mark i. visscher and gerrit e. w. bauer- mesoscopic charge-density-wave junctions

8/3/2019 Mark I. Visscher and Gerrit E. W. Bauer- Mesoscopic charge-density-wave junctions

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Mesoscopic charge-density-wave junctions

Mark I. Visscher and Gerrit E. W. BauerDelft University of Technology, Faculty of Applied Physics/Delft Institute for Microelectronics and Submicron Technology (DIMES),

Department of Theoretical Physics,

Lorentzweg 1, 2628 CJ Delft, The Netherlands

Received 18 December 1995; revised manuscript received 29 February 1996

We investigate theoretically charge-density waves CDWs in mesoscopic heterostructures for the configu-ration where the one-dimensional chains are oriented normal to the interfaces. Based on Bogoliubovde

Gennes equations and suitable boundary conditions, ground state properties of phase coherent CDW systems

are calculated in the mean-field approximation. It is shown that in a charge-density-wave/normal-metal/charge-

density-wave junction the CDW condensates couple through the normal metal region by means of Friedel

oscillations. Geometrical resonance effects are predicted. We relate the phase-dependent energy of the junction

to a momentum current, carried by electron-hole pairs. The effects of pinning potentials at the interfaces are

also considered. Finally, we discuss the analogies with superconductor and ferromagnet junctions. S0163-

18299607924-6

I. INTRODUCTION

The coupling of spatially separated ground states in

phase-coherent heterostructures is known to cause many re-

markable effects. The Josephson effect and Andreev reflec-

tion in superconductor junctions,1,2 the nonlocal exchangecoupling,3 and the associated giant magnetoresistance4 inmagnetic multilayers are only a few examples of the strikingphenomena that occurr in such systems. In this context it issurprising that the concept of charge-density waves CDWsin mesoscopic junctions has apparently escaped attention. Inthe mesoscopic regime the coherence length of the CDWbecomes comparable to the sample size. A search for alter-nate quantum size effects in mesocopic devices like hetero-

structures of CDWs and normal metals might lead to funda-mental insights into the physics of charge-density waves.

In the 1950s, Peierls suggested that strongly anisotropicmetals are unstable with respect to lattice deformations. Thisinstability is known as the Peierls transition. The groundstate consists of a periodic electron density modulation,called the charge-density wave, accompanied by a static pe-riodic lattice distortion. Frohlich5 noted that in incommensu-rate systems the degeneracy of the condensate with respect torigid translations generates a symmetry restoring collectivemode of transport. The sliding motion of the CDW producesremarkable electrical behavior, like non-Ohmic conductivity6

and narrow band noise.7

Charge-density waves in bulk systems have already re-ceived much attention in the last few decades for an over-view see Gruner8. Unlike mesoscopic superconducting de-vices, there has not been much work reported on CDWs inmesoscopic systems so far. We are only aware of sporadiccontributions in charge-density-wave tunnel junctions withina tunneling Hamiltonian approach.911 Experiments on me-soscopic CDW samples are expected to be conducted in thenear future.14 Stimulated by these experimental efforts, weinvestigate theoretically the ground state properties of vari-ous heterostructures in the mean-field approximation. Trans-port properties such as quasiparticle conductance and sliding

CDW motion will be treated subsequently in Refs. 12 and

13.The basis for our calculations are the Bogoliubovde

Gennes15 BdG equations for CDWs. The BdG equationsare known to successfully explain many experiments on spa-tially inhomogeneous superconductor structures. In Sec. IIthe Bogoliubovde Gennes equations for CDW systems arederived from the mean-field approximation. We also proposeboundary conditions which are necessary to solve the BdGequations in heterostructures. In the geometry where the one-dimensional chains are oriented normal to the interfaces, wecalculate in Sec. III the density of states and the pinningenergy of normal-metal/charge-density waves N/C andC/N/C junctions with barrier potentials of arbitrary strengthat the interfaces. We consider the CDW proximity effect and

the mechanism of phase coupling by Friedel oscillations. InSec. IV we will point out the analogies with ferromagnet andsuperconductor junctions. Section V summarizes our conclu-sions.

