Resonant reflection of interacting electrons from an impurity in a quantum wire: interplay of Zeeman and spin-orbit effects Rajesh K. Malla and M. E. Raikh arXiv:1804.10283v1 [cond-mat.mes-hall] 26 Apr 2018 Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112 A single-channel quantum wire with two well-separated Zeeman subbands and in the presence of a weak spin-orbit coupling is considered. An impurity level which is split off the upper subband is degenerate with the continuum of the lower subband. We show that, when the Fermi level lies in the vicinity of the impurity level, the transport is completely blocked. This is the manifestation of the effect of resonant reflection and can be viewed as resonant tunneling between left-moving and right-moving electrons via the impurity level. We incorporate electron-electron interactions and study their effect on the shape of the resonant-reflection profile. This profile becomes a two-peak structure, where one peak is caused by resonant reflection itself, while the origin of the other peak is reflection from the Friedel oscillations of the electron density surrounding the impurity. PACS numbers: 73.50.-h, 75.47.-m I. INTRODUCTION Electron states in a ballistic wire in the presence of spin-orbit coupling became the subject of intensive theoretical, see e.g. Refs. 1–6, and experimental7–10 studies almost three decades ago. Initial motivation for these studies was the proposal of a spin transistor by Das and Datta.11 The motivation for the later studies was the proposal12,13 that, in the proximity to a superconductor, the interplay of spin-orbit coupling and Zeeman splitting can lead to the formation of zero-energy bound states at the wire ends. Yet another motivation for the research on the combined action of Zeeman and spin-orbit fields comes from the recent experiments on cold gases.14 Nontriviality of the interplay of spin-orbit coupling and Zeeman splitting manifests itself already in the ballistic transport through the wire. It was predicted1,2 and confirmed experimentally7 that, as a result of this interplay, the dependence of the conductance on the Fermi level can become non-monotonic. Such a “spin gap” develops when the spin-orbit minimum in energy spectrum of a free electron is comparable to the Zeeman splitting. Another nontrivial consequence of the interplay shows up when the spin-orbit coupling is inhomogeneous3–6 . Namely, a step-like inhomogeneity can lead to a full reflection of the incident electron. The underlying physics of the full reflection is the same as the physics of the resonant reflection in the twosubband wire first studied in Refs. 15, 16. It does not require either Zeeman field or spin-orbit coupling. An attractive impurity in a two-subband wire splits off an energy level from the bottom of both subbands. If the Fermi level, lying in the lower subband, coincides with the level split from the upper subband, see Fig. 1, the transport involves multiple virtual visits to this level. As it was first shown in Ref. 15, the outcome of these visits is a reflection rather than resonant transmission as one would naively expect. In a single-channel wire the role of the size-quantization subbands is played by the spin subbands, while the visits to the split-off level are enabled by the spin-orbit coupling. The goal of the present paper is to study the effect of electron-electron interactions on the resonant reflection. For a single-channel interacting wire it is accepted that any weak potential impurity blocks completely the zerotemperature transport through the wire. The theories17 which capture this phenomenon are Luttinger-liquid description and backscattering by the Friedel oscillations in electron gas imposed by an impurity. In the latter case, the role of interactions is simply a conversion of the oscillations of electron density into the oscillations of the potential. As it was first pointed out in Refs. 18,19 (see also later papers Refs. 20,21), the period of the Friedel oscillations matches the Bragg condition for electron at the Fermi level. Thus, the electron is scattered by a compound object consisting of the impurity itself and the oscillating potential, which it creates. The theory of Refs. 18,19 was later generalized to the case of a pair of impurities.22,23 Specifics of the pair is that electron can bounce between the constituting impurities for a long time. As a result of this bouncing, a quasi-local level degenerate with the continuum is formed. For incident electron with energy in resonance with this quasi-local