Analysis of dichroism in the electromagnetic response of superconductors

K.�

CapelleandE. K. U. GrossInstitut�

fur� Theoretische Physik, Universitat Wurzburg,� Am Hubland, D-97074 Wurzburg,� Germany

B. L. GyorffyH.�

H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom�Received�

13 May 1997�The absorptionof polarizedlight in superconductorsis studiedwithin the frameworkof the Bogolubov–de

Gennes�

approachto inhomogeneoussuperconductorsin magneticfields.Severalmechanismswhich give riseto�

a polarization-dependentabsorption i.e.,

dichroism� in superconductorsareanalyzedin detail.The relationto�

the absorptionof unpolarizedlight in superconductorsand to the absorptionof polarizedlight in normalconductors� is investigatedandseveraleffects,not known from eitherof thesecases,arefound.Theseeffectsarise from the interplay of broken chiral symmetry,which producesdichroism, with the superconductingcoherence.� Onepotentialsourcefor dichroism,namelyspin-orbit coupling, is investigatednumericallyfor asimple� modelsuperconductor.� S0163-1829

� �98� 01122-9�

I. INTRODUCTION

In�

this paper we presenta systematicapproachto thepolarization-dependent� absorptionof light in superconduct-ors.� The phenomenonthat left-handedcircularly polarizedlight�

andright-handedcircularly polarizedlight areabsorbeddifferently�

by many substancesis commonlyknown as di-chroism.� Dichroismin the presenceof an externalmagneticfield is usually referredto as the Faradayeffect or the Kerreffect,� dependingon the geometryof the experiment.Di-chroism� in the absenceof externalmagneticfields is oftenreferredto asspontaneousdichroism.It is found in magneti-cally� ordered systems,such as iron, but also in systemswhich� break inversion symmetry, such as sugar. In thepresent� paperwe usethe term dichroismto refer to all situ-ations� in which the absorptionof light dependson its polar-ization.

In�

normal � i.e.,�

not superconducting� and� magneticallyor-dered�

metalstheseeffectshavebeenthe subjectof intensestudy for manyyears.! See

"Refs.1 and2 for recentreviews.#

It is by now unanimouslyacceptedthat dichroismin thesesystems arisesmainly from the simultaneouspresenceofspin-orbit couplingandthe spin magnetization.Moderncal-culations� of the optical responseto polarizedlight arethere-fore usuallyperformedin a relativistic framework.1,2

However,$

similar calculationsfor superconductorsdo notexist� yet. It is the purposeof the presentwork to provideabasis%

for suchcalculationsandto presentsomefirst results.The motivation for this investigationarisespartly from

the&

fact that severalrecentexperimentsreport the observa-tion&

of dichroicphenomenain superconductors,3–8'

and� partlyfrom the fundamentalinterest in the role of the spin-orbitcoupling� in superconductors.9–1

(1 Spin-orbit"

coupling,beingarelativistic effect of secondorder in ) /

*c+ ,, becomesmoreim-

portant� in systemswith heavy elements- atomic� numberZ./

40) in thelattice.12 Many interestingsuperconductors,e.g.,the&

heavy-fermioncompoundsand the high-temperaturesu-perconductors,� do indeedcontainvery heavyelements,suchas� mercury (Z 0 80)

1, uranium (Z 2 92)

3, bismuth (Z 4 83)

1,

lanthanum�

(Z. 5

57)6

, platinum(Z. 7

78)8

, etc.In�

view of the fact that a qualitativeandquantitativeun-derstanding�

of dichroismin normalandmagneticallyorderedmetalsnecessarilyrequiresa relativistic theory,it is of obvi-ous� relevancefor thepresentinvestigationsto employa rela-tivistic&

theory of superconductivity.Such a theory has re-cently� beenconstructed.9–1

(1 Fromthis theorythefull form of

the&

spin-orbit operatorin superconductorsis known to con-tain&

not only gradientsof the lattice potential 9 as� is the casefor:

the conventionalspin-orbit operator; ,, but also gradientsof� thepair potentialof thesuperconductor.The latter typeofspin-orbit couplingis referredto astheanomalousspin-orbitcoupling� < ASOC

= >.

In�

the presenttheoryof dichroismwe takeboth typesofspin-orbit coupling into account. Several other potentialsources for dichroismin superconductorsarealsoconsideredin�

detail. To this endwe usea perturbativeapproach,basedon� theBogolubov–de Gennesequationsin thepresenceof amagneticfield. The various sourcesfor dichroism are in-cluded� via first-order stationaryperturbationtheory for thequasiparticle? wave function. The resultingperturbedsingle-particle� statesarethen,in a secondstep,usedasunperturbedstates betweenwhich the transitionscausedby the polarizedlight�

takeplace.The absorptionof light is treatedby a gen-eralization� of the standardgolden rule of first-order time-dependent�

perturbationtheory.First resultsfrom this inves-tigation&

werealreadypresentedin a recentpaper.13

The@

presentpaperis organizedasfollows: Sec.II A con-tains&

an outline of the perturbativeapproachwhich we em-ploy� for the calculations.The unperturbedsystem is de-scribed by the spin-Bogolubov–de Gennesequations,whichare� a generalizationof the conventional Bogolubov–deGennesA

equations,designedto treatspin-dependentphenom-ena.� Sinceperturbationtheoryhasup to now beendevelopedonly� for the conventionalBogolubov–de Gennesequations,we� presentthe correspondinggeneralizationsfor the spin-BogolubovB

–de Gennesequationsin somedetail.The result-ing expressionscanbeusedfor any typeof perturbativecal-culation� for the spin-Bogolubov–de Gennesequations,not

PHYSICAL REVIEW B 1 JULY 1998-IVOLUMEC

58, NUMBER 1

PRB 580163-1829/98/58D 1E /473F G

17H /$15.00F

473 © 1998The AmericanPhysicalSociety

only� for investigationsof dichroism. Next, we explicitlyspecify the perturbationsto be includedin our approachtodichroism.�

These include the above-mentionedspin-orbitterms,&

the effect of orbital currentsand,if necessary,order-parameter� inhomogeneities.

In Sec. II B we employ the perturbativeexpressionstoderive�

a formulafor thepowerabsorptionin superconductorsas� a function of the polarizationof the light. In Sec. II Cseveral distinct mechanismsfor dichroismin superconduct-ors� are identified.A detailedanalysisof the physicsbehindthe&

variousmechanismsis performedandthe circumstancesunderI which they canproducedichroismin superconductorsare� discussed.

Section"

III containsmodel calculationsfor one of thesemechanisms,J namely the conventionalspin-orbit coupling.UsingK

simple approximationsfor the relevant matrix ele-mentsandthedensityof states,which arediscussedin Secs.III�

A andIII C, respectively,we drawfurtheranalyticalcon-clusions� about the physicsbehind this mechanismin Sec.III B, and evaluatethe formulas numerically in Sec. III D.The numericalresultsareanalyzedasfunctionsof tempera-ture,&

frequency,andmagnetic-fieldstrength.WeL

emphasizethat thesecalculationsare not meantasquantitative? predictionsfor experiments,but ratherasmodelcalculations� illustrating the analyticalresultsof the previoussections andexhibiting surprisingqualitativeM features

:of di-

chroism� in superconductors.Thepaperendswith a brief dis-cussion� of somerecentexperimentsin light of our theoryinSec."

IV anda summaryin Sec.V.

II. PERTURBATIVE APPROACH TO DICHROISMIN SUPERCONDUCTORS

A.N

Perturbation theory

1.O

The spin-Bogolubov–deP

Gennes equations

Thepropermicroscopicdescriptionof inhomogeneoussu-perconductors� is provided by the Bogolubov–de Gennesequations� Q BdGE

B RhS

0T UWV rX YZ † [ rX \^] h

S0T*

u_ n` a rX bcn` d rX e f E

gn` u_ n` h rX ij

n` k rX l ,, m 1nwhere�

hS

0T o pp 2

q2r

ms tvuxw rX y{z}| ~ 2r �

is the normal-statesingle-particleHamiltonian, � (�r)�

is thepair� potential,andthe u_ n` (� rX )� and � n` (� rX )� areparticleandholeamplitudes.� The rX dependence

�of the pair potential � (

�rX )� de-

scribes the center-of-massmotion of the Cooperpair. Theinternal degrees-of-freedomof the pair would be describedby%

the dependenceof the pair potentialon the relativecoor-dinate�

of thetwo electrons.This would requiretheuseof thenonlocal� version of the Bogolubov–de Gennesequations.� 14,15 In the presentpaperwe limit our attentiontothe&

local versionof the Bogolubov–de Gennesequation,asspecified above.This meansthatwe cannot adequatelytreatthe&

effects of the internal degreesof freedom,such as thedifference�

between s� -wave and d�

-wave superconductors.The@

local Bogolubov–de Gennesequationshave been the

starting point for manymicroscopicinvestigationsof super-conductors� � Refs. 14–25 amongothers� . However, for thepresent� purposetheir form is too restrictivebecausewe needto&

incorporatethe spin degreesof freedomof the quasiparti-cles� andspin-orbitcouplingterms.Thereexistsa generaliza-tion&

of the BdGE in which thesecan be includedproperly,the&

spin-Bogolubov–de Gennesequations � SBdGE" �

. Theyread14,16,26,27

hS �

0 0� �

0�

hS � ���

0�

0� ���

* � hS �

* 0�

�* 0 0

� �hS �

*

u_ � n` ��� rX �u_ � n`  ¢¡ r£¤ ¦ ¥

n` §�¨ rX ©ª¦«n` ¬�­ r®

¯ En` °u_ ± n` ²�³ ru_ µ n` ¶�· rX ¸¹xº

n` »�¼ r½¾ x ¿n` À�Á rX Â

,, Ã 3Ä Å

where�hS ÆÈÇ

hS

0T ÉËÊÍÌ

BB. Î 4ÏТÑÓÒ 1 for h

S Ôand� Õ¢ÖÓ× 1 for h

S Ø. We havealreadyincluded

the&

Zeemancoupling of the spins to the externalmagneticfieldÙ

BÚ Û

which� is assumedto be spatially constantand topoint� along the zÜ direction

� Ý,, but not yet the couplingof the

orbital� degreesof freedomto the vector potentialand spin-orbit� coupling.The latter two effectsareincludedaspertur-bations,%

in Sec.II A 3.WeL

take theseequationsto describethe unperturbedsu-perconductor.� They can be obtainedfrom a spin-dependentBogolubov-ValatinB

transformation.14,16,26,27 Alternatively,=

they&

are found as the nonrelativisticlimit of the relativisticBogolubov–de Gennesequations.9–1

(1 Theyrelateto thecon-

ventionalÞ BdGE in exactly the sameway asthe Pauli equa-tion&

relatesto the Schrodinger�

equation.WeL

now summarizea numberof propertiesof theSBdGEwhich� areessentialfor the following considerations.To ev-ery� eigenvector

u_ ß n` à�á râu_ ã n` ä�å ræç¦è

n` é�ê rX ëì¦ín` î�ï rð

ñ56 ò

of� the SBdGE with eigenvalueEn` ó¢ô 0�

belongsa secondeigenvector�

u_ õ n`¯ ö ÷ røu_ ù n`¯ ú û rX üýxþ

n`¯ ÿ � r����n`¯ � � r�

��

n` � � r����n` � � rX �

u_ � n` � � r�u_ � n` � � r�

* �6� �

with� eigenvalueEg

n` � "! Eg

n` # . Thenegativeenergysolutionsdescribe�

boundstatesof theBogolubovquasiparticles$ bogo-%

lons% ,, or, equivalently,electronscondensedin Cooperpairs.BelowB

we considerpair-breakingprocessesasthe dominantsource for absorptionat low temperatures.Theseprocessescan� simply be describedas transitionsfrom a negativeen-ergy� stateof the form & 6� ' to

&a positive energystateof the

form: (

56 )

.