II. MEAN-FIELD THEORY

A. Bogoliubovde Gennes equations

In strongly anisotropic or quasi-one-dimensional materi-als, the particular shape of the band structure causes a nest-ing of electron-hole states near the Fermi surface. The back-scattering of electrons near the Fermi energy produces adivergence in the response of the charge-density to lattice

deformations. As a consequence, a charge-density-waveground state is formed spontaneously at a critical tempera-ture Tc , which is rigidly coupled to a periodic lattice distor-tion. The ground state consists of a condensate of electron-hole pairs and is characterized by a complex order parameter. Because the effective mass of the electrons in the trans-verse direction is much larger than the effective mass in thedirection along the chains, it is reasonable to assume that theonly effect of neighboring chains is the suppression of ther-mal fluctuations of the order parameter. In this approxima-tion, the main features of the Peierls transition and of theassociated collective mode can be captured by the mean-field

PHYSICAL REVIEW B 15 JULY 1996-IIVOLUME 54, NUMBER 4

540163-1829/96/544/27988 /$10.00 2798 1996 The American Physical Society


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dE

d

2Zsin

F

u kvk vk2uk

2

vk2u k

22Z2 12u kvkcos2 d,

23

where we have used the Fermi energy F as integration cut-off. For weak impurity potentials, Z1, we may disregardthe Z2 term in the denominator. Carrying out the integral inthis limit by using Eq. 5 the classical result

dE

d

Z

gsin 24

is recovered, where the dimensionless coupling constantgg ep/2vF is introduced. If the scattering potential is re-pulsive (Z0) , the energy minimum configuration corre-sponds to and the electron density at the impurity siteis minimal. For stronger impurity potentials the integral hasto be calculated numerically. Figure 2 shows the pinningforce for different scattering strengths. For small Z the pin-ning force is sinusoidal. For stronger potentials a deviationoccurs, which leads to higher harmonics in the Fourier spec-trum. The behavior of the pinning force is qualitatively thesame as calculated by Ref. 17.

B. N/C junction

Consider a normal-metal/charge-density wave N/C junc-tion with an interface at x0. In the CDW region the orderparameter (x) near the interface is not constant, but decayssmoothly over a finite length of the order of the coherencelength, as indicated schematically in Fig. 3. This is the CDWproximity effect. To determine (x) near the interface onehas to solve the gap equation 5.18 Here we disregard theproximity effect and assume a step function pair potential,which is a common assumption in S/N junctions. A func-tion potential is added to the interface to model an interfacedefect or impurity. For simplicity we assume the CDW andthe normal-metal to be one dimensional and the averageelectron densities to be equal. Mismatch in electron densities

will result in an additional potential barrier, which canreadily be included by using the transfer matrix M as definedin Eq. 15.

The wave functions in both regions are

x

A

k 1

0 e ikx B

k 0

1 eikx, x0

C

kvk

u kei e

ik x

D

ku ke

i

vk e

ik x, x0

25

with wave vector k/vF in the normal-metal and

k22/vF in the CDW region. The wave function inthe normal-metal region is decoupled ( 0 implies vk1and uk0). Matching the wave functions using Eq. 13 weobtain after some algebra

r11 1 iZu ke

ii Zvk

1 iZvki Zukei ,

r22

1iZu kei iZvk

1i ZvkiZukei ,

t12t21vk

2uk

2

1iZvkiZukei , 26

where we have used the relation k/kvk2u k

2. Figure 4shows the reflection and transmission probabilities forZ0 and Z1 (0). Electron states in the gap(0)are totally reflected. If 0 the normal reflec-tion and transmission probabilities are obtained:T(1Z2)1 and R1T. Note that in contrast to theBlonder-Tinkham-Klapwijk model1 of superconductor junc-tions the transmission probability depends on the macro-

scopic phase of the CDW. This phase dependence has im-portant consequences for the quasiparticle conductance,which will be treated in Ref. 12. The determinant of thescattering matrix is

detS, 1i ZvkiZuke

i

1i ZvkiZukei . 27

For a perfect interface with Z0 the expression obviouslydoes not depend on . Therefore, the CDW is not pinnedand the translational degeneracy of the CDW is conserved.For Z0 the CDW is pinned.