level the transmission coefficient is close to 1. Physically, the results of Refs. 22,23 can be interpreted as follows. When the incident electron is resonantly transmitted, the Friedel oscillations do not form, so that the interactions suppress the transmission only when the Fermi level is spaced away from the resonant level. Contrary to the resonant transmission, in the case of the resonant reflection the Friedel oscillations are the strongest when the Fermi level lies close to the impurity level. Thus, the modification of the resonant reflection profile due to interactions is also strong. This demands a more detailed treatment of partial reflection of electron on the way to the impurity than the renormalizationgroup scheme adopted in Refs. 18–23. Our most spectacular finding is that, for certain phases accumulated by the electron on the way to the impurity, the resonant re- 2 flection from the bare impurity can turn into the resonant transmission. where the characteristic length, 2 x0 = 2m∆ 1/2 , (4) is the de Broglie wave length of the electron with energy ε = ∆. In the dimensionless variables the system Eq. (2) takes the form ∂ 2 ψ1 ∂ψ2 + U0 δ(z)ψ1 − (E + 1)ψ1 = α , 2 ∂z ∂z ∂ψ1 ∂ 2 ψ2 + U0 δ(z)ψ2 − (E − 1)ψ2 = −α . − ∂z 2 ∂z − FIG. 1: (Color online) Schematic illustration of the resonant reflection. An attractive impurity creates bound states under the bottoms of ↓ (red) and ↑ (blue) sub-bands. The binding energy, measured in the units of ∆, is 1 − E0 Weak spinorbit coupling mixes ↓ and ↑ wave functions. As a result, an incident ↑ electron undergoes a resonant scattering, illustrated by a green line. The result of the scattering is almost full reflection rather than conventional resonant transmission. II. In the presence of the Zeeman field and spin-orbit coupling, the Hamiltonian of a wire has the form   2 2 k iγkx − 2mx − ∆  ˆ  (1) H= , 2 2 kx −iγkx − 2m + ∆ where m is the electron mass, 2∆ is the Zeeman splitting, and γ is the spin-orbit coupling strength. We assume that the impurity potential is short-ranged, V (x) = V0 δ(x). The system of coupled equations for ↑ and ↓ components of the spinor reads ∂ 2 ψ1 ∂ψ2 , + V0 δ(x)ψ1 − (ε + ∆)ψ1 = γ 2 2m ∂x ∂x 2 ∂ 2 ψ2 ∂ψ1 − + V0 δ(x)ψ2 − (ε − ∆)ψ2 = −γ . 2m ∂x2 ∂x 2 − (2) Since the energy of the incident ↑ electron in resonance with impurity level of ↓ electron is close to ∆, see Fig. 1, it is convenient to introduce the following dimensionless variables z= α= 2mx0 2 x , x0 γ, U0 = E= 2mx0 2 ε , ∆ V0 , Without impurity, the solutions of the system Eq. (5) in the domain −1 < E < 1 correspond to propagation of ↑ spin component and the decay of ↓ spin component, see Fig. 1. Due to spin-orbit coupling, both components of corresponding spinors are nonzero,         ψ1 ψ1 1 D   =   eiqz ,   =   e−κz , (6) ψ2 iC ψ2 1 where the wave vector, q, the decay constant, κ, and the components, C and D, of the spinors are given by q(E) = (1 + E)1/2 , κ(E) = (1 − E)1/2 , 1 1 C = αq, D = ακ. 2 2 RESONANT REFLECTION (3) (5) (7) Coefficients C and D describe the admixture of the opposite spin projection due to spin-orbit coupling. In the presence of impurity, the general solution at z < 0 has a form         D ψ1 1 1  e−iqz + r2   eκz ,   =   eiqz + r1  1 −iC ψ2 iC (8) which is the combination of the solutions Eq. (6). First two terms describe the incident and the reflected ↑ waves, while the third term describes the solution corresponding to ↓, which decays at z → −∞. The corresponding solution for z > 0 reads       ψ1 1 −D   = t1   eiqz + t2   e−κz . (9) ψ2 iC 1 The first term describes the transmitted ↑ wave, while the second term describes the decay of ↓ component. Although the parameters C and D are proportional to α, and thus are small due to the weakness of the spinorbit coupling, it is these admixtures that are responsible for the resonant reflection. To capture this effect, we follow the standard procedure and calculate the reflection and transmission coefficients from the system of boundary conditions at z = 0. 