474 PRB 58K.*

CAPELLE, E. K. U. GROSS,AND B. L. GYORFFY�

In the absenceof all spin-dependentinteractions theSBdGE"

rigorouslyreduceto theBdGE.Therelationbetweenthe&

respectiveeigenfunctionsis simply u_ + n` ,0T

(�r)� -/.1032

u_ n` (� r)�

and� 465n` 70T

(�rX )� 8:9<;>=@? A n` (� rX )� , whereB and� C are� D 1. Theinclu-

sion of the Zeemanterm in Eq. E 3Ä F does�

not changetheserelations,G aslong asB

Úis�

spatiallyconstant.If thepair field His spatially constantas well, the spatial dependenceof theeigenfunctions� of the SBdGE is governedby the normal-state Hamiltonian h

S0T . The eigenfunctionsare thus propor-

tional&

to normal-stateeigenfunctionsI n` ,, which aredefinedthrough&

hS

0T J

n` KML n` N n` . The constantsof proportionality aregivenO by the BCS amplitudes14 u_ n` : P"Q 1/2(1 RTS n` /

*Eg

n` )� andUn` : V"W 1/2(1 XTY n` /

*Eg

n` )� . The full form of the eigenfunctionsof� the SBdGEunderthesecircumstancesthus is

u_ Z n` [0T \

r]3^ u_ n` _1`3acb n` d re3f 1

21 gih n`

En`j1kmlcn

n` o rp ,, q 78 r

sutn` v0T w

rx3y:z|{ n` }>~�� � n` � r�3�:� 1

21 ��� n`

Eg

n`�>��� � n` � r� ,, �

81 �

and� the correspondingeigenvalueis given by

En` ��� En` �:��� B� B ��� � n`2q ��� 2  :¡�¢

B� B. £ 93 ¤

If the pair potentialis not spatially constant,the spatialde-pendence� of theSBdGEeigenfunctionsis not determinedbythe&

normal-stateHamiltoniananymore.In this casethe par-ticle&

andhole amplitudesarenot proportionalto ¥ n` (� rX )� , buthave¦

to bedeterminedby solving the full SBdGE.Equations§3Ä ¨

– © 93 ª define�

the unperturbedsuperconductor.In the nextstep we developperturbationtheory for suchsuperconduct-ors.�

2. Perturbation theory for the spin-Bogolubov–deP

Gennesequations«

Perturbation¬

theory for the conventionalBdGE hasbeendeveloped�

by deGennes14 and,� in a slightly different formu-lation, by Kummel andco-workers.28,29

qWhileL

beingequiva-lent�

to the former approachon the exact level, the latterapproach� hasthe advantagethat in every order of approxi-mationthe formulasareof the samestructureas in conven-tional&

perturbationtheory for the Schrodinger�

equation.Inthe&

following, we generalize the latter approach to theSBdGE,"

wherewe allow for perturbationsof thenormalandof� the pair potential and discussboth stationaryand time-dependent�

perturbations.Sincethederivationof theformulasis�

very similar to that of perturbationtheory for the Schro-dinger�

equationand to that for the conventionalBdGE, weonly� presentthe final resultsand point out the main differ-ences� to the conventionalcase.

The@

wavefunctionsin thepresenceof a stationarypertur-bation% ­

H are,� to first order in ® H,, given by

u_ ¯ n` ° ± r²3³ u_ ´ n` µ0T ¶

r·3¸º¹»m¼ ½¿¾ÀÂÁ

n` ÃÅÄÆ

Hm¼ Ç ,n` ÈEn` É�Ê Em¼ Ë u_ Ì m¼ Í0

T ÎrÏ

ÐÒÑ Hm¼¯ Ó ,n` ÔEg

n` Õ×Ö Eg

m¼¯ Ø u_ Ù m¼¯ Ú0T Û

rÜ ,, Ý 10Þßuà

n` á â rã3ä u_ å n` æ0T ç

rè3éºêëm¼ ì¿íîÂï

n` ðÅñò

Hó m¼ ô ,n` õ

En` ö�÷ Em¼ øuùuú m¼ û0T ü

rýþÒÿ H

ó m¼¯ � ,n` �En` ��� Em¼¯ � ��� m¼¯ 0

T r� � 11

with� correspondingequationsfor u_ � n� � and� ���n� � . � H

ócan� be

any� perturbing4 � 4�

matrix andmay thuscontainperturba-tions&

of the lattice potential and of the pair potential. Alllabels(ms � )

�refer to positive energystatesof the form � 56 � .

The barredlabels,such as (ms¯ � )�, refer to negativeenergy

states of theform � 6� � . For notationalsimplicity we havesup-pressed� an upperindex 0 on the energies,althoughthey are,of� course,unperturbedenergies.The changeof the energiesis,�

to first order,given by the usualresult � Eg

n� �! #" Hó n� $ ,n� % .

A time-dependentperturbationgives rise to transitionsfrom onestateof the systemto another.We takethe pertur-bation%

to be

&�')(t* ,, r+-,

.hS /

t* ,, r021 0�

0� 3#4

hS 5

t* ,, r6 * 7 ,, 8 129where� we condenseda 4 : 4

�matrix in 2 ; 2

rform ( < is

�the

2 = 2 unit matrix> and� assumedthat there is no time-dependent�

perturbingpair potential ? such a potentialwouldappear� off the diagonalin Eq. @ 12A and� could, if necessary,be%

included without difficultiesB . C hS

(�t* ,, rD )� is of the general

form:

EhS F

t* ,, rD G-H#I hS J

rD K eL i M tN O#PhS Q

rD R †eL S i T tN . U 13VThe transitionprobability per unit time for a transitionfromthe&

single-particlestate(nW X ) t�

o (nW Y[Z]\ )� is then

w^ n_ `ba n_ cedgfEh i 2 jkmlon�p˜ n_ q ,n_ resgtvu 2q wyx En_ ze{g|[} En_ ~�������� ,, �

14�w^ n_ ��� n_ �����A � 2 ��m�o���˜ †n_ � ,n_ ������� 2q �y� E� n_ �e g¡[¢ E

�n_ £!¤�¥�¦�§ ,, ¨

15©where� thefirst expressiondescribesemissionandthesecondabsorption.� The labelsnW ª ,, nW «­¬]® refer to arbitrary solutionsof� the SBdGE.Thereis oneimportantdifferenceto the con-ventionalÞ goldenrule: the matrix elementsin Eqs. ¯ 14° and�±15² are� not simply thoseof the perturbation ³ 12 and� its

Hermitianconjugate,but thoseof

µ�¶˜ · ¸hS ¹

rº­» 0�

0� ¼#½

hS ¾

r¿ À TÁ Â 16Ãand�

PRB 58 475ANALYSIS OF DICHROISM IN THE . . .Ä

Å�Ƙ † Ç È hS É

r¿ Ê †Ë 0�

0� Ì#Í

hS Î

rÏ * Ð ,, Ñ 17Òwhich� differ in the locationof the conjugateoperators.Thereasonfor this is that the coefficientof eL i Ó tN which� givesriseto&

emission,andthatof eL Ô i Õ tN which� producesabsorption,aremixed alreadyin Eq. Ö 12× .

Theaboveformulascanof coursebeusedfor anykind ofperturbation� of the SBdGE and should thereforebe usefulnot� only for investigationsof dichroism,but for a largeva-riety of calculations.

3.Ø

Explicit form of the perturbations

In specifyingtheSBdGEeigenfunctionsaccordingto Eqs.Ù78 Ú

and� Û 81 Ü ,, we assumedthe pair potential to be spatiallyconstant.� While this is a good approximationfor supercon-ductors�

with a large coherencelength, it is inadequateforthose&

with a shortcoherencelengthandfor superconductingheterostructures.In order to take this into accountwe writethe&

full pair potentialas

ÝßÞrà-á�⯠ã�ä˜ å ræ ,, ç 18è

where� é¯ is a suitableaverageof ê (�r)� ë

e.g.,� takenovera unitcell� ì and� í˜ (

�r¿ )� is thelocal deviationfrom thataverage.In the

SBdGE"

frameworkit is includedby replacingî (�r¿ ) b�

y ï¯ in�

the&

unperturbedSBdGEandaddingthe term

ðH ñ 0T ò­ó 0

�iô õ ˆ

yö ÷˜øiô ù ˆ

yö ú †û˜ * 0� . ü 19ý

To@

simplify the notationwe haveemployedthe Pauli matrixþ ˆyö to&

write the 4 ÿ 4�

equation � 19� a� s a 2� 2r

equation.Forsmall �˜ (

�r),� �

H (0)�

can� be treatedas a small perturbation.�The@

caseof large �˜ (�r¿ )� is discussedbelow Eq. � 22

r .

In�

Sec.II A 1 we alsoassumedthat the magneticfield BÚ

is�

spatiallyconstantandactsonly on the electronspins.Wedo�

not considerthecaseof an inhomogeneousexternalmag-neticfield. However,evenfor constantfields, thereis a cou-pling� of theorbital degreesof freedomof theelectronsto thevectorÞ potential. To first order in the vector potential thiscoupling� is describedby the term

�H� 1 ����� qM

mc��A�

p� ˆ ��� 0�

0� ���

A�

p� ˆ � * � ,, � 20r

where� BÚ !

(� "$#

A�

)�

z% and� p� ˆ is�

the momentumoperator.Wechoose� A in the Coulombgauge,so that & A ' 0.

� (H (1)�

isconsidered� a small perturbationas well. For realistic vectorpotentials� this is certainly justified. In the caseof the con-ventionalÞ BdGE this is the standardway to treat vectorpo-tentials,&

for instancein deriving the Meissnereffect.14

Finally,)

we also consider the spin-orbit coupling as asmall perturbation. The relativistic theory ofsuperconductivity 9–1

(1 predicts� that there are two distinct

types&

of spin-orbitcoupling * SOC" +

in superconductors.One,the&

conventional SOC, appearson the diagonal of theSBdGE"

and containsgradientsof the lattice potential , (�r).�

The other,the anomalousSOC - ASOC. ,, appearsoff the di-

agonal� and containsgradientsof the pair potentialwith re-spect to the center-of-massand relative coordinates / C-

0ASOC=

andR-ASOC1 . In the following calculationsonly theC-ASOC0

term is included.This amountsto neglectingtheinternaldegreesof freedomof the two electronsin the Coo-per� pair andretainingonly thespin-orbitcouplingdueto thecenter-of� massmotion. The perturbationcorrespondingtothe&

SOCandC-ASOCtermsis given by

2H 3 2 4�5 6

4m� 2qc+ 2q

7 8:9 ;=<?> r@BA p� ˆ C�D EGF HJILK rMBN p� ˆ O:P iô Q ˆ yö RSUT:V WYX[Zr\B] p� ˆ ^G_ iô ` ˆ yö acb † d�eGf gih?j rkBl p� ˆ m * n .

o21p

Furthertermscould be includedin q H (2)�

,, e.g., the conven-tional&

and anomalous Darwin terms,9,11(

and� the(�qM 2q/2*

mc� 2q)�

nW A2q

term&

which goesalongwith Eq. r 20s to&

sec-ond� orderof ( t /

*c+ )� . Both of theseareknown not to produce

dichroism�

in the normalstateandarethereforenot includedhere among the relevant perturbations.We return to thispoint� in the discussionof existencecriteria in Sec.II C 2.

The@

full u stationary v perturbation� to be consideredin thispaper� is the sumof all threeterms

wH xzy H { 0T |�}z~ H � 1 ���z� H � 2q � . � 22�

The@

upperindex,k�,, of eachterm � H (

�k�

)�

refersto theorderin1/c+ of� the respectiveterm. The relative magnitudeof thethree&

termsdependson theparticularsystemunderstudyanddoes�

not necessarilycorrelatewith this order.If�

thepair potentialinhomogeneity�˜ (�r¿ )� is small enough,

the&

ASOC term, which alsocontains�˜ (�r)�, canbe dropped.