Calculation of the electron density in the normal-metalregion (x0) for Z0 yields

FIG. 2. The dependence of the pinning force from a single im-

purity as a function of the CDW position for different barrier

strengths Z.

FIG. 3. The proximity effect at a N/C interface schematic. In

the CDW (x) recovers its bulk value at a distance of the order

of 0 from the interface.

54 2801MESOSCOPIC CHARGE-DENSITY-WAVE JUNCTIONS


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particle interference effect caused by scattering from the gapstructure. In the field of superconductivity similar geometri-cal resonances are known as Tomasch oscillations.20 Themodulation of the quasiparticle conductance could be ob-served in the d2V/dI2 analysis of an I-V transport measure-ment. It is emphasized that these expressions are very similarto those for the Josephson junction as calculated in Refs.2123, where the resonant states correspond to the transferof a Cooper pair. One crucial difference is that in our system,dE/d, does not express a zero bias electrical Josephsoncurrent. The electrical current is zero in a C/N/C junction,since the transmission coefficient from left to right equals thetransmission coefficient from right to left. In Sec. IV we willrelate dE/d to a flow of electron-hole pairs, which carryzero charge but momentum Q .

Phase coupling is determined by scattering states in thecontinuous spectrum () and by localized states(0). The total energy depends only on the phasedifference over the junction. This means that at an ideal in-terface there is no intrinsic pinning. If one CDW is movedadiabatically, dissipationless sliding CDW transport throughthe normal region is possible. This is equivalent to a super-current flowing in a S/N/S junction. As shown in Sec. III B,an N/C contact will modulate the electron density in thenormal metal, which resembles a Friedel oscillation, but de-pends on the phase of the CDW. The phase coupling in theC/N/C junction can thus be understood as a phase locking ofFriedel oscillations, arising from both interfaces. The mecha-nism is similar to the nonlocal exchange Ruderman-Kittel-

Kasuyan-Yosida RKKY interaction in magnetic multilay-ers via spin-density oscillations.3

For future experiments it should be relevant to includeimpurities or defects at the interfaces. We now include two function potentials V(x)H1(x)H2(xL) at the in-terfaces of the C/N/C heterostructure, with L50 . Figure 7shows the response of phase 2 as a function of phase 1 forwhich the energy is minimal (H1H2). If 1 is changedadiabatically slow in time, the change of 2 in time is non-linear and shows a phase jump.

IV. THREE MESOSCOPIC JUNCTIONS

We have already mentioned similarities between thephase coupling in the S/N/S and the C/N/C junctions. We

will now investigate these analogies further and also com-pare them with the ferromagnet/normal-metal/ferromagnetF/N/F junction. Mathematically the phase coupling is cal-culated almost identically for all three systems, namely bysolving Bogoliubovde Gennes type of equations in thethree regions and by matching of the wave functions. Thephysics is very different, however. A phase difference overthe junction results in a current. In S/N/S junctions, Andreevscattering generates a supercurrent through the normal metalregion in response to a phase difference over the junction.2 InF/N/F junctions, depending on the width of the normal-metal, nonlocal exchange coupling of the magnetization di-rections may cause a stable antiparallel () coupling viaspin-density oscillations.24 In Refs. 3 and 24 it has beenshown that a difference in magnetization moments results ina net spin current flowing through the normal layer. We ar-gue that the phase coupling in the C/N/C junction can beformulated in terms of a momentum current, or to put itdifferently, by coherent transport of electron-hole pairs.

In Sec. II we have linearized the energy spectrum near theFermi energy. If we keep the quadratic dependence of theenergy dispersion, the BdG equations are given by

H * H

fg f

git f

g , 35

where H2x2/2mFU(x) and (x)exp(iQx).

The quasiparticle current j qp can directly be calculated as

j qp

mImf*xfIm g*xg . 36

In the approximation that the one-dimensional electron gas isperturbed only in the vicinity of the Fermi energy, a widelyused expression for the momentum p per particle is

pkFf2g2, 37

where the terms in brackets are the difference between rightand left going electron states. The force tp can be calcu-lated using the BdG equation and this procedure results in

FIG. 6. The transmission probability as a function of energy.

Geometrical resonances with period 20 /L are identified as Tom-

asch oscillations.

