Inhomogeneous States of Nonequilibrium Superconductors: Quasiparticle Bags and Antiphase Domain Walls

8 Pages·0.33 MB·English

by  M. I. Salkola

Inhomogeneous States of Nonequilibrium Superconductors:
Quasiparticle Bags and Antiphase Domain Walls
M.I. Salkola and J.R. Schrieffer
NHMFL and Department of Physics, Florida State University, Tallahassee, Florida 32310
(December19, 1997)
8
9
9
Nonequilibrium properties of short-coherence-length s-wave superconductors are analyzed in the
1
presence of extrinsic and intrinsic inhomogeneities. In general, the lowest-energy configurations

n of quasiparticle excitations are topological textures where quasiparticles segregate into antiphase
a domainwallsbetweensuperconductingregionswhoseorder-parameterphasesdifferbyπ. Antiphase
J
domain wallscanbeprobedbyvariousexperimentaltechniques,forexample,byopticalabsorption
0
and NMR.At zero temperature, quasiparticles seldom appear as self-trapped bag states. However,
1
for low concentrations of quasiparticles, they may be stabilized in superconductors by extrinsic

defects.
]
n
o PACS numbers: 74.80.-g, 74.20.-z, 74.25.Jb, 74.25.G7
c
-
r
p I. INTRODUCTION
u
s
. In general,studies of superconductorsemphasize their Ferromagnetic Superconductor
t metal
a equilibrium properties as probed by linear response.
m 1 eiπ 1 eiπ 1
Equallyimportantareconditionswherethesuperconduc-
- torisdrivenfarfromequilibrium. Anonequilibriumstate
d
may be achieved, for example, by photoexciting quasi-
n
o particles [1,2,3] or injecting them into the superconduc-
c tor througha tunnel junction [4,5,6]. These experiments FIG.1. Schematic description of a tunnel-junction exper-
[ have revealed a variety of interesting phenomena, rang- iment where spin-polarized quasiparticles are injected from