If it is large, �˜ (�r)�

must be includedalreadyin the unper-turbed&

Hamiltonian.The latter situationarises,e.g., for su-perconducting� heterostructuresand in the vortex phaseof atype-II&

superconductor.Thepair potential� (�r)�

nearthevor-tices&

is strongly inhomogeneousand the deviationsfrom itsaverage� �¯ are� so large that �˜ (

�r)�

cannotbe consideredasasmall perturbation.With the full pair potentialin the unper-turbed&

Hamiltonian,the spatialdependenceof its eigenfunc-tions&

is, of course,not determinedby normal-stateeigen-functionsanymore.As a consequence,the particleandholeamplitudes� in Eqs. � 78 � and� � 81 � mustbe found by solving theSBdGE."

The ASOC term then is a first-orderperturbation,just�

as the conventionalSOC. Indeed,the vortex lattice isamong� the situationsin which this term is expectedto bemost important.

In�

orderto facilitate the treatmentof stronglyandweaklyinhomogeneoussituations,we will keepboth �˜ (

�r)�

and theASOC=

term in the expressionfor the perturbation.Depend-ing�

on the particular systemunder study, one of the twoterms&

canbe droppedin the final result,Eq. � 34Ä �

.The time-dependentperturbationis given by the interac-

tion&

of thepolarizedlight with thequasiparticles.It is of thesame form as the stationaryperturbation� H (1)

�,, but with a

time-dependent&

vector potential AL(�t* ,, r)�. The electromag-

netic� radiationis specifiedby its electric-fieldamplitudeE�

0T ,,

476 PRB 58K.�

CAPELLE, E. K. U. GROSS,AND B. L. GYORFFY�

the&

frequency � ,, the wave vector q� ,, and the polarizationvectorÞ � ,, leadingto the vectorpotential

A�

L� � t* ,, r  ¡B¢ ic

ô2r E0

T£¥¤ eL i ¦�§ tN ¨ qr© ª�« c.c.� ¬ 23

r ­

In theCoulombgauge( ® AL� ¯ 0),� °

hS

(�r)�, asneededin Eqs.±

13² ,, ³ 16 ,, and µ 17¶ ,, is given by

·hS ¸

r¹Bº�» eL2mi�

E�

0T¼ eL ½ iqr© ¾ p� ˆ ,, ¿ 24À

where� Á eL is�

the chargeon the electron.We assumeperpen-dicular�

incidenceof thelight, sothatq� is�

parallelto thestaticmagneticJ field B

Âand� perpendicularto the samplesurface,

while� Ã is a unit vector in the plane of the surface.ThisgeometryO correspondsto very commonexperimentalsetups.

B. Absorption of polarized light in superconductors

The@

quantity we use to describethe interactionof lightwith� a superconductoris the power absorptionP. It can becalculated� within the aboveperturbationtheoreticalframe-work� by evaluating

PÄ ÅÇÆ

f iÈ É E f

È Ê E i Ë fÌ Í

E i ÎGÏ 1 Ð fÌ Ñ

E fÈ ÒcÓ w^ i Ô f

ÈAÕ

,, Ö 25×

where� iô

and� fÌ

denote�

the initial and the final single-particlestates, respectively,w^ i Ø f

ÈAÕ

is given by Eq. Ù 15Ú and� fÌ

(�E)�

stands for the Fermi function. The power emissioncan becalculated� in a similar way from w^ i Û f

ÈE . In the following wewill� consideronly powerabsorption.Theemitted Ü scattered Ýpower,� being diluted over the entire solid angle, is usuallynot measuredexperimentally.Furthermore,as long as weconsider� the groundstateor situationsin which only a fewexcited� single-particlestatesare occupied,absorptionis byfar:

the dominantprocess.At sufficiently low temperaturesalmostall electronsare

condensed� in Cooperpairs.The most importantmechanismfor:

absorptionunder thesecircumstancesis pair breaking.Scattering"

from unpairedelectronsÞ i.e.,�

from thermally ex-cited� quasiparticlesß can� safelybe neglectedat low tempera-tures.&

The inclusion of single-particlescatteringnormallyproduces� only someadditionalabsorptionbelow the absorp-tion&

edge.30'

Therefore@

we will limit ourselvesto consideringonly� pair breaking à i.e., bogoloncreationá as� the mechanismfor absorption.It shouldbestressedthatneglectingemissionprocesses� andscatteringfrom brokenpairsis physicallyjus-tified,&

but in no way necessaryfor the further development.WithinL

the linear-responseregime,the absorptionof lightis�

generallydescribedby the conductivity tensorâ ˆ . Expres-sions for the conductivity tensor in superconductorsareknown in a numberof different approximations.3

'0–33

For)

any systemthe elementsof the conductivity tensorcan� berelatedin a simplefashionto thepowerabsorptionforvariousÞ polarization directions.34

'For systemswith cubic

symmetry onefinds within the dipole approximation,q� ã 0,�

Im äæå xyç èæéëêcìîí 1

VE0T2 ï P ðæñ ,, ò LóBô P õæö ,, ÷ R øcù ú 26û

and�

Reü�ý xxç þ ÿ ����� 1

VE0T2 � P �� ,, L� � P �� ,, � R ��� ,, � 27�

where� V is thesamplevolume,P(� �

,, � )� is thepowerabsorp-tion&

for polarization � and� frequency � ,, and E�

0T is�

theelectric-field� strengthof the light. In the frequencyrangerelevantfor pair breaking,the dipole approximationis gen-erally� valid andwill be usedexclusively.For light with left-handed¦

polarization � LHP� �

we� have

�L � 1�

2r

1iô0� ,, � 28

while� light with right-handedpolarization ! RHP" hasa po-larizationvector

#R$ % 1&

2r

1' iô

0� . ( 29

r )

Equations* +

26r ,

and� - 27r .

hold¦

regardlessof whetherthe sys-tem&

is superconducting,magnetic,or in the normal state,because%

the microscopicpropertiesof the systemdo not en-ter&

thederivation.34'

Similar"

formulascanalsobederivedfornoncubic� crystals2,35 or� beyond the dipoleapproximation.� 2,36,37

qEquation / 260 shows that the imaginarypart of the off-

diagonal�

elementsof theconductivitytensoris nonzeroonlyif�

light with left- andright-handedpolarizationis absorbedina� different way. Theseelementsthus provide a direct mea-sure for dichroismwhich canbe usedto relateexperimentalresultsG to theoreticalcalculations.For the caseof magneticand� normal metalsthey are routinely usedfor this purpose.1See"

thereviewRef.2 andtheproceedingsvolumeRef.1 forfurther:

discussionandapplications.2ByB

substitutingEqs. 3 24r 4

and� 5 176 in�

Eq. 7 158 and� theresult in Eq. 9 25: ,, we arrive at

PÄ ;=<?> @BA eC 2E

�0D2

2E

m� 2 FGNNH I f

Ì JLKE�

NH M fÌ N?O

E�

NH PRQTSVU uW N

H XZY * p� []\ NH ^* _

`badcNH eZf * p� g uW N

H h* iTj 2 kml E� NH n E

�NH oqpsrutwv . x 30

y zHerewe employedthe particle-holeconvention,i.e., all en-ergies{ are positive, the sumsare limited to positive energysingle-particle| statesand the explicit form of the negativeenergy{ states,as given by Eq. } 6~ � was� used.In writing Eq.�30y �

we� useda short-handnotationfor the matrix elements.Written�

out in completedetail, the first matrix elementisgiven� by

�uW NH �Z� * p� �]� N

H �* � � d� 3�r

uW � n_ ��� r� �uW � n_ ��� r�

T �i� � * � r� � � n_ ¡V¢¤£Z¥ r� ¦§ ¨

n_ ©Vª¤«Z¬ r­*

®31y ¯

and° the secondonereads

PRB 58 477ANALYSIS OF DICHROISM IN THE . . .±

²d³NH ´Zµ * p� ¶ uW N

H ·* ¸ ¹ d� 3�r º » n_ ¼�½ r� ¾¿ À

n_ Á� rÃT Ä

i� Å * Æ r

uW Ç n_ ÈÊÉÌËÎÍ r� ÏuW Ð n_ ÑÊÒÌÓÎÔ rÕ

*.Ö32y ×

IfØ

we now substitute the unperturbedsolutions to theSBdGE,Ù

uW Ú n_ Û0D

andÜ ÝßÞn_ à0D

,á as givenby Eqs. â 7ã ä andÜ å 8æ ç ,á in Eqs.è31é ê

andÜ ë 32é ì

,á we cancalculatethe powerabsorptionin thesuperconductorí asa function of polarization.Evaluatingtheresultoncefor î givenï by Eq. ð 28ñ andÜ oncefor ò givenï byEq.ó ô

29E õ

,á then leadsto a direct measurefor dichroism,and,accordingÜ to Eq. ö 26

E ÷,á an expressionfor the off-diagonalel-

ementsø of the imaginarypartof theconductivitytensor.Pro-ceedingù in this way it is immediatelyfound from Eq. ú 30

é ûthatü

ýP þ ÿ

nn_ � � p� yönn_ ¯ ��� p� x�nn_ ¯ � * � p� x�nn_ ¯ �� p� yönn_ ¯ �� * ��� 0,� �

33é �

i.e., thereis no dichroism.The p� ikl�

in Eq. � 33é �

areÜ normal-stateí matrix elementsof Cartesiancomponentsof the mo-mentum� operator.A barred index standsfor the complexconjugateù normal-state wave function � n_¯(

�r� ):� �! n_ (� r� )� *"$#

n_ (� % r)�. The latter equality holds if the original normal-

stateí Hamiltonian,h&

0D ,á doesnot breaktime-reversaland in-

version' symmetry.38,39(

As)

a consequenceall momentumma-trixü

elementsarereal, evenif the individual wavefunctionsareÜ not, and * P

Ävanishes' identically. This result, which

holds independentlyof the explicit form of the normal-statewave+ functions,was of courseto be expected.Indeed,it iswell+ known from normal and magneticallyorderedmetalsthatü

to find dichroism one needsto include mechanismswhich+ break chiral symmetry,so that the systembecomessusceptibleí to the differencebetweenleft- and right-handedpolarization., 1,2,34

In-

thenextstepwe includeall thestationaryperturbationswhich+ werediscussedabovein orderto find out if, andunderwhich+ circumstances,theseproducedichroismin supercon-ductors..