3 Continuity of the wave function Eqs. (8) and (9) yields two conditions 1 + r1 + r2 D = t1 − t2 D, iC(1 − r1 ) + r2 = iCt1 + t2 . (10) The other two conditions come from the discontinuity of the derivatives, ∂ψ1 and ∂ψ2 , at z = 0. Integrating the ∂z ∂z system Eq. (5) near z = 0, we get is quadratic in spin-orbit coupling strength. This allows one to simplify Φ+ to tan−1 U0 . Then Φ+ can be iden2q tified with the scattering phase of ↑ electron from the impurity in the absence of spin-orbit coupling. Turning to the phase Φ− , we note that the small parameter α2 in the expression for λ allows one to neglect the term qλ in the denominator. Then we see that, for attractive impurity, U0 < 0, this denominator turns to zero at energy E = E0 determined by the condition iqt1 + κt2 D − iq(1 − r1 ) + κr2 D = U0 (t1 − t2 D), −qCt1 − κt2 − − qC − qCr1 + κr2 = U0 (iCt1 + t2 ). (11) Simplifying the above boundary conditions by introducing R2 = Dr2 , T2 = Dt2 , and λ = CD, we get R2 + T2 = t1 − r1 − 1, R2 − T2 = iλ(t1 + r1 − 1), (12) iq(t1 + r1 ) − U0 t1 − iq = R2 κ − (κ + U0 )T2 , (κ + U0 )T2 + κR2 = −λ − it1 (iq − U0 ) − q(1 + r1 ) . (13) Since we are interested in the reflection and transmission coefficients, r1 and t1 , it is convenient to express R2 and T2 from the system Eq. (12) and substitute them into the system Eq. (13), which assumes the form κ(E0 ) = |U0 | . 2 This condition expresses the fact that in the absence of spin-orbit coupling, the energy position of the level of ↓ electron in the potential U0 δ(z) is E = E0 , see Fig. 1. To establish the energy width, Γ, of the resonance, we recast the expression for tan Φ− into the form (1 − E 2 )1/2 + |U0 | 1 2 2 α (1 − E 2 )1/2 |U0 | . U2 8 1 − E − 40 (20) 2 U0 Near the resonance, E = E0 = 1 − 4 , the expression Eq. (20) assumes the conventional Breit-Wigner form tan [Φ− (E)] = tan [Φ− (E)] = i U0 2 q − λ(κ + U0 2 ) + q − λ(κ + U0 2 ) − i U0 2 t1 + r1 = t1 − r1 = κ+ κ+ U0 2 U0 2 , (14) . U0 (15) + qλ − iλ U0 2 + qλ + iλ |r1 |2 = sin2 (Φ− − Φ+ ), |t1 |2 = cos2 (Φ− − Φ+ ), (16) where U0 1 2 tan−1 2 q − λ(κ + Φ− = 1 tan−1 2 κ+ U0 2 ) λU0 2 U0 2 + , (21) where Γ is given by Γ= α2 |U0 |3 . 16 (22) qλ . 1 2 α (1 − E 2 )1/2 4 U2 With binding energy of ↓ electron being 40 , we see that the width, Γ, is much smaller than this binding energy, which justifies the expansion near the resonance. If the bound state in the potential U0 δ(z) is shallow, i.e. U0 1, we can replace tan−1 in expression for Φ+ by the argument. After that, the final expression for the energy-dependent reflection coefficient assumes the form |r1 (E)|2 = sin2 tan−1 (17) Γ− = Until now the calculation was exact. Weakness of spinorbit coupling, quantified by the condition α 1, was used in the explicit expressions for q and κ. We will now use this condition to simplify the phases Φ+ and Φ− . First, we note that the dimensionless parameter λ = CD = Γ , E0 − E 2 We see that the absolute values of t1 + r1 and t1 − r1 are equal to 1. Then it is convenient to cast the solution of the system Eq. (14) into the form Φ+ = (19) (18) |U0 | 2q (E0 Γ E0 − E − |U0 | 2q 2 − E) (E0 − E)2 + Γ2 . (23) It follows from Eq. (23) that |r1 (E)|2 has a characteristic Fano shape24 . Near the resonance, E = E0 , it is a Lorentzian with the width, Γ. As the energy is swept through E0 , the reflection coefficient passes through zero (antiresonace) before returning to its non-resonant value |2 |r1 |2 = |U02 . 4q 4 III. INCORPORATING THE ELECTRON-ELECTRON INTERACTIONS As it was explained in the Introduction, the effect of interactions is more pronounced in the case of resonant reflection than in the case of resonant transmission.22,23 The reason is that the amplitude of the Friedel oscillations is proportional to the reflection amplitude18,19 which, for resonant reflection, is close to 1. On the other hand, the Friedel oscillation of electron density creates perturbations which play the role of the “Bragg mirrors” for incident and transmitted electron waves. As a result of Friedel oscillations being strong, each Bragg mirror is highly “reflective”. This suggests to incorporate the effect of attenuation, caused by the mirrors, more accurately than in Refs. 22, 23. The process of electron reflection from a compound object consisting of three scatterers, two Bragg mirrors and impurity between them, is illustrated in Fig. 2. The rigorous way to describe this reflection analytically is by employing the scattering matrices of each scatterer relating the amplitudes of the incoming and outgoing partial waves. These matrices are defined as follows Left mirror Right mirror FIG. 2: (Color online) Schematic illustration of the electron scattering from impurity “dressed” by Friedel oscillations, which play the role of the Bragg mirrors. The incident electron, i, can be reflected by the left mirror, by the impurity, or by the right mirror.    