln (1F ) ≡ t0 + δt(1F ) eikz z φ0y (x, y) + ϕk k · eikz z φ0y (x, y) k (x, sΛ0 ). (E.11) 1 s (E.12) with δt(1F ) = −α2 (B)t0 (1 − |t0 |2 ) ln . Therefore the transmitted wave within the present approximation is of the form (1H) ϕk(x) = {t0 + δt} eikz z φ0y (x, y) + ϕk k (1F ) (x, sΛ0 ) + ϕk where δt = −α(B)t0 (1 − |t0 |2 ) ln 1 s (x, sΛ0 ) , . 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