To this endwe substituteEqs. / 190 ,á 1 20E 2

,á and 3 21E 4

in5

Eq.ó 6

22E 7

. Equation 8 22E 9

is5

thenused,togetherwith Eqs. : 7ã ;andÜ < 8æ = ,á in Eqs. > 10? andÜ @ 11A to

üdeterminethe form of the

perturbed, wave functions. Thesewave functions are, in anextB step,usedasunperturbedW single-particleí stateswith re-spectí to the time-dependentperturbationin Eqs. C 30

é D– E 32é F

.MultiplyingG

out all termsto first order in H H�

we+ findIP: J P KML L

N OQP P RTS R$ U

VXW eC 2YE0D2

m� 2Y Z\[

nn_ ]_^ p� ` nW ,á nW acb fÌ dMe En_ fhg fÌ iMj En_ k_l mnporq

E�

n_ sut E�

n_ v_w xzy|{~}�$�

m� Re� p� � m� ,á nW �c� Cnn_ �m�

E�

n_ � E�

m� T�

m� ���� l� �

m� ,á nW �c� Cnn_ �m�¯E�

n_ � E�

m� T�

m� �� ,á�34é �

where+ Re denotesthe real part. All sumsare restrictedtosingle-particleí states with positive energies � particle-hole,conventionù � . The first term under the sum on m� representstheü

contributionof the brokenpairs � for�

this term the caseEm� � En_ is excludedfrom the sum� ,á while the secondrepre-

sentsí that of the condensedquasiparticles.The matrix ele-ments of the stationaryperturbationsare containedin thequantities�

T�

m� ��� uW n_ uW m� h

& ¡�¡nm_ ¢uW n_ £ m� d

¤ ¥�¥nm_ ¦¨§n_ uW m� d

¤ ©ª©* mn� «¨¬n_ ­ m� h

& ® ¯* nm_ °35é ±

andÜTm� ²³�´

uW n_ µ m� h& ¶�¶nm_ ¯ ·

uW n_ uW m� d¤ ¸�¸nm_ ¯ ¹»º

n_ ¼ m� d¤ ½ª½* m�¯ n¾ ¿¨À

n¾ uW m� h& Á Â* nm¾ ¯ ,á Ã

36é Ä

where+h& Å�Ånm¾ Æ ÇÉÈ 2

4im� 2cÊ 2 Ë nÌ Í_ÎcÏÑÐÓÒzÔÑÕ z% Ö m� ×MØ iq

� Ùmc� Ú nÌ Û A ÜÞÝ m� ß ,á à

37é á

h& â�â

* nmã äæåh& ç�çnmã ,á è 38

é éd¤ ê�ênmã ë ìÉí 2

Y4im� 2cî 2 ï nð ñ_òcóÉôzõzöÑ÷ z% ø m� ùMúüû nð ý þ˜ ÿ m� � ,á � 39

é �

d¤ ���* nmã ��� � 2

Y4�

im� 2cî 2 � nð ����� * ����� z% � m� ����� n� � ˜ * ! m" # . $ 40

� %A is the vector potentialof the static magneticfield B,á andshould& not be confusedwith that of the light wave, A

�LN .'

Normally,(

AL(�t) ,á r)� *

A(�r).� +

The matrix elementsof thetime-dependent,

perturbationenterthrough

Cnnã -m� .2E

i/ 0

p1 yönã¯nã 2 p1 x�mn� ¯ 3�4p1 x�nã¯nã 5 p1 yömn� ¯ 687

,á 9 41� :

which+ generalizesthe term found in Eq. ; 33é <

. The combina-tion,

of momentummatrix elementsin Eq. = 41� >

is?

typical ofdichroism@

andalsoappearsin manyapproachesto dichroismin the normal state and in magnetically orderedmaterials.2,34,35,40

Yp1 A n� ,á n� B�CED uF nG H nG IKJML nG uF nG N O 42P

andÜlQ R

n� ,á n� S�TEU uF nG uF nG V�WYX nG Z nG [ \ 43]areÜ coherencefactors,with theBCSamplitudesuF nG andÜ ^

nG asÜdefined@

in Eqs. _ 7ã ` andÜ a 8æ b . Theseare the coherencefactorsnormallyB found in treatments of optical absorption insuperconductors.& 30,41

(The additional factors uF nG c nG ,á etc., in

Eqs. d 35é e

andÜ f 36g h

resultfrom theeffectof thecoherenceonthei

stationaryperturbations.Wej

can rewrite Eq. k 34g l

inm

a more compactform, bynotingn that,dueto the ansatzo 7p q ,r s 8t u ,r the amplitudesuF nG andvw

nG arev solutionsof the BdGE x not the SBdGEy . They havethei

propertythat if

uF nG z r{|nG } r~ � � uF nG�

nG � n� � r~ � � 44� �

ism

a solutionwith eigenvalueE�

n� ,r then

u� n� � r��n�¯ � r� �

�Y�n�

u� n� � n� � r� * � �Y� n�u� n� � n� � r� � 45�

478 PRB 58K.�

CAPELLE, E. K. U. GROSS,AND B. L. GYORFFY�

ism

a solution with eigenvalue  E�

n� . ¡ This¢

is a well-knownproperty£ of theBdGE,which is thecounterpartto Eq. ¤ 6~ ¥ for

¦thei

SBdGE.§Hence¨

the secondterm underthe sum on m© inm

Eq. ª 34g «

can¬ be written in the sameform asthe first term,with everyindexm

m© replaced­ by m©¯ . It representsthe contributionof thenegativeenergystates.We canthereforeallow ® m¯ to

irange

over° all± m,r not just over thosewith positive energiesandfind, after slight rearrangements,

²P³ ´¶µ e· 2

YE�

0D2

m© 2 ¸ ¹nn� º�»

¼m¯

p½ ¾ n¿ ,r n¿ À�Á p½  m© ,r n¿ Ã�ÄE�

n� Å E�

m¯ ReÆ Ç

Cnn� Èm¯ TÉ

m¯ ÊËÍÌÎ

fÏ ÐÒÑ

E�

n� ÓÕÔ fÏ ÖÒ×

E�

n� Ø�Ù ÚÕÛÝÜ E� n� Þàß E�

n� áãâ ä�åçæéè ,r ê 46� ë

whereì the sumson n¿ andv n¿ í arev still restrictedto positiveenergies.î In orderto obtainthis convenientandcompactver-sionï of Eq. ð 34

g ñ,r theparticle-holeconventionis usedonly for

En� andv En� ò ,r but not for Em¯ .The¢

argumentsof the ó andv Fermi functions,in Eqs. ô 34g õ

or° ö 46� ÷

implym

that two quasiparticles,one with spin up, theother° with spin down,arecreatedby the absorptionprocess.This expressesthe fact that absorptionof light doesnot in-duceø

spin flips, so that the original spin directionof the par-ticlesi

in theCooperpair is not affectedby breakingthepair.ùThis is in contrastto the paramagneticpair breakingdis-

cussed¬ below.úC.û

Analytical results

Before explicitly evaluatingthe aboveformulasnumeri-cally¬ for a simplemodelwe first want to demonstratethat anumbern of general conclusionsfollow already from theirstructure.ï

1.ü

Mechanisms for dichroism in superconductors

Oný

the basisof Eqs. þ 34g ÿ

or° � 46� weì canidentify severaldistinctø

mechanismsfor dichroismin superconductors.Mechanism�

1 is the conventional spin-orbit coupling(SOC)�

produced£ by gradientsof the lattice potential ��� (�r).�

The relevantmatrix elementfor this mechanismis

� 2Y

4�

im� 2c 2

�n¿ ������������� z% � m© � ,r � 47

� �

whichì contributesto h� ! nm� andv h

� "!"* nm� in

mEqs. # 37

g $andv % 38

g &.

This mechanismis easyto interpretphysically: the mag-neticn field breakstime-reversalinvarianceand leadsto a fi-niten spinmagnetization.Sincetheinitial single-particlestatesarev occupiedin pairswith total spinzero,themaincontribu-tioni

to this magnetizationarisesfrom the final states.Thespin-orbitï couplingconvertsthebrokenorientationalsymme-tryi

in spinspaceinto a brokenchiral symmetryin realspace.The polarizationof the light is sensitiveto the chiral sym-metry' of the system,hencedichroismarises.

This¢

mechanismis known from normalandmagneticallyordered° metalsto give rise to the Faradayand Kerr effectsandv to x-ray magneticdichroism, respectively.1,2 We

jthus

find(

that the samemechanismis operativein superconduct-ors° aswell, stronglymodified,though,by thepresenceof the

energyî gapandthe coherencefactors.We presentnumericalresultsfor this mechanismin Sec.III D.

From)

the abovephysical explanationit is seenthat thepresence£ of an * externalî or internal+ magnetic' field is a nec-essaryî condition for the mechanismto work. This is easilyverified, from the equations.In the absenceof any type ofmagnetic' field, theenergiesin Eq. - 34

g .doø

not dependon thespin.ï Theonly spindependencethencomesfrom thefactor /inm

front of the matrix elements0 47� 1

. The sumover the spinstheni

reducesto

235476 1 8:9 0.

; <48=

Hence,¨

to havedichroismfrom the spin-orbit coupling, thepresence£ of magneticfieldsandthusof brokentime-reversalinvarianceis mandatory.This is confirmedby manyinvesti-gations> on normalandmagneticmaterials.1,2

Mechanism�

2 is the anomalous spin-orbit coupling(ASOC)�

produced£ by gradientsof the pair potential ?:@ (�r~ ).�

The¢

relevantmatrix elementsfor this mechanismare

A�B 2

4im� 2Yc 2Y C n¿ D�EGFIH�J�KIL z% M m© N O 49P

andQ thecorrespondingtermcontainingR * appearingQ in Eqs.S39T U

andQ V 40W . The ASOC term producesdichroismfor thesameX reasonasthe SOCterm andalsorequiresthe presenceofY magneticfields. The temperaturebehaviorof this term isveryZ different from that of the SOC term becausethe pairpotential[ itself is stronglytemperaturedependent.Moreover,the\

coherencefactors for this mechanism,in Eqs. ] 35T ^

andQ_36T `

,a aredifferent from thosefor the SOCmechanism.Sinceb

boththeSOCandtheASOCmechanismdependongradientsc of the respectivepotentials,they do not operateinhomogeneoussystems.On theotherhand,themoreinhomo-geneousc the system,the more important they become.TheASOCd

term,for instance,beingproducedby gradientsof thepair[ potential,becomeslarge in the vortex stateof a type-IIsuperconductorX and for superconductorswith a short coher-encee length.

Mechanism�

3 is provided by orbital currents,a which flowin thepresenceof themagneticfield B f�g�h A. Therelevantmatrixi elementis

iq� jmc© k n¿ l A mon m© p . q 50

r sThet

physicalinterpretationis that thesecurrents,circulatinginu

the plane perpendicularto the magneticfield, give themateriala definitehandednessandhenceresultin dichroism.This mechanismis knownalsofrom thenormalstate,whereitu

is usually much smaller than the SOC mechanismandtherefore\

neglectedin mostcalculations.In superconductorsthe\

screeningcurrentswhich give rise to the MeissnereffectareQ strongerthantypical screeningcurrentsin normalmetals.Current-inducedv

dichroism may thus be strongerin super-conductorsw than in normal conductors.Indeed,this type ofdichroismx

in superconductorswas alreadyobservedexperi-mentally.We discusstheseexperimentsin Sec.IV.

Sinceb

all of theabovemechanismsrequirethepresenceofaQ magneticfield, we concludethatdichroismin theMeissner

PRB 58 479ANALYSIS OF DICHROISM IN THE . . .±

phase[ is a surfaceeffect becausein the bulk the magneticfield doesnot coexistwith superconductivity.The region inwhichy coexistenceis founddependson thepenetrationdepth.Thet

largerthe penetrationdepth,the largerthe regionof thesampleX in which dichroismis produced.In the vortex phasethe\

field penetratesinto the bulk of the superconductorandcoexistsw with the order parameternear the vortices.EveryvortexZ is thusa sourcefor dichroismdueto eachof thethreeaboveQ mechanisms.

Mechanism 4 is caused by inhomogeneities of the super-conducting order parameter itself. The relevantmatrix ele-mentsi are

zn¿ { |˜ } m© ~ andQ � n¿ � �˜ * � m© � . � 51

r �To seehow theorderparametercanleadto a finite resultfor�

P wey first look at the quantity Cnn� �m¯ ,a as definedby Eq.�41� �

. As mentioned above, if the original normal-stateHamiltonian¨

doesnot breaktime-reversalandinversionsym-metry all momentummatrix elementsare real. Under thesecircumstancesw Cnn� �m¯ is purely imaginary.Since in Eq. � 34

T �the\

real part of the expressionunderthe sumon m© isu

taken,Cnn� �m¯ musti bemultiplied by a quantitywith a finite imaginarypart[ in orderto leadto a finite � P

³. Sincethematrix elements

for�

the SOC,ASOC,andorbital mechanismsexplicitly con-tain\

a factor i�,a this condition is normally satisfiedin these

cases.w The matrix elementsof the pair potential are easilyseenX to be purely real � andQ thusnot to producedichroism� if

�˜ * � r� �����˜ ��� r� � . � 52r  

Thet

condition ¡ 52r ¢

isu

violated if £˜ (�r� )� is real and breaks

inversionsymmetry¤ mechanism4a¥ , oa r if ¦˜ (�r)�

is symmetricunder§ inversion,but complex,in which caseit breakstime-reversal¨ symmetry © mechanismi 4bª .