i1 tL  = ∗ o −rL      i2 t1 i  ,   =  ∗ t∗ o1 o1 −r1 L rL The amplitude r1 in Eq. (24) was found in the previous section. Two remaining amplitudes, rL and rR , will be calculated later. Excluding the intermediate amplitudes, i1 , i2 , o1 , o2 from Eq. (24), we find the expression for the net amplitude reflection coefficient of the compound scatterer ref f = − ∗ ∗ ∗ ∗ o r∗ + r1 + rR + rL r1 rR = L ∗ ∗ ∗ . i 1 + r1 rR + rL r1 + rL rR (25) To analyze this expression, we express the power reflection coefficient, |ref f |2 , via the magnitudes of the reflection coefficients r1 , rL , and rR and obtain |ref f |2 = 1−|tef f |2 = 1− 1 − |rBragg |2 2 1 − |r1 |2 1 + |rBragg |2 + 2|rBragg ||r1 | cos β 2. (26) In Eq. (26) we took into account that, unlike Refs. 22, 23, there is a symmetry between the left and right mirrors, so that the magnitudes, |rL | and |rR | are equal to each other and are denoted with |rBragg |. The phase, β, is the combination of the phase Φ− , defined by Eq. (16), and the phase, ΦBragg , accumulated in the course of the reflection from the mirror. We will see that this phase      i1 o tR  ,   =  ∗ o2 t∗ o2 −rR 1   i2  . t∗ 0 R r1 rR (24) is big and depends strongly on the energy. Thus, we average Eq. (26) over β using the identity 1 (a + cos β)2 = β (a2 a . − 1)3/2 (27) The result of this averaging reads 2 |ref f | = 1− 1 − |rBragg |2 2 1 + |rBragg |2 1 − |r1 |2 2 (1 − |rBragg |2 ) + 4|rBragg |2 (1 − |r1 |2 ) 3/2 . (28) It is also instructive to express the effective power transmission coefficient via the partial transmission coefficients |t1 |2 and |tBragg |2 . One obtains |tef f |2 = |tBragg |4 2 − |tBragg |2 |t1 |2 |tBragg |4 + 4 (1 − |tBragg |2 ) |t1 |2 3/2 . (29) Since the transmission, |tBragg |2 , is strongly dependent on the position of the Fermi level, EF , with respect to the resonant energy level, E0 , the magnitude of |tBragg |2 falls off with increasing (EF − E0 ). Then one would expect 5 0.10 ˆ where VH (z) and Vex are the Hartree and the exchange terms, respectively. When the interaction is shortranged, one can consider only the Hartree term, since the exchange term causes only a modification of the interaction constant.18 The other consequence of the interaction being short-ranged is that the Hartree potential is proportional to the modulation of the electron density created by the Friedel oscillations18 , i.e. it has the from 0.08 0.06 0.04 0.02 0.00 0.0 0.2 0.4 0.6 0.8 1.0 VH (z) = FIG. 3: (Color online) Effective power transmission coefficient of the impurity dressed by the Friedel oscillations is plotted from Eq. (29) versus the transmission of the Bragg mirrors for |t1 |2 = 0.01 (blue) and |t1 |2 = 0.04 (red). that |tef f |2 grows monotonically with increasing |tBragg |2 and approaches |t1 |2 . The reasoning behind this expectation is that the scattering by the Bragg mirrors becomes inefficient for large (EF − E0 ). Remarkably, the dependence of |tef f |2 , described by Eq. (29), is non-monotonic. As illustrated in Fig. 3, this dependence has a maximum. For small transmission of the impurity, |t1 |2 1, the position of maximum is easy to calculate analytically. It is |tBragg |4 = 8t2 . Note that the value |tBragg |4 has a mean1 ing of the net transmission of two mirrors. Thus, the maximum occurs when the transmissions of the impurity and of the two mirrors are equal within numerical factor. Substituting |tBragg |4 = 8t2 into Eq. (29), we find the 1 maximal value of the effective power transmission |tef f |2 max = 2 |t1 |. 33/2 µ(EF ) cos(2qF |z|), qF |z| (32) where qF is the Fermi momentum. The magnitude of the electron-electron interactions as well as the energy dependence of |r1 |, responsible for the Friedel oscillations, are encoded into the constant, µ, which we will specify later. The main difference between our approach and the approach of Ref. 18 is that we find an asymptotically exact solution of Eq. (31), while in Ref. 18 it was solved perturbatively. The reason why asymptotically exact solution can be found is that the amplitude of VH (z) falls off slowly with z, so that the relevant values of qF z are big. This, in turn, suggests to search for ψ1 (z) in the form ψ1 (z) = A+ (z)eiqF z + A− (z)e−iqF z , (33) where the functions A+ and A− change slowly with z, so that their second derivatives can be neglected. Upon substituting Eq. (33) into Eq. (31), and neglecting nonresonant terms exp(±3iqF z), we arrive to a coupled system of the first-order equations (30) 2 We see that this value is much bigger than |t1 | . The origin of the maximum is that the dominant contribution to the phase-averaged transmission, |tef f |2 , comes from the phases, β, in Eq. (26) for which the denominator is close to zero. In other words, while the impurity alone acts as a reflector, adding of the two Bragg mirrors