FIG. 7. The response of the phase 2 to an adiabatic change of

the phase 1 for different impurity strengths Z. The change of2 is

nonlinear for Z0.



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tpxjp4kFIm f*g , 38

where jp is defined as

jp2kF

mImf*xfIm g*xg . 39

In this one-dimensional model, jp corresponds to the Txxcomponent of the stress tensor and is equivalent to the flowof x-momentum density in the x direction. Note that thisequation only differs by a minus sign from Eq. 36. In bulkCDW systems, j qp0 at the gap edge , whereas themomentum current jpkF . This is similar to bulk supercon-ductors where j qp0, but the electrical current j ekF.Equation 38 is equivalent to the conservation law for thequasiparticle charge in a superconductor.1 Here it is the con-servation law for the quasiparticle momentum. The right-hand side is a source or drain term connecting the quasi-particles with the condensate. In one dimension jp is relatedto the pinning force Eq. 18 via

dE

d

2 Jp , 40

where the total momentum current Jpis the sum over alloccupied states Jp1/kFjp.

The phase coupling in the three junctions under consider-ation can be summarized by the following universal iden-tities:

F/N/F:dE

d

2Js ,

S/N/S:dE

d

2Je ,

C/N/C:dE

d

2Jp . 41

The currents are determined by

Js

mk Imf*xg Im g*xf,

Je

mk Imf*xfIm g*xg ,

Jp

mk Imf*xfIm g*xg , 42

where f and g represent spin up and spin down electrons inmagnetic junctions, electron and hole like states in the super-

conductor junctions and right and left going electrons in thecharge-density wave junctions. These wave functions are cal-culated from the following equations

F/N/F: H * H

fg f

g ,

S/N/S:

H

* H*

f

g

f

g , 43

C/N/C: H * H

fg f

g ,

where H , H , and in the F/N/F junction are defined byHHhz and hxi hy , with hx,y ,z the components ofthe magnetization vector. The magnetic junction is simpleras far as the self-consistency condition is concerned, becausespin-up and spin-down electrons have the same mass,whereas in the superconductor and charge-density-wave

junction the effective mass of the quasiparticles and quasi-holes has to be taken into account.

V. SUMMARY

We have investigated coherence effects of charge-densitywaves in mesoscopic systems by solving the CDWBogoliubovde Gennes equations with suitable boundaryconditions. The scattering matrix theory, using the BdG so-lutions as a basis, provides a simple method for ground stateenergy calculations. From this model, transport propertieslike quasiparticle conductances can easily be extracted.12 Wehave studied the N/C interface, with and without an impurity.It is found that an ideal interface will not pin the CDW. Theinteraction between the impurity potential and the CDW hasa strong effect on the transmission and reflection probabili-ties. The electron density in the normal region is modulatedlike a Friedel oscillation, but is induced by the CDW and

depends on its phase. These Friedel oscillations are the ori-gin of the phase coupling in a C/N/C junction, analogous toexchange coupling in magnetic systems via spin-density os-cillations. Tomasch-like oscillations due to quantum size ef-fects are identified in the C/N/C junction. The phase-dependent energy is related to a momentum current,indicating a coherent flow of electron-hole pairs.

ACKNOWLEDGMENTS

The authors would like to thank C. Dekker, J.E. Mooij, T.Stoof, Y. Tanaka, and especially Y. Nazarov and B. Rejaeifor valuable discussions. This work is part of the researchprogram of the Stichting voor Fundamenteel Onderzoek

der Materie FOM, which is financially supported by theNederlandse Organisatie voor Wetenschappelijk Onder-zoek NWO.

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Rouxel, Phys. Rev. Lett. 37, 602 1976.7 R. Flemming and C. C. Grimes, Phys. Rev. Lett. 42, 1423 1979.8 G. Gruner, Density Waves in Solids Addison Wesley, Reading,

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Rev. Lett. 66, 76 1991.17 I. Tutto and A. Zawadowski, Phys. Rev. B 32, 2449 1985.18 M. I. Visscher et al. unpublished.19 G. Gruner, A. Zawadowski, and P. M. Chaikin, Phys. Rev. Lett

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mark i. visscher and gerrit e. w. bauer- mesoscopic charge-density-wave junctions

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