Orderý

parameterswhich breakinversionsymmetry,whilethe\

underlying normal-stateHamiltonian doesnot, are ex-amplesQ of ‘‘unconventionalorder parameters’’in the sensethat\

the superconductingphasehasdifferent spatialsymme-tries\

from thenormalphase.42«

Thefact thatbrokeninversionsymmetryX cangive rise to dichroismis well known, e.g., inchemistry,w wheremoleculeswithout a centerof inversionarecalledw ‘‘optically active’’ since their optical propertiesde-pend[ on the polarizationof the light. In principle, inversionsymmetryX is alsobrokenat everysurface.However,aslongasQ the light penetratessufficientlydeeplyinto thematerialtosampleX the bulk properties,this contribution is expectedtobe¬

very small. This expectationis corroboratedby detailedinvestigationsof dichroism in the normal state,where thistype\

of surface-induceddichroismwasnot found to play anappreciableQ role in explainingexperimentalobservations.

A complex order parameteris found in the presenceofexternale magneticfields,wherethegradientof its phasecor-responds¨ to supercurrents.14 This

taspectof mechanism4b is

thus\

closelyrelatedto mechanism3. Complexorderparam-eterse are also possiblein the absenceof magneticfields, ifthe\

superconductingphaseitself breakstime-reversalsym-metry.This is thecase,for example,for the‘‘ d

­ ®is�

’’ typeoforderY parameterdiscussedin connectionwith the heavyfer-mions,i the high-temperaturesuperconductors,and anionicsuperconductivity.X 42–44

Dichroism producedby order parameterswhich breakinversion- or time-reversalsymmetrywas discussedprevi-ouslyY by severalauthors.In Ref. 45, a phenomenologicalanalysisQ of this mechanismfor the high-temperaturesuper-conductorsw is performed.Our analysisprovidespossiblemi-croscopicw mechanismsunderlying the phenomenologicaltreatment\

of Ref. 45. In Ref. 46, the breakingof thesesym-metriesi is discussedin thecontextof collectivemodesof theorderY parameter anQ effect we do not discussin the presentpaper[ ° . In Ref. 47, the effect of brokenchiral symmetryinthe\

normalstateof superconductorsis analyzed.This effectfollows in our framework if for the wave functions ± n� (� r)

�oneY usesthe eigenstatesof a normal-stateHamiltonianwithbroken¬

chiral symmetry.In this casethe momentummatrixelementse arecomplex,anddichroismis found from Eq. ² 33

T ³alreadyQ in zerothorder,i.e., evenin theabsenceof theabovefour�

mechanisms.Mechanism�

5 is therefore provided by a normal statewhich´ already displays dichroism. In the presentpaperwefocuson thesuperconductingstateanddo not discussfurtherthis\

type of normal-state-induceddichroism.Inµ

conclusion,we haveidentifiedfive distinctmechanismsfor dichroism in the superconductingphase.Two of thesemechanisms,¶ 1· andQ ¸ 3T ¹ ,a arealreadyknownfrom thenormalstate,X strongly modified, though,by the presenceof the su-perconducting[ order parameter.Mechanismsº 2E » ,a ¼ 4a

� ½,a and¾

4b¿ onY the other hand,exist only in superconductors.Onemechanismof eachtype,namely À 1Á andQ Â 2Ã ,a is of relativisticorigin,Y arising from the interplay betweenrelativistic sym-metryi breakingandsuperconductingcoherence.MechanismÄ5r Å

operatesY only in superconductorsin which the normalstateX alreadybreakstime-reversalor inversionsymmetry.

2.Æ

Existence criteria

WeÇ

can extract from the generalformula a number of‘‘existencecriteria’’ which determineunderwhich circum-stancesX onecanexpectdichroismat all. Fromtheabovedis-cussionw of the mechanismswe alreadyhave

Existence criterium 1: Thereis no dichroismdueto spin-orbitY coupling in the absenceof magneticfields.

ExistenceÈ

criterium 2: Thereis no dichroismdueto BCS-type\

order parameters,i.e., order parameterswhich are realandQ spatiallyconstant.

Existence criterium 3: Any perturbationwhosecontribu-tion\

to Eqs. É 37T Ê

– Ë 40� Ì

isu

purely real doesnot give rise todichroism.x

Thethird criteriumappliesin particularto thenormalandanomalousQ Darwin terms,containingÍ 2

Y Î(�r� )� and Ï 2

Y Ð(�r� ).� If

the\

potentialsÑ (�r)�

and Ò (�r)�

arereal andinversionsymmet-ric,¨ these terms lead only to real contributions underReÓ i� . . . Ô in Eq. Õ 34

T ÖandQ thereforedo not producedichro-

ism.u

If × (�r� )� and Ø (

�r� )� breakinversionsymmetryor arecom-

plex,[ dichroism already arises from the zero-orderwavefunctions� Ù

inu

the caseof Ú (�r� )� ] or from the matrix elements

ofY Û˜ in Eqs. Ü 37T Ý

– Þ 40ß . In this case,the Darwin termsaresmallX additionalcorrectionswhich do not breakany furthersymmetries.X Hence,the Darwin termsthemselvesare not asourceX of dichroism.A similar argumentappliesto the n¿ A

à 2

couplingw of theexternalvectorpotentialto thedensity.Theseconclusionsw hadbeenanticipated,in discussingthe relevantperturbations[ below Eq. á 21â .

480 PRB 58K.ã

CAPELLE, E. K. U. GROSS,AND B. L. GYORFFY�

A numberof further criteria canbe found from the vari-ousY ingredientsof Eq. ä 34

T å. Since the minimum value the

unperturbed§ energiescantakeis æ¯ ,a it follows from theargu-mentsof the ç function that there is an absorptionedgeatè�éëê

2ì¯ . This edge is a consequenceof the fact that welimited ourselvesto the considerationof pair-breakingpro-cesses.w For sufficiently low temperatures,wherepair break-ingu

is the only absorptionprocess,we thushaveExistence criterium 4: At low temperaturesthere is no

dichroismx

for frequenciesbelow the absorptionedge.Thet

SOCandASOC mechanismsrequirethe presenceofgradientsc of the lattice- and pair potentials.However, noteverye gradientproducesdichroism.In both caseswe haveaterm\

of the form

í îfÏ ï

r� ð ñ�òôó zõ öø÷ fÏ ù

rúûxü

ýþ

yÿ ��

fÏ �

r��yÿ

��

xü ,a � 53r �

wherey fÏ

(�r� ) i�

s (�r� ) o�

r (�r� )� , respectively.Obviously,thegra-

dientsx

in the z� direction,x

i.e., the direction of the magneticfield(

andthe incident light, do not enter.This leadstoExistence criterium 5: In the polar geometry(B � q )

�di-

chroismw arisesentirely from the lateral � in-plane� inhomoge-neity,� not from the perpendicularinhomogeneity.

Thist

observationis particularlyrelevantfor surfacegeom-etries.e As long aslight incidenceandmagneticfield areper-pendicular[ to thesurface,thesurfacegradientsthemselvesdonot� producedichroismdueto mechanisms1 or 2 � SOC

band

ASOCd �

,a only the muchsmallerlateralgradientsdo. Mecha-nism 4b � broken

¬inversionsymmetry� ,a on the otherhand,is

operativeY at surfaces,as inversion symmetryis necessarilybroken.¬

However, as mentionedabove, this mechanismisexpectede to yield only a very small contribution,as long asinversionsymmetryis not brokenin thebulk of thematerialasQ well.

Finally,)

we note that all the energiesin Eq. � 34T �

areQ un-perturbed[ energies.The stationaryperturbationsenter onlythrough\

their effecton thewavefunctions,asdeterminedbyEqs.� �

7� �

andQ � 8� � . Thereasonfor this is that in calculating� P³

to\

first orderin theperturbations,thecoefficientof the termscontainingw theenergyshift is, of course,thedifferenceof thezero-order� matrixi elements.This difference,specifiedin Eq.�33T

,a is zero.Thus thesecontributionsvanishidentically. InsecondX orderin thestationaryperturbationstheenergyshiftsreappear.This givesus

ExistenceÈ

criterium 6: For sufficientlysmallperturbations!suchX that a first-order treatmentis justified" perturbations[

leavingthezero-orderwavefunctionsunchangeddo not giverise to dichroism.

Thist

existencecriteriumalsoprovidesa differentpoint ofviewZ of the abovediscussedfact that the Zeeman(m B)

�couplingw of a constantmagneticfield to the electronspinsdoesx

not producedichroismwithout the simultaneouspres-encee of SOC,while the j

#Aà

couplingw does:Namely,unlikethe\

coupling to the orbital currents,the Zeemanterm alonedoesx

not changethe form of the wavefunctions.

3.$

The normal-state limit

In thenormalstate,thepair potential% (�r)�

vanishesiden-tically.\

Thereforethematrix elementsof theASOCtermand

ofY &˜ areQ zero, while Cnn� 'm¯ isu

unaffected.The transitionweconsiderw is from an occupiedstaten¿ (*) whichy in the normalstateX lies below the Fermi surface+ to

\an unoccupiedstaten¿,

lying abovethe Fermi surfacein the normal state- . In theparticle-hole[ convention the energiesof statesbelow theFermi)

surfaceareinterpretedasholeenergiesandtakento bepositive.[ It is only the quantumnumber n¿ whichy distin-guishesc betweenoccupiedandunoccupiedstates.

Using.

this convention,the /10 0;

limit of the coherencefactors�

can be evaluatedstraightforwardly. Explicitly wehave2

p½ 3 n¿ ,a n¿ 46587 u9 n� : n� ;=<?> n� u9 n� @A 1

21 B C

n�EÈ

n�1

21 D E

n� FEÈ

n� GH 1

21 I J

n�En�

1

21 K L

n� MEn� NO 1. P 54

r QSimilarlyb

we find

p½ (�mR ,a n¿ S )� T 1 if U mR V is above W F

0 i;

f X mR Y isu

below Z F,a [ 55

r \

lQ ]

mR ,a n¿ ^6_8` 0 i;

f a mR b isu

above c Fd

1 if e mR f is below g Fd .

h56r i

As jlk 0;

, Eq. m 34T n

therefore\

becomes

oP³ p NH qsrut e· 2

YEÈ

0D2

mR 2Y vxw

nn� y{z fÏ |~}��

n� ��� fÏ �����

n� �{� ������� n� ����� n� ��� �1��������

m¯ Re Cnn�  m¯ h� ¡¢¡nm�£

n� ¤�¥ m¯ ¦ Cnn� §m¯¯ h� ¨¢¨nm� ¯©

n� ª�« m¯ . ¬ 57r ­

Similarb

to the superconductingcase,the secondterm in thesumX on mR represents¨ the contributionof the single-particlestatesX belowtheFermisurface,while thefirst termrepresentsthat\

of statesaboveit. This formuladescribesdichroismdueto\

the creation of particle-hole excitations in the normalmetal.i It is readilyverifiedthat thesameresultis obtainedinaQ normal-statecalculation:if the unperturbedHamiltonianistaken\

to be Eq. ® 4 ,a simple first-order perturbationtheoryleadsto Eq. ° 57

r ±. The ²1³ 0

;limit of our formula thus cor-

rectly reproducesthe correspondingnormal-stateresult.Re-sultsX for nonsuperconductingmaterials,which areof thetypeofY Eq. ´ 57

r µ,a were derived previously by several

authors.Q 34,40,48,49¶

Many of the above-mentionedconclusionsaboutmecha-nisms� andexistencecriteriaarevalid alsoin thenormalstate·wherey theyaremostlywell known . We just mentiona few:¹i º Thereis no effect to zero order in the stationarypertur-

bations,¬

evenin the presenceof polarizedlight andZeemansplitting.X » iiu ¼ The

tconstraintson the geometryof the experi-

menti arethesame.½ iiiu ¾ SOCb

alone,without ZeemansplittingofY the energies,doesnot lead to dichroism. ¿ iv À The argu-menti that the Darwin term andthe A