can lead to the resonant transmission. Naturally, the values of |t1 |2 and |tBragg |2 are not independent. It is the reflection from the impurity that controls the magnitude of the Friedel oscillations. To analyze the behavior the effective transmission with energy, E, of the incident electron and with EF , we need to specify the analytical form of |tBragg |2 . This is done in the next section. µ ∂A+ (z) 2 + A− (z) = E + 1 − qF A+ (z), ∂z 2z ∂A− (z) µ 2 2iqF + A+ (z) = E + 1 − qF A− (z). (34) ∂z 2z −2iqF It appears that this system can be solved exactly for arbitrary interaction strength, µ. To see this, we first perform a rescaling y=z 2 E + 1 − qF 2qF , (35) and then introduce the auxiliary functions a(y) = A+ (y) + iA− (y), b(y) = A+ (y) − iA− (y). (36) Then the system Eq. (34) reduces to IV. TRANSMISSION OF THE BRAGG MIRROR In the presence of electron-electron interactions, propagation of electron through the mirror is described by the Schr¨dinger equation o − ∂ 2 ψ1 ˆ + VH (z)ψ1 + Vex ψ1 = (E + 1) ψ1 , ∂z 2 (31) ∂a µ + a(y) = ib(y), ∂y 4qF y ∂b µ − b(y) = ia(y). ∂y 4qF y (37) In the rescaled form, the system contains a single dimenµ sionless parameter, 4qF . As a next step, we substitute 6 b(y) from the first equation into the second equation and arrive to the following second-order differential equation µ 1 − 4( 4qF + 1 )2 ∂2a 2 + 1+ a(y) = 0. ∂y 2 4y 2 (38) To conclude this Section, we present the microscopic expression for the parameter µ in terms of the Fourier components of the interaction potential. This expression follows from the expression for the amplitude of the oscillations of the electron density, calculated in Appendix A, and has the form The general solution of this equation can be presented as a linear combination µ a(y) = y 1/2 c1 J 4q F +1 2 µ (y) + c2 J− 4q F −1 2 (y) , (39) µ µ 1 where J 4q + 2 and J− 4q − 1 are the Bessel functions. At 2 F F large y both Bessel functions oscillate, so that the value of the transmission coefficient is governed by the ratio c1 /c2 . This ratio is determined by the condition that at small y = yc , where the Friedel oscillations are terminated (see Appendix A), the amplitude of the reflected wave vanishes. The final expression for the transmission coefficient, reads 1/2 (2πyc ) tBragg = µ J 4q −1 2 F µ J 4q (yc )e F πµ i 8q F −1 2 µ (yc )J− 4q F µ + J− 4q −1 2 F −1 2 (yc ) πµ −i 8q (yc )e . (40) F The details of the derivation are presented in the Appendix B. The result Eq. (40) can be simplified when yc is small. Then we can use the small-argument asymptotes of the Bessel functions and obtain 1 tBragg = cosh µ 4qF . (41) ln yc In deriving this expression we took into account that the interactions are weak in the usual sense, namely that the typical interaction energy is much smaller than the Fermi energy. This condition ensures that qµ is small. F Concerning the value of yc , in Appendix A it is demonstrated that the Friedel oscillations are terminated at z = zc ∼ q0 . Using the relation Eq. (35), we find that, Γ within a numerical factor, yc is given by yc = E − EF . Γ (42) We see that in the interesting limit when the Fermi level is close to the resonance yc is indeed small. Equations (41) and (42) describe how the transmission of the Bragg mirror evolves with energy. Indeed, the argument of the hyperbolic cosine is the product of a µ small factor 4qF and a big factor ln yc . If this product is small, e.g. when the interactions are weak, then the transmission coefficient is close to 1. On the contrary, if the product is big, we have 2|E − EF | Γ i.e. the mirror is highly reflective. (44) V (0) − V (2qF ) . 2π vF (45) where ν is given by ν= The term V (0) comes from the exchange potential, while V (2qF ) comes from the Hartree potential; vF stands for the Fermi velocity. Note that the transmission, tBragg , is full not only in the absence of electron-electron interactions. If the interactions are present, but there is no reflection from the impurity, r1 (EF ) = 0, then transmission is also full. This is natural, since in the absence of reflection, the Friedel oscillations do not form. V. ENERGY DEPENDENCE OF THE EFFECTIVE REFLECTION In Eq. (29) both t1 and tBragg are the functions of energy. While t1 is a growing function of energy, tBragg grows µ with increasing |E − EF |. In addition, the power, 4qF , in Eq. (43) depends on the difference |E0 − EF |, see Appendix A. Concerning the overall dependence |ref f (E)|2 , the situation is most transparent when the Fermi level lies away from the resonance. Then the presence of the Bragg