1 ing from first-ordersuperconductor-metal transitions, to a ferromagnetic metal through an insulating barrier to a su-
perconductor. In the superconductor, injected quasiparticles
v various instabilities and spatially inhomogeneous states
4 with a laminar structure where either superconducting haveformedalaminarstructureofinterveningsuperconduct-
8 ing domains separated by antiphase domain walls between
phaseswithdistinctenergygapsorsuperconductingand
0 regions where the neighboring order-parameter phases are
normal phases coexist.
1 shifted byπ.
0 At finite temperature, quasiparticle dynamics in the
8 nonequilibrium state is dominated by scattering with
9 phonons and decay with phonon emission. These pro- spins are aligned along the same direction may be ob-
t/ cesses are characterized by the scattering time τs and tained by using a ferromagnetic metal as a source [9].
a the lifetime τ . While these time scales are long enough In this Note, we examine various meta-stable configu-
m ∗
compared to h¯/∆ so that the superconducting energy rations of quasiparticles and their signatures that might
0
d- gap ∆0 is sharply defined [7], they are also typically of develop when spin-polarized quasiparticles are excited
n the same order of magnitude, τs τ∗ [8]. As a con- in s-wave superconductors. Our most important find-
o sequence, it is not obvious that th∼e quasiparticles will ing is that superconductors are unstable against a for-
c equilibratetoameta-stablestateevenundersteady-state mation of antiphase domain walls into which the quasi-
:
v conditions. In contrast, when the lifetime of quasiparti- particleslocalizeandthatthelocalstructureandnonuni-
Xi cles is the longest time scale, quasiparticles will reach formspindensitymakesthesetopologicaltexturesacces-
such a state making it possible for new phenomena to sible to various experimental probes. In particular, they
r
a emerge. A meta-stable state is obtained, if the system produce a distinctive optical absorption spectrum that
hasasymmetrygroupwhichmakesitpossiblefortheex- may serve as a unique signature of their presence. In
cited and groundstates to transformaccordingto differ- addition, any probe that is sensitive to a local magneti-
ent irreducible representations so that the quasiparticle zation would lend further support. For example, NMR
excitationscannotdecay. Itisclearthatintheabsenceof and µSR might be suitable for this purpose. Figure 1 il-
spin-orbit coupling the only practically meaningful sym- lustrates schematically a tunnel-junction experiment for
metry group is spin-rotationalsymmetry, because in the generatinganddetectingantiphasedomain-walltextures.
superconducting phase broken gauge symmetry destroys Similar experimentalconstruction has been suggestedto
the chargeconservation. Quasiparticleexcitationswhose demonstrate that spin and charge are transported by
1
separate quasiparticle excitations in a superconductor singletchannel. The operatorΨ(r)=[ψ (r) ψ†(r)]T is a
↑ ↓
[10]. We also consider quasiparticle bags which are non- two-component Gor’kov-Nambu spinor, τˆ (α = 1,2,3)
α
topologicalstatesofquasiparticlesassociatedwithalocal are the Pauli matrices for particle-hole degrees of free-
suppression of the order parameter that may appear in dom, and τˆ is the unit matrix. In a translationally in-
0
the presence of defects when the quasiparticle density is variant system, the BCS Hamiltonian reduces to
small enough.
Our work is partially motivated by the fact that only H0 = Ψ†k(ǫkτˆ3−∆0τˆ1)Ψk, (3)
X
few studies exist on self-trapped quasiparticle states in k
superconductors[11,12]andthateitherquasiparticle-bag where ∆ =∆(R) and Ψ =(ψ ψ† )T. The fermion
or antiphase domain-wall excitations are usually consid- 0 k k↑ −k↓
operators in real and momentum spaces are related by
ered as a curiosity and often disregarded as unphysical
the unitary transformation,ψ (r)=N−1/2 ψ eik·r,
[12,13]. Our purpose is to address the question of their σ k kσ
where N is the number of sites in the sysPtem. For a
existence in mean-field approximation and to examine
square lattice with the nearest-neighbor hopping, the
possible experimental implications by focusing on quasi-
single-particle energy relative to the chemical potential
one and two-dimensional superconductors which are re- in the normal state is ǫ = 1W(cosk a+cosk a) µ;
alizedinwiresandfilms. Analogousquestionshavebeen k −2 x y −
aisthelatticespacing. Inauniforms-wavesuperconduc-
studied in the context of antiferromagnets, although no
tor,theenergyspectrumofbarequasiparticleexcitations
detailedpredictionsregardingsuperconductorshavebeen
is E = ǫ2 +∆2. Allowing the excited quasiparticles
made [13,14]. k k 0
to relax,pthe energy spectrum and the order parameter
must be modified, as will be discussed below.
The interaction between the conduction electrons and
II. FORMALISM
the impurities in the superconductor is given by the
Hamiltonian
Our starting point in describing quasiparticle excita-
tions in an s-wavesuperconductor is the lattice formula- H = [V(r)n(r)+JS(r) s(r)], (4)
imp
·
tion of electrons hopping between nearest-neighbor sites Xr
and interacting via an effective two-particle interaction,
wheren(r)= n (r)ands(r)= 1 ψ†(r)τˆ ψ (r)
σ σ 2 σν σ σν ν
H = 1W ψ†(R+r)ψ (R) µ n (R) aretheconducPtionelectronnumberdePnsityandspinden-
−4 σ σ − σ
hRXriσ XRσ sity operators. In the case of point like impurities lo-
cated at sites r , the potential (scalar) and magnetic
+U n (R)n (R). (1) n
↑ ↓ scattering terms have the forms V(r) = V δ and
X n n rrn
R S(r) = S δ . Typically, the distribPution of impu-
n n rrn
Here,ψσ(r)istheelectronoperatorwithspinσ, Rr de- rities isPassumed to be random whereas the magnitude
h i
notesnearest-neighborsitesseparatedbyr,W isthehalf of scalar and magnetic scattering are constant, V = V
n
bandwidth(onasquarelattice),µisthechemicalpoten- and w = JS/2, where S = S . For later emphasis, it
n
tial, andthe operatornσ(r)=ψσ†(r)ψσ(r) is the conduc- is useful to introduce here a|par|ticle-hole transformation
tion electron number density for spin σ. The strength generated by the operator τˆ :
1
of the pairing interaction U (< 0) is assumed to be in-
Ψ(r) Ψ′(r)=( 1)rτˆ Ψ(r). (5)
termediate so that the mean-field approximation gives a 1
→ −
qualitatively reliable description of the superconducting
At half filling (µ = 0), the BCS Hamiltonian H on
0
groundstateandthelow-energyexcitations. Specifically,
a square lattice is invariant under this transformation.
consideratwo-dimensionallatticemodelwhereelectrons
Moreover,ifthe impurity moments arealignedalongthe
can interact with randomly distributed defects. These
same direction and there is no potential scattering, the
defects can either have a magnetic moment or be non-
impurity Hamiltonian H will also be invariant un-
imp
magnetic. The model is defined by the effective Hamil-
der the same transformation. Potential scattering and
tonian H = H + H , where H describes a BCS
eff 0 imp 0 randomlyorientedimpuritymomentsbreakparticle-hole
superconductor [15] and H is the contribution due to
imp symmetry of this nature.
impurities. In the mean-field approximation,
Given that the pairing of electrons occurs in the spin-
H = 1W Ψ†(R+r)τˆ Ψ(R) µ Ψ†(R)τˆ Ψ(R) singlet state, the superconducting order parameter (am-
0 −4 X 3 − X 3 plitude) can be expressed in the form
hRri R
∆(R)Ψ†(R)τˆ Ψ(R), (2) F(R,r)= 1 (iτˆ ) ψ (R+r)ψ (R) . (6)
−X 1 2X 2 σνh ν σ i
Rr σν
where∆(R)is the superconductinggapfunctionandas- The relation between the order parameter and the gap
suming that the pairing of electrons occurs in the spin- function is given by the equation
2
∆(R)= UF(R,r=0). (7) Therefore,inthegroundstate, n =1and s <0. Re-
z
− h i h i
versing the transformation, the transverse antiferromag-
The on-site pairing interaction U is assumed to be in- netic order parameter is mapped to a superconducting
stantaneous in time. Thus, the energy cutoff in the gap order parameter in the attractive Hubbard model.
equation is set by the bandwidth. In our numerical ap- Next, consider the attractive Hubbard model away
proach, the gap equation is solved self-consistently with from half filling but now in the magnetic field so that
a given number of quasiparticle excitations on finite size n < 1 and s < 0. This model is mapped by the
z
lattices with periodic boundary conditions. In our nu- phairticle-hole thranisformation to the repulsive Hubbard
merical examples, the strength of the interaction U is modelawayfromhalffilling. Ithasagroundstatewhich
chosensothatinthe absenceofimpuritiesandquasipar- isdescribedbyantiphasedomainwallsbetweenantiferro-
ticle excitations the energy gap is ∆0/W =0.1. magneticallyorderedspins[13]. Byvirtueoftheparticle-
hole transformation, it is clear then that the attractive
Hubbard model with spin-polarized quasiparticles has a
III. MAPPING TO AN ANTIFERROMAGNET
superconductinggroundstatewherethesuperconducting
domains with the opposite