à 2 term\

do not, on their

PRB 58 481ANALYSIS OF DICHROISM IN THE . . .Á

own,Y produce dichroism, remains valid. Â vZ Ã The above-mentionedresultthat theeffectarisesto first orderonly fromthe\

changeof thewavefunctionsdueto theperturbationandnot� from the changein the energiesstill holds. Ä Inµ higher-orderY perturbationtheory,of course,the energyshifts reap-pear.[ Å Conclusions

v Æi Ç – È iv É areQ well known in the theory of

dichroismx

in normal and magneticallyorderedmetals,butareQ usuallyarrivedat usingmorecomplicatedmethods.Thedominancex

of the changeof the wave function over that ofthe\

energiesÊ conclusionw Ë vZ Ì=Í wasy notedby severalauthors,whoy arrivedat this or very similar conclusionsin a varietyofways.y 40,48,50–52

III.Î

MODEL CALCULATIONS

A. A model for the SOC mechanism

After discussinggeneralconsequencesof the resultswenow� turn to their approximatenumericalevaluation.A fullevaluatione of Eq. Ï 34

T Ðrequires¨ self-consistentsolutionsof

the\

SBdGEin the presenceof magneticfields andremainsaproject[ for the future.Herewe only look for a simplemodelwhichy illustratesthe main physicalaspectsof the theory.Tothis\

endwe specializeto mechanism1.By imposinga numberof physicalconditionson the su-

perconductor[ understudywe canmakesurethat the contri-butions¬

of theremainingmechanismsaresmall.MechanismsÑ2E Ò

andQ Ó 4� Ô dox

not contributeif the pair potentialis real andspatiallyX constant,i.e., for orderparametersof the type con-sideredX in the original BCS model. Moreover, in materialswithy heavyelementsÕ high

2atomicnumberZ

Ö)�

in the lattice,the\

SOC × whichy increasesroughly as ZÖ 4)�

is strongly en-hanced,ascomparedto low Z materials.

For superconductorswith constant pair potential andheavy2

elementsin the lattice, in sufficiently weak externalfields,(

we canthereforelimit ourselves,for the purposeof amodelcalculation,to the SOCmechanismØ 1Ù . We thusne-glectc thematrix elementsfor theothermechanismsandkeeponlyY

h� Ú¢Únm� ÛÝÜ

M�

nm� Þàßâá ã 58r ä

withy

M�

nm� åàæâç : è 1

4�

im� 2c 2 é n¿ ê{ë6ìîíðï1ñîò zõ ó mô õ ö 59

r ÷in the equations.ø In this sectionwe set ùûú 1.ü The variouscoherencew factorscanall be expressedin termsof the func-tion\

p½ ý E,a E þ6ÿ : � 1

21� �

E

1

21 � ���

E �� 1

2E 1

EÈ 1

2E 1 � ��

EÈ � ,a � 60

� �whichy in the respectiveenergyintervals reducesto the ap-propriate[ coherencefactors. � is

udefinedthrough

� : ��� E2 ��� 2. � 61� �

Finally,)

we define

F� �

EÈ

,a EÈ � ,a H ! : " fÏ #%$

EÈ &('

fÏ )%*

EÈ + ,.-0/

fÏ 1%2

EÈ 3(4

fÏ 576

EÈ 8 9.:

. ;62� <

By introducingthe densityof states,N=

(�E)�, in the supercon-

ductingx

state> SDOSb ?

,a we canconvertthesumsin Eq. @ 46� A

inu

integrals.u

Oneof the integralscanimmediatelybeperformedduex

to the B function.Utilizing the abovedefinedquantities,Eq.� C

46� D

isu

written as

EP³ FHGJILKNM

2E OQPLR

CM S FT U e· 2E

È0D2Y

mô 2 VW XZY\[^]

dE­

1 _\`a^b dE­

2Y c d e dE

­2Y F f E1 ,a gih E1 ,a H j

EÈ

1 k EÈ

2Y

lp½ m E1 ,a nNo E1 p p½ q E1 ,a E2

Y r p½ s E2Y ,a tNu E1 vw

N= x

E1 y N= z E2 { N= |L}N~ E1 � ,a � 63� �

wherey theE1 integrationarisesfrom thesumon n¿ andQ theE2integrationu

from the sum on mô . The signatureof supercon-ductivityx

in this equationis the appearanceof the coherencefunctions�

and the SDOS. The matrix elementsM�

nm� andQCnn� �m¯ ,a on the other hand,are normal-state¿ matrixi elements.WeÇ

haveapproximatedtheseby their average� CM � FT overYthe\

Fermi surface.Since superconductivityhappensessen-tially\

within a few meV aroundthe Fermi surface,this is averyZ reasonableapproximation.TheaboveprocedurehastheadvantageQ that the remainingtwo integralscanbe evaluatednumerically,� oncea modelfor theSDOSis chosen.By form-ingu

the ratio of thesuperconductingresultto its normal-statelimit,

�PS�

�P³ NH : �

�P

lim� ���

0D � P³ ,a � 64

� �

the\

averagematrix elementscanceland we obtain a directmeasurei for the interplay betweenthe superconductingco-herence2

and spin-orbit coupling. The samestrategyis alsoappliedQ in a large numberof similar model calculationsforotherY propertiesof superconductors,e.g., for the absorptionofY sound,30,41

�the\

nuclear-spinrelaxationrate,53�

the\

thermalconductivity,w 54

�the\

spinsusceptibility,55�

the\

absorptionof un-polarized[ light,31

�etc.e

B.�

Qualitative analysis of the SOC mechanism

The simple form of Eq. � 63� �

allowsQ us to draw somefurtherconclusionsaboutthenatureof SOC-induceddichro-ismu

in superconductors,beforeproceedingto its numericalevaluation.e Closeto T � 0

;we can replacethe pair potential�

(�T)�

by its zero-temperaturevalue � (0),�

and the Fermifunctions�

in F�

(�EÈ

1 ,a �i� EÈ

1 ,a H )�

by stepfunctions,accordingtofÏ

(� �

En� � )� � fÏ

(� �

En� �¡ £¢ BH)� ¤¦¥

(�En� §¡¨£© BH)

�. These step

functionsimposerestrictionson the integrationlimits of theE1 integral,which cansymbolicallybe written as

482 PRB 58K.ª

CAPELLE, E. K. U. GROSS,AND B. L. GYORFFY«

¬­\®^¯

dE­

1F ° E1 ,a ±i² E1 ,a H ³´ µ Min[

¶ ·\¸^¹, º\»½¼ B

¾ H¿

]ÀdE­

1 ÁMax[ Â , Ã B

¾ H]À

Ä\Å^ÆdE­

1 .

Ç65� È

Inµ

the É usual§ Ê casew that Ë BH Ì�Í

wey immediatelyseethatthetwo\

integralson the right-handsideof Eq. Î 65� Ï

cancelw eachotherY exactly. If Ð B

Ñ H

exceedse Ò there\

is no suchcancella-tion.\

Thereis thus no SOC-induceddichroismin supercon-ductorsx

at TÉ Ó

0;

, unlessthe magneticfield is of the orderofmagnitudei of the energygap.Physically,this canbe under-stoodX asfollows: at T

É Ô0;

all electronsarecondensedin pairswithy net spin zero. The systemis thus not spin polarized,evene for finite magneticfields. In this casethe symmetryinspinX spaceis not brokenandhenceSOCcannotinducebro-kenÕ

chiral symmetry.Hence,no dichroismresults.A moregeneralc viewpoint is that time-reversalinvariancemust bebroken¬

to havedichroismfrom the SOCmechanism.Since,by¬

construction,the groundstateof a superconductorat TÉÖ 0

;consistsof pairsof mutually time-conjugatestates,it is

invariantu

undertime reversal,evenin the presenceof finitefields,(

so that no dichroismcanarise.If, on the otherhand,the\

magneticfield is strong enoughto break Cooperpairsparamagnetically,[ there exist unpairedelectronseven at T

É× 0;

. In this caseone finds a finite spin polarization Ø i.e.,u

aground-statec breakingtime reversalÙ atQ T Ú 0

;anddichroism

results.¨ Thus,for TÉ Û

0;

and Ü BÑ H Ý�Þ there

\is no dichroism,

whiley for T ß 0;

and à BH á�â there\

is a finite amount ofdichroism,x

in accordancewith the aboveanalyticalfindings.Thist

is a direct manifestationof paramagneticpair breaking,aQ phenomenonwhich is hard to observeexperimentallybyotherY means.Of course,the details of the phasetransitioninducedby paramagneticpair breakingcannotbe describedwithy our simple model for the unperturbedsuperconductorandQ our approximatetreatmentof the matrix elements.Inparticular,[ real superconductorsdisplay paramagneticlimit-ingu

alreadyat H ã�ä

/å æ

2ç

andnot at H è�é

.30�

Thet

factorof ê 2ç

follows from a comparisonof total ground-stateenergiesandis not containedin our single-particlemodel.The merefact,however,2

that thereis no spin-orbit-induceddichroismat TÉë 0

;until thefield is of theorderof magnitudeof thegap,is

aQ very definiteconsequenceof our results.Interestingly,if we performthecorrespondinganalysisfor

the\

orbital currentmechanismì 3T í ,a we find that the two inte-gralsc in Eq. î 65

� ïareQ not subtracted,but added,henceleading

to\

a finite resultevenfor T ð 0;

and ñ BH ò�ó . Physically,thiscanw be interpretedby noting that eventhe smallestmagneticfield(

givesrise to orbital currentsin the superconductor,re-gardlessc of temperature.The

Édifference between the orbital

and± the spin response of a superconductor thus has a deci-siveô influence on the low-temperature behavior of dichroismin�

superconductors.WeÇ

now return to the caseof finite temperatureand in-vestigateZ thebehaviorof õ P

³asQ a functionof the ö externale or

internalu ÷

magnetici field. The field enters only throughF(�E,a E ø ,a H)

�, which can be expanded,for sufficiently small

H,a aboutH ù 0;

. Onefinds

úP³ û

Hü ý

fÏ þ ÿ � E

È �fÏ �

EÈ �����

fÏ � �

EÈ �����

fÏ ���

EÈ �����

O� �

Hü 2 � ,a�

66� �

wherey fÏ �

(�EÈ

)�

standsfor the derivativeof the Fermi function.Hence,for small fields, P,a and the imaginarypart of theoff-diagonalY elementsof the conductivity tensorfor super-conductors,w are linear functions of H. The absenceof thezero-orderterm meansthat the SOC mechanismalonewillnot� give riseto dichroismwithout magneticfields.This is thesameX conclusionwe alreadyarrivedat in Sec.II C 1 by fol-lowing a somewhatdifferent line of reasoning.