mirrors manifests itself only near E = E F . Bragg mirrors cause a spike in the reflection. When the spacing between EF and E0 is much smaller than the width of the resonance, there are two features in |ref f (E)|2 -dependence that are present for any interaction strength. Firstly, the reflection is full for any position of the Fermi level when the energy of the incident electron is E = E0 . This is because the electron is fully reflected even in the absence of the Friedel oscillations. Secondly, |ref f (E)|2 = 1 at E = EF due to full reflection from the mirror. Thus, in the domain −EF < E < 0, the reflection coefficient should pass through a minimum. Indeed, this minimum is present in the curves |ref f (E)|2 plotted from Eqs. (28) and (41) in Fig. 4. VI. tBragg = νqF |r1 (EF )|2 , 2 µ= DISCUSSION |µ| 4qF 1, (43) (i) To establish the relation between our results and those obtained within the renormalization-group approach18–23 we assume that the reflection of the Bragg 7 a. 1.0 0.8 0.6 0.4 0.2 b. -2 -1 0 1 2 -1 0 1 2 1.0 0.8 0.6 0.4 0.2 0.0 -2 FIG. 4: (Color online) (a) In the absence of interactions, the effective power reflection coefficient is a Lorentzian, |ref f |2 = 2 −1 1 + (E−E0 ) (black dashed line). With interactions, full Γ2 reflection takes place at two energies E = E0 as a result of scattering from the impurity and at E = EF as a result of scattering from the Bragg mirror. This is illustrated by red and blue curves plotted from Eqs. (28) and (41) for (E0 − EF ) = 0.8Γ and (E0 − EF ) = 0.6Γ, respectively. The µ interaction strength in both curves is chosen to be 4qF = 0.4. (b) Scattering by two Bragg mirrors can, for certain energies, transform the resonant reflection into the resonant transmission. While the plot (a) shows the average over the phase, β, the plot (b) shows the reflection profile for the same parameters prior to averaging. mirrors is weak and expand Eq. (26) with respect to |rBragg |2 . This yields |ref f |2 − |r1 |2 = 4 1 − |r1 |2 |rBragg |2 + |r1 ||rBragg | cos β . (46) The second term in the brackets contains the first power of |rBragg |, unlike the first term which contains |rBragg |2 . This second term comes from interference of incident and reflected waves passing through the Bragg mirror. If we average Eq. (46) over β, the second term will disappear. Then it is the first term, 1 − t2 , that will deBragg scribe the reduction of the transmission of the impurity due to electron-electron interactions. As follows from Eq. 41, |rBragg |2 is proportional to |r1 |2 and contains µ ln yc . Then Eq. (46) reproduces the main result of Ref. 18. In Ref. 18 this result is subsequently converted to the renormalization-group equation. We studied the limit in which both |r1 | and |rBragg | are close to 1. Then the denominator in Eq. (26) is close to zero when cos β = −1. Definitely, the expansion with respect to |rBragg | and subsequent summation of the leading terms, which is the essence of the renormalization-group approach, does not capture this resonant transmission. (ii) Adopting of the renormalization group approach in Refs. 18–23 relies on the assumption that the coefficients of the expansion of |tef f |2 is powers of ln(|E − EF |) fall 1 off as n! . Our calculation is equivalent to the summation of all the orders of the expansion and confirms this assumption. (iii) The form Eq. (23) of the resonant reflection is the same as for the resonant tunneling between the two electrodes via a localized state located between the electrons. This suggests the interpretation of the resonant transmission as resonant tunneling between left-moving and right-moving electrons. If this interpretation is correct, the width, Γ, calculated from the golden rule should coincide with Eq. (22), and, in particular, should be pro3 portional to U0 . Taking into account that the normalized wave function of the localized state has the form ∂ ψ2 (z) = κ1/2 exp(−κ|z|), the matrix element of α ∂z between ψ2 (z) and the right-moving plane wave, exp(iqz), is given by ∞ 1/2 iακ q dz exp iqz − κ|z| = 2i qκ3/2 . q 2 + κ2 (47) −∞ One can neglect κ2 in the denominator. Then the square of the matrix element is proportional to κ3 and thus to 3 U0 , since, at resonance, κ = U0 . 