C. A BCS superconductor in the Meissner phase

Now!

we turn to thenumericalevaluationof Eq. " 63� #

. Forthis\

purposewe have to assumea particular form for theSDOS.b

The simplestchoice,the BCS form

N$

BCS% EÈ &' N$

N( E)

EÈ 2 *,+ 2

,a - 67� .

wherey N$

N( isu

theDOSof thenormalmetal,leadsto unphysi-calw singularitiesat E /,0 . In everyreal superconductorthesesingularitiesX are smoothedby a large numberof processes,suchX as gap anisotropy,residualinteractionswith phonons,strong-couplingX effects,inelasticscatteringfrom impurities,etc.e 30,53,56,57

�Many parametrizationsof theSDOSwhich take

these\

effects into account have been suggestedin theliterature.� 41,53,56–58

Inµ

the following, we modeltheSDOSin a way similar tothat\

of HebelandSlichter in their seminalwork on NMR insuperconductors.X 53,57,58

1The SDOSis written as a weighted

averageQ over the BCS-DOS,

N$ 2

E 3 : 4 Re 576879

dE­ :

N$

BCS; E <>= w´ ? E,a E @>A . B 68� C

Thet

weightingfunctionw´ (�EÈ

,a EÈ D )� cantakea varietyof forms,but¬

in the caseof NMR experimentson BCS superconduct-orsY the simplesquareform

w´ E EÈ ,a EÈ F>GH 1

2ç IKJ E

È L�MONEÈ POQSR T U

EÈ VXWSYZ

EÈ [

,a \ 69� ]

whichy makesw´ constantw for EÈ ^

between¬

EÈ _X`

andQ EÈ aXb

,aandQ zero everywhereelse, already leads to quantitativeagreementQ with experiments.53,56

1–58 The broadening c is

typically\

chosento be dfe 0.1; g

(�T)�. Equationsh 68

� iandQ j 69

� kconvertw the singularity into a peak.In the caseof NMR thispeak,[ in conjunctionwith thecoherencefactors,givesrise tothe\

well-known Hebel-Slichterpeak in the nuclear-spinre-laxation rate.30,53,57,58

lThe form, Eqs. m 68

� nandQ o 69

� p, oa f the

SDOSb

canbe physically justified in termsof the anisotropyofY the gap,53,56

1–58 but

¬for the presentpurposeit canalsobe

regarded¨ as just a simple phenomenologicalmodel. Apartfrom this simple form we also consideredseveralmore so-phisticated[ expressionsfor the SDOS,but it turnedout thatitsu

detailedform doesnot qualitatively q andQ quantitativelyonlyY up to within 5–10%r affectQ our results,as long as thesingularityX is smoothedout in whateverfashion.We havetherefore\

chosento work with Eqs. s 68� t

andQ u 69� v

for numeri-calw simplicity.

PRB 58 483ANALYSIS OF DICHROISM IN THE . . .w

In orderto investigatedichroismasa functionof tempera-ture,\

we alsoneedto specifythe temperaturedependenceofthe\

pair potential.For BCS superconductorsthis is found byinterpolatingu

betweentwo analyticallyknown limiting cases.Closev

to TÉ x

0;

we usetheformuladerivedin Ref.59 in termsofY Besselfunctions, while close to T y Tcz wey employ theparametrization[ of Ref. 56 which very close to T

É {TÉ

cz re-¨ducesx

to the standardBCS squareroot form. | More detailscanw be found in thesereferences.}

In~

theactualcalculationwe needto assignphysicallyrea-sonableX valuesto the parametersenteringEq. � 63

� �. For the

zero-temperaturepair potential we choosea value of � (�T� 0)

; �1 meV. Assuminga BCS-typerelation betweenthe

pair[ potentialat TÉ �

0;

andthe critical temperaturethis leadsto\

Tcz � 6.6�

K, which is closeto that of Pb.The normal-stateDOS� �

NDOS! �

isu

assumedto be constantbetween� 0.3;

and�0.3;

eV and zero everywhereelse.This correspondsto asquare-shapedX NDOScenteredaroundtheFermienergy.TheabsoluteQ valueof theNDOSdoesnot enterbecauseit cancelswheny forming the ratio � 64

� �. Of course,all thesenumbers

canw be modifiedeasily,without changingany of our conclu-sionsX in an essentialway.

The assumptionsmadeand the numericalvalueschosen,specifyX thesuperconductorunderconsiderationto bea BCS-type\

sô -wavesuperconductorin theMeissnerphase.Thusthedichroismx

we discussin the following is inducedin the re-gionc wherethe magneticfield penetrates.The more this su-perconductor[ is typeII � i.e.,

uthelargertheratio of penetration

depthx

to coherencelength� ,a the larger is the region of thesampleX where the magneticfield and the order parametercoexistw and dichroism results. We therefore assumethemodeli superconductorto bea strongtype-II superconductor.�Note!

that this assumptiondoes not¿ underly§ the generaltheory\

leadingto Eq. � 34T ���

. Sinceunderthesecircumstancesthe\

penetrationdepthis muchlargerthanboth thecoherencelength�

and the lattice constant,the light probesthe bulk ofthe\

superconductor.At this point we thereforeneglectanysurface-specificX sourcesfor dichroismand use typical bulkvaluesZ for the energygap,the SDOS,etc.

In~

concludingthis sectionwe stressthat our calculationsareQ meantonly to illustrate the physicsof the SOC mecha-nism andnot to providequantitativepredictionsfor real su-perconductors.[ For the latter purpose,a self-consistentsolu-tion\

of the SBdGEis required.

D. Numerical results

Thet

integrationsin Eq. � 63� �

areQ performednumerically.Inthe\

following subsectionswe displaytheresultsof thesecal-culationsw as a function of temperature,magneticfield, andfrequency.

For)

eachof thesewe first presentthe result for a normalconductor.w Although the correspondingcalculationscan bedonex

for a normal conductorin a self-consistentand fullyrelativistic¨ manner,1,2 wey perform them using the sameap-proximations[ as discussedabovefor the superconductor,inorderY to facilitate the comparisonwith the results for thesuperconductor.X Strictly speaking,the calculationfor a nor-mali conductor loses its meaningbelow T

Écz . However, as

comparedw to the superconductor,the responseof the normalconductorw doesnot vary significantly with temperature.In

anyQ case,it constitutesa convenientnormalization.We dis-play[ the result in arbitrary units, becausethe prefactorsaredeterminedx

by the absolutevaluesof the matrix elementsM�

nm� andQ Cnn� �m� andQ the NDOS at the Fermi surface,noneofwhichy can be calculatedwithin the simple model of thepresent[ section.

WeÇ

alsoshowthe ratio of the superconductingresultstothose\

for the normal conductor.Here the matrix elementscancelw andno ambiguity is left. We thenhavea direct illus-tration\

of thechangein theresponseof themetalto polarizedlight�

due to the presenceof the superconductingcoherence.This strategy,of course,is standard,andwasusedsuccess-fully in many similar calculations� cf.w the referencesbelowEq.� �

64� ���

.

1. Dependence on temperature

Figure)

1 showsthat dichroism in the normal conductorincreasesu

with decreasingtemperature.Figure 2 showsthecorrespondingw calculationfor thesuperconductor,dividedbythe\

valuesfor the normal conductor.The calculationswere

FIG. 1. Dichroism in the normal statevs temperatureT.� Thisandall otherfiguresdisplayonly the contributionof SOC-induceddichroism.The other mechanismsare excludedfrom the calcula-tion, as discussedin the main text. The numericalvaluesof theparametersspecifyingthe systemaregiven in the main text.

FIG. 2. Dichroismratio vs T at small magneticfields.A strongcoherencepeak is seenclose to Tc� ,� while near T � 0

 the curve

approaches0.

484 PRB¡

58K.¢

CAPELLE, E. K. U. GROSS,AND B. L. GYORFFY£

donex

for a frequencyof 4.5meV anda magneticfield of 0.05T. Figure3 repeatsFig. 2, but for a magneticfield closetothe\

paramagneticlimit of superconductivity.¤ Thist

plot cor-responds¨ to a superconductorin which the critical field fororbitalY pair breakingH

¥cz 2¦ isu

high enoughthat the paramag-netic� limit is actuallyreached.§

Sinceb

Tcz ¨ 6.6©

K, thedatapointsbetween6.6 and 8 K ax

rethe\

samein Figs. 1 and 2. The above statementthat theresponse¨ of the normal conductor does not changeverymuch,if comparedwith thesuperconductingresponse,is im-mediatelyverified.

Clearly,v

thedominatingfeatureof Fig. 2 is thehugepeakjustª

belowTcz . Similar peaksin otherobservablesarisefromthe\

peak in the SDOS in conjunction with the coherencefactors.�

Thesepeaksare known as Hebel-Slichteror coher-encee peaks.30,41

lHowever, the conventionalHebel-Slichter

peak,[ as seen,e.g., in NMR experiments,53,57,58,60«

typically\

reaches¨ valuesin the range ¬ 1–5­ ,a while the peakin Fig. 2goesc up to almost33.

Someb

part of this additionalenhancementmay be duetothe\

oversimplifiedapproximationsfor the matrix elements,the\

NDOS and the SDOS.On the other hand,the approxi-mationsi for the matrix elementsare the samein the super-conductorw and in the normal conductorand do not lead topeaks[ in thelattercase.Furthermore,differentmodelsfor theSDOSb

leadto enhancementsof thesameorderof magnitude.Thist

suggeststhat there is a secondmechanismat workwhichy is responsiblefor the additionalenhancementof thepeak.[ Numerically, this effect arisesfrom the regionsof in-tegration\

in Eq. ® 63© ¯

whichy are excludeddue to the energygapc in the superconductor,but contributeto the integralsforthe\

normal conductor.Performinga numericalexperiment,wey can integratethe normal conductorwith the integrationlimits�

of thesuperconductor.It turnsout that it is mainly thelimits of the inner integration ° overY E2)

�which in this case

lead to a strongenhancementof the normal-conductingre-sultsX aswell.

Thist

effect is independentof the form of the DOS andthus\

not thesameastheconventionalHebel-Slichtermecha-nism.It canbeinterpretedby notingthat,apartfrom produc-ingu

thepeakin theSDOS,thegapalsoactsasanabsorptionedgee which is presentin the superconductorbut not in thenormal conductor.Experimentsand theoreticalcalculations

for�

magneticallyorderedmaterialsshow that dichroism isalwaysQ greatlyenhancednearabsorptionedges.1,2 Theabovenumericalexperimentandthe additionalenhancementof di-chroismw in superconductors,seenin Fig. 2, indicatethat thisisu

the casefor superconductorsaswell.WeÇ

canthus identify two± independentways in which di-chroismw in superconductorsis enhancedascomparedto thenormal� conductor:Thepileup in thedensityof states,givingrise to the Hebel-Slichter-typeenhancementand the gap it-self,X producingan absorptionedge.Both the original Hebel-Slichterb

mechanismand the additional ‘‘gap-enhancement’’cruciallyw dependon the existenceof an energygap.For su-perconductors[ in which theenergygapis zeroon somepartsofY the Fermi surface,such as in d

­-wave superconductors,

they\

will be stronglyreducedor evennot presentat all.61²

The physicsat low temperaturesis very different fromthat\

at higher temperatures.By comparing the low-temperature\

behaviorof Figs. 1, 2, and3 we canverify theanalyticalQ resultsobtainedin Sec.III B. In the normal con-ductorx

andthe superconductorin the realmof paramagneticpair[ breakingthere is a finite amountof SOC-induceddi-chroismw at T ³ 0

;, while in thesuperconductorat lower fields

dichroismx

is quenchedat TÉ ´

0.;

2. Dependence on the magnetic field

Figure)

4 displaysthebehaviorof dichroismin thenormalconductorw asa function of the magneticfield. In accordancewithy the discussionin Sec.III B it startsfrom zeroat H

¥ µ0;

andQ risesalmostlinearly. ¶ For)

fields about10 timesaslargeasQ in the figure a slight deviation from linearity is found.·Figure 5 illustratesthat, althoughboth the normal and thesuperconductorX are approximatelyproportional to H

¥,a their

ratio is not constant.This is dueto thehigher-ordertermsinH¥

. Both figures are for a temperatureof 2.5 K and a fre-quency¸ of 4 meV.

Figure6 displaystheratio for thecaseof zerotemperatureandQ very largemagneticfields.All datapointsat H ¹ 17.5TareQ zero,while above17.5T suddenlyfinite valuesshowup.Thist

reflectsthe paramagneticlimit, asdiscussedabove.In-deed,x

H¥ º

17.5T correspondsto an energyof 1 meV, whichis just thevaluechosenfor thepair potentialat T » 0

;. Super-

FIG. 3. Dichroismratio vs T at largemagneticfields.Paramag-netic pair breakingleadsto a finite valueat T

¼ ½0.¾ FIG. 4. Dichroism in the normal statevs magneticfield H at

finite temperature.¿ P risesalmostexactly linearly with H.

PRB 58 485ANALYSIS OF DICHROISM IN THE . . .w

conductorsw where the upper critical field is thought to beinfluencedby paramagneticlimiting are, e.g.,62

²the\

heavy-fermion�

compoundsUBe13 andQ CeCu2¦ Sib

2¦ ,a theChevrelphase

material Gd0.2À PbMo6

² Sb 8,a the A-15 superconductorNb!

3l Ald

0.75À Ge

Á0.25À andQ thin films of, e.g.,Al or Sn.63,64

²WhileÇ

plausible,[ this is not supportedby direct experimentalevi-dence.x

Clearly,undersuchcircumstances,theobservationofdichroismx

at H Â Hcz 2,a as in Fig. 6, could be decisive.

3.$

Dependence on frequency

Figure 7 shows à P in a normal conductorversusfre-quency.¸ At ÄÆÅ 0

;thereis, of course,no dichroismbecause

no� transitionscantakeplace.At higher frequenciesÇ PÈ

ap-Qproaches[ an almostconstantvalue.This reflectsthe feature-lessNDOS which was usedin the calculationsÉ namely itsaverageQ valueat the Fermi surfaceÊ .

Thet

correspondingplot for the ratio, Fig. 8, displaysanabsorptionQ edgeat ËÍÌ 2Î . This edgeis dueto our limitationto\

pair-breakingprocesses.At low temperaturesvery fewexcitede quasiparticlesare presentand pair breaking is theonlyY availablemechanismfor absorption.The inclusion of

scatteringX from excitationsÏ i.e.,u

brokenpairsÐ isu

knownfrominvestigationsu

of the absorptionof unpolarizedlight,30Ñ

to\

givec rise to some additional absorptionbelow the edge,whichy doesnot significantlyaffect the part of the curvedueto\

pair breaking.Thet

shapeof thepeakdirectly abovetheedgereflectsthebehavior¬

of the perturbation,acting on the system,undertime\

reversalÒ cf.w thediscussionin Ref. 30, in particularFig.2-9Ó Ô

. The curve in Fig. 8 is of a mixed type, which reflectsthat\

we have two perturbationsacting on the system:thespin-orbitX coupling,which is evenundertime reversal,andthe\

magneticfield, which is odd.BothÕ

plots weredonefor a temperatureof 3 K Ö welly be-low�

the strongpeakseenin Fig. 1× andQ a Ø relatively¨ strongÙmagneticfield of 0.1 T.

IV.Ú

EXPERIMENTAL ASPECTS

A number of experimentalresults on dichroism in thevortexZ state of high-temperature superconductorsareavailable.Q 3,4

ÑBelowÕ

we offer somespeculativeremarkson theinterpretationu

of theseexperimentson thebasisof thepresenttheory.\

First,we notethat thedifferenceLHP-RHP,which canbe

FIG.Û

5. Dichroism ratio vs HÜ

at finite temperature.The super-conductordisplaysa strongervariation with the magneticfield ascomparedto the normalconductor.

FIG. 6. Dichroismratio vs H at zerotemperature.A finite-spinpolarizationis not produceduntil the magneticfield is comparableto the energygap.

FIG. 7. Dichroismin the normalstatevs frequencyÝ .

FIG. 8. Dichroism ratio vs Þ . The absorptionedgeat ßáà 2âhasa mixed type I-II character,reflectingthe behaviorof the per-turbationsundertime reversal.

486 PRBã

58K.ä

CAPELLE, E. K. U. GROSS,AND B. L. GYORFFYå

readoff from Fig. 3 of Ref.4, displaysa strongenhancementbetween¬

Tæ ç

Tæ

cz andQ Tæ è

Tæ

cz /2,é

which is consistentwith ourgeneralc predictionof enhanceddichroismbelow Tcz .

Thist

differencewastheoreticallyanalyzedon thebasisofthe\

cyclotron motion of the orbital currents circulatingaroundQ the vortex core.25,65

¦In our framework this corre-

spondsX to mechanismê 3T ë . Theorbital mechanismcanbedis-tinguished\

experimentally from the SOC mechanismbe-cause,w asexplainedin Sec.III B, it leadsto a finite valueforì

PÈ

andQ Im íïî ˆ xyð ñ atQ Tæ ò

0;

. By contrast,the SOC-induceddichroismx

vanishesat T ó 0;

, asdemonstratedanalytically inSec.b

III B andnumericallyin Sec.III D 1. Thedataof Ref. 4dox

indeedextrapolateto a finite valueat T ô 0;

. The presentanalysisQ thus supports the assumptionsmade, as to themechanismi responsiblefor the dichroism,in evaluatingtheabove-mentionedQ experiments.

Thereareat leasttwo featuresof theseexperimentswhichsuggestX that the spin-orbit mechanismsare presentas well.First,)

theobserveddifferenceLHP-RHPdisplayschangesofsignX 3,4

õbetween¬

T ö Tcz andQ T ÷ 0;

. Such a behaviorfinds anatural explanationin the presenceof two distinct mecha-nisms� for dichroismwhich havea different temperaturebe-havior,2

but areoperativefor roughly thesamevaluesof tem-perature,[ magneticfield, andfrequency.This interpretationisfurther�

supportedby the fact that, in order to phenomeno-logically�

fit their measureddata,theauthorsof Refs.3 and4had to assumetwo± independentLorentzianoscillators.Evi-dently,x

coexistenceof theSOCor theASOCmechanismø 1ùandQ ú 2Ó û ,a with the orbital mechanismü 3T ý ,a providesa possibleexplanatione for theseobservations.

Second,b

as statedin Ref. 3, the orbital mechanismalonedoesx

not fully explainthe observationsbecausetherearein-dicationsx

for ‘‘other þ electronlikee ÿ chiralw resonancesin addi-tion\

to the simple cyclotron resonance.’’3õ

Clearly,v

the spin-orbitY mechanismsSOCandASOCcanplay therole of theseelectronlikee chiral resonances.

In~

a differentsetof experiments5–7«

spontaneousX dichroisminu

high-temperaturesuperconductors,both aboveandbelowTcz ,a was observedto exist at frequenciesmuch higher thanthose\

consideredin the presentpaper.In view of this differ-encee in frequencyit is unlikely that theseeffects are pro-ducedx

by the pair-breakingcontributionto the SOC,ASOC,orY orbital mechanismsdiscussedabove.Thefact thatdichro-ism is observedalso above Tcz points[ to mechanism5�normal-statedichroismwhich continuesto bepresentbelow

Tæ

cz )� , in which casethe energyscaleis not setby the energygap,c but by normal-state parameters. More recentexperimentse 8 indeedsuggestthat this is the correctexplana-tion.\

Althoughd

the resultsobtainedin this paperare thuscon-sistentX with all availableexperiments,moredetailedexperi-mentalandtheoreticalinvestigationsarecalled for, in orderto\

identify the variousmechanismsanalyzedin this paperinanQ unequivocalway.

The SOCmechanismmay be easiestto identify in super-conductorsw which satisfyasmanyof thefollowing criteriaaspossible:[ � iu �

heavy2

atomsin the lattice � this\

favors the con-ventionalZ SOC, which rises approximatelyas Z4

«),� �

ii � nozeros� of the energygap � suchX zeroscaneliminatethe coher-encee peak� ,a iiiu

shortX coherencelengthandlargepenetrationdepthx �

to\

maximizethe regionin which the orderparameter

andQ themagneticfield arepresentsimultaneously� ,a iv � weakyshieldingX currents � to\ minimize the influenceof the orbitalmechanismi � . Theseconditions are approximatelysatisfied,e.g.,e by lead.

For the ASOC mechanismto be observedwe suggest:� i �light�

atomsin the lattice � to\ minimize the effect of the SOCterm\ �

,a � iiu �large�

gradientof the pair potential � i.e.,u

largeen-ergye gapandshortcoherencelength� ,a � iii � aQ largepenetrationdepth,x �

iv � weaky shieldingcurrents,as for the SOC mecha-nism.� It shouldbestressedthat theobservationof theASOCmechanismi would beof generalsignificancebecauseit couldconfirmw or reject the form of the relativistic BCS Hamil-tonian\

which hasbeenproposedonly recently9–1�

1 andQ is notyet� experimentallyverified.

Thet

pair potentialmechanism� 4� isu

uniquein that it doesnot dependon theexistenceof an ! externale or internal" mag-netic� field. The observationof dichroism below T

æcz inu

theabsenceQ of suchfields would be a stronghint at mechanism#4$ .

V.%

CONCLUSION AND OUTLOOK

WeÇ

havepresenteda perturbativeapproachto theabsorp-tion\

of polarizedlight in superconductors.Severaldistinctmechanismsfor dichroism in superconductorswere identi-fied andinterpreted.Theseare & 1' the

\conventionalspin-orbit

coupling,w ( 2Ó )the\

anomalousspin-orbit coupling, * 3T +orbitalY

currents,w , 4a� -

complexw order parameters, . 4b� /

inversionu

symmetry-breakingX orderparameters,and 0 5r 1aQ normalstate

whichy alreadydisplaysdichroism.Using.

perturbation theory for the spin-Bogolubov–deGennesÁ

equations,we deriveda generalformula which con-tains\

thecontributionsof all thesemechanisms.On thebasisofY this resultseveralanalyticalconclusionsconcerningthesemechanismsi were drawn.Theseinclude an investigationofthe\

circumstancesunderwhich the variousmechanismscanbe¬

operativeandan analysisof their behaviorasa functionofY temperatureandmagneticfield.

Thet

important role playedby time-reversalsymmetryispointed[ out. While a Cooperpair consistsof two mutuallytime\

conjugatesingle-particlestatesandthegroundstateof asuperconductorX is thusinvariantundertime reversal,severalmechanismsi for dichroismrequirethe breakingof this sym-metry.

For moredetailedillustrationswe havechosenthemecha-nism� basedon the conventionalspin-orbit couplingbecauseitu

is known to be the dominantmechanismin the normalstate.X Numericalcalculationsfor a simple model supercon-ductorx

confirmtheanalyticalresultsandprovidefirst hintsatexperimentale signaturesof SOC-induceddichroismin super-conductors.w

The influenceof paramagneticlimiting on dichroism atlow temperaturesand high fields, unconventionalorder pa-rameters¨ as a sourcefor dichroism,the quenchingof spin-orbit-inducedY dichroismat T 2 0,

;andan additionalenhance-

ment of the coherencepeaksare definite predictionsof ourtheory\

which can be verified experimentally.None of thesephenomena[ showup in theabsorptionof unpolarizedlight insuperconductorsX or in dichroismin the normalstate.

PRB 58 487ANALYSIS OF DICHROISM IN THE . . .3

ACKNOWLEDGMENTS

WeÇ

want to thankM. Luders,x

R. Kummel,i H. Plehn,andH.4

Ebertfor very usefuldiscussionsaboutvariousaspectsofthis\

work. This work has benefited from collaborationswithin,y andhasbeenpartially fundedby, the HCM network

Ab-initio (from electronic structure) calculation of complexprocesses5 in materials (Contract: ERBCHRXCT930369),and6 the programRelativistic

7effects in heavy-element chem-

istry8

and physics of9 the EuropeanScienceFoundation.Par-tial:

financial support by the DeutscheForschungsgemein-schaftX is gratefully acknowledged.

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PRB 58 489ANALYSIS OF DICHROISM IN THE . . .g