2 (iv) There is a question whether the attenuation of electron wave functions upon passage of the Bragg mirrors disturbs the shape of the Friedel oscillations. It is important that this disturbance is negligible. Qualitatively, this follows from the fact that many states with E < EF are responsible for the formation of the Bragg mirrors, while only the states with |E − EF | νΓ are strongly affected by the Bragg mirrors. (v) Another question is why we did not take into account the Friedel oscillations originating from the electron reflection within the same sub-band. Indeed, while the Friedel oscillations caused by the resonant reflection develop at large distances zc ∼ q0 , “non-resonant” Γ Friedel oscillations start at much smaller z ∼ 1. To answer this question one should estimate the contribution to the reflection coefficient within the domain 1 < z < zc , where non-resonant Friedel oscillations dominate. The U0 amplitude of the these oscillations is ∼ qF and they fall U0 off as 1/z. This leads to the estimate qF ln(zc ) as in Ref. U0 1, the weakness of non-resonant reflec18. Since qF tion cannot be compensated by the logarithmically big factor ln q0 , it is for this reason we have neglected the Γ Friedel oscillations originating from the reflection within the same sub-band. 8 (vi) Our main finding is that, for weak transmission through a single Bragg mirror, the net transmission from two Bragg mirrors and the impurity can be close to one. This enhancement of the net transmission takes place when the “Fabry-Perot” condition cos β ≈ −1 is met. Then the denominator in Eq. (26) becomes small. This happens near certain distinct energies of incident electron. Averaging over the phase, β, employed above, requires that there are many such energies within the interval |E0 − EF |. To verify that this is the case, consider the contribution to β coming from the factor exp(iqF z) in Eq. (33). As an estimate for z in this factor one should take the effective length of the Bragg mirror where the reflection is formed. From Eq. (38) we see that this length is determined by the condition y 1. At these values of µ y the product y 1/2 J 4q + 1 (y) saturates meaning that the 2 F formation of the Bragg reflection is complete. The conqF dition y 1 transforms into the condition z E−EF . Thus, the contribution to β from the phase, ΦBragg , accumulation in the course of traveling through the mirror is of the order of (E − EF )−1 . In the relevant domain |E0 − EF | Γ this phase goes through (2n + 1)π many times. Acknowledgements We are strongly grateful to E. G. Mishchenko for a number of illuminating discussions. The work was supported by the Department of Energy, Office of Basic Energy Sciences, Grant No. DE- FG02-06ER46313. where q0 = (1 + E0 )1/2 , see Eq. (7). Upon measuring q from qF and introducing new variables, u = 2q0 qF − q q0 − qF , u0 = 2q0 , Γ Γ Eq. (A2) assumes the form 2q0 qF Γ Γ δn(z) = 2πq0 The scattering of electrons from the impurity modifies the electron densities around the impurity. In the presence of electron-electron interaction this modulation of density leads to an additional scattering, which we call “Bragg mirror” in the main text. This scattering barrier is also called Hartree potential, ∞ V (z − y)δn(y)dy, VH (z) = 1 + (u + u0 ) 0 × cos 2|z| qF − where V (z − y) is the interaction potential and δn(y) is the fluctuation of the density. Assuming interaction to be short ranged, V (z − y) = ν δ(z − y), we see that the Hartree potential takes the form, VH (z) = ν δn(z). Now, the modulation of the electron density, δn(z), which depends on the reflection coefficient, r1 , reads 1/2 2 1 Γ u + tan−1 2q0 u + u0 . (A4) It is convenient to separate the contributions proportional to sin(2q|z|) and to cos(2q|z|). This yields 2q0 qF Γ δn(z) = Γ 2πq0 du 0 (u + u0 ) cos 2|z| qF − 1 2× 1 + (u + u0 ) Γ u 2q0 −sin 2|z| qF − Γ u 2q0 . (A5) The shift, Γ 2q0 u, of the arguments of both cosine and sine Γ|z| q0 u and cos Γ|z| q0 u in the nu- merator. For Γ|z| 1, both terms rapidly oscillate with q0 u. Without u-dependence of the prefactor, the contribution from the cosine term will vanish. With the prefactor the contribution of this term remains much smaller than the contribution of the sine term. Retaining only the sine-term we get 2q0 qF Γ Γ δn(z) = cos (2qF |z|) 2πq0 sin du 0 (A1) −∞ 1 du leads to the factors sin Appendix A: Magnitude of the Friedel Oscillations (A3) Γ|z| q0 u 2. 1 + (u + u0 ) (A6) For Γ|z| 1 we can replace the upper limit of the inq0 tegral by infinity and neglect the u-dependence of the denominator. This leads to the final answer δn(z) = |r1 (EF )|2 cos (2qF |z|) , 2π|z| (A7) where we have used the fact that |r1 (EF )|2 is (1 + u2 )−1 . 0 Note that, unlike the conventional Friedel oscillations18 , dq δn(z) = 2Re(r1 (q) e2iqz ) Eq. (A7) contains the second power of |r1 (EF |. Extra π power originates from the phase of the cosine in Eq. (A4), 0 qF which is strongly energy-dependent. dq Γ Γ −1 The most important outcome of the above analysis is = cos 2q|z| + tan , 2 1/2 π q0 − q 2 that the Friedel oscillations are terminated at rather large 2 + (q 2 − q 2 )2 Γ 0 0 distances z = zc ∼ q0 . We have used this value as a cutoff (A2) of log-divergence inΓthe main text. qF 9 Appendix B: Calculation of transmission coefficient from more rigorous approach Substituting the general form Eq. (39) of a(y) in the system Eq. (37) we find the following general form of b(y) µ b(y) = −iy 1/2 c1 J 4q F −1 2 µ (y) − c2 J− 4q +1 2 F (y) . (B1) Once a(y) and b(y) are known, the incident amplitude, 1 A+ (y) = 2 [a(y) + b(y)], and the reflected amplitude 1 A− (y) = 2i [a(y) − b(y)] can be expressed as a combination of the Bessel functions A+ = y By definition, the amplitude transmission coefficient of the mirror, tBragg , is the ratio of the values of A+ at y = yc and at large y. Using the ratio Eq. (B6) and Eqs. (B4), (B5) we arrive to Eq. (40) of the main text. Appendix C: Alternative derivation of resonant reflection It is instructive to trace how the resonant reflection of ↑ electrons emerges from the closed equation for the spin component ψ1 (z). To derive this equation, we introduce the Fourier transform, ∞ 1 ϕ2 (p) = 2π 1/2 2 µ c1 J 4q F µ +c2 J− 4q F −1 2 +1 2 µ (y) − iJ 4q −1 2 F µ (y) + iJ− 4q F (y) (y) +1 2 , (B2) F −1 2 µ (y) − iJ− 4q ∞ +1 2 F (C1) we rewrite the second equation of the system Eq. (5) in the form (y) U0 α ψ2 (0) = − 2π 2π (p2 + κ2 )ϕ2 (p) + y 1/2 µ µ A− = c1 J 4q + 1 (y) + iJ 4q − 1 (y) 2 2 F F 2i µ +c2 J− 4q dz ψ2 (z) exp(−ipz), −∞ dz ∂ψ1 exp(−ipz). ∂dz −∞ . (B3) (C2) Expressing ϕ2 (p) and substituting it into the selfconsistency condition ∞ In the limit y → ∞, the behavior of A+ and A− is the following ψ2 (0) = dp ϕ2 (p), (C3) −∞ 1 −i πµ i πµ c2 e 8qF − ic1 e 8qF eiy , (2π)1/2 −i i πµ −i πµ A− = c2 e 8qF + ic1 e 8qF e−iy . 1/2 (2π) A+ = we find (B4) µ µ 1 J± 4q − 2 (y), so the For small y, we have J± 4q + 1 (y) 2 F F asymptotic expressions for A+ and A− can be written as 1/2 A− = A+ = y y 2i µ ic1 J 4q F −1 2 µ (y) + c2 J− 4q F −1 2 2 − ic1 J (y) + c2 J µ − −1 4qF 2 dz ∂ψ1 −κ|z| e . ∂z (C4) −∞ Substituting Eq. (C4) into Eq. (C2), we express ϕ2 (p) in terms of ψ1 (z) (y) , ∞ 1/2 µ −1 4qF 2 ∞ α ψ2 (0) = − U0 + 2κ (y) (B5) α ϕ2 (p) = − 2 + κ2 ) 2π(p dz −∞ To find the transmission of the Bragg mirror we need to know the ratio c1 /c2 . This ratio is determined by the condition that the Bragg mirror exists only for y > yc . Correspondingly, the amplitude A− at y = yc is zero. This yields µ J− 4q − 1 (yc ) c1 2 F =i . µ 1 (yc ) c2 J 4q − 2 ∂ψ1 −ipz U0 e − e−κ|z| ∂z U0 + 2κ (C5) Multiplying Eq. (C5) by exp(ipz) and integrating over p, we get the following expression for ψ2 (z) ∞ α − ψ2 (z) = 2κ −∞ ∂ψ1 −κ|z−z1 | U0 e−κ|z| dz1 e + ∂z1 U0 + 2κ ∞ dz1 −∞ (B6) ∂ψ1 −κ|z1 | e . ∂z1 (C6) F  ∂ 2 ψ1 α2 ∂  − + U0 δ(z)ψ1 − (E + 1)ψ1 = ∂z 2 2κ ∂z ∞ −∞ .  ∂ψ1 −κ|z−z1 | U0 e−κ|z|  dz1 e − ∂z1 U0 + 2κ ∞ −∞  ∂ψ1 −κ|z1 |  dz1 e . ∂z1 (C7) 10 The term responsible for the resonant reflection is the second term in the right-hand side. Near the resonance, it is much bigger than the first term. The term U0 δ(z) in the left-hand side describes a non-resonant scattering from the impurity. Neglecting these terms we get  ∂ 2 ψ1 α2 U0  − − (E + 1)ψ1 = ∂z 2 2 U0 + 2κ ∞ −∞ We see that the right-hand side is a discontinuous function of z. This fact constitutes the origin of the resonant reflection. For example, if we integrate Eq. (C8) near z = 0, we will see that, unlike conventional scattering, the derivative, ∂ψ1 , is continuous at the position of im∂z purity. This translates into the relation t1 = 1−r1 , which is nothing but Eq. (14). To derive the second equation, Eq. (15), one should notice that ψ1 (z) is present in the right-hand side only under the integral, so that the explicit solution of Eq. (C8) can be readily found. This solution also contains t1 and r1 . Then Eq. (15) emerges as a self-consistency condition. Appendix D: Smallness of the transmission through the Bragg mirror  ∂ψ1 −κ|z1 |  −κ|z| dz1 e e sign(z). ∂z1 (C8) It is seen from Eq. (D1) that the functions A± oscillate at z > zt , where the turning point zt is given by zt = |µ| . 2 2|E + 1 − qF | (D2) For smaller z, A± (z) are the combinations of growing and decaying exponents. This behavior is sustained in the interval zc < z < zt , where zc ∼ 1/Γ is the point where the Friedel oscillations are terminated (see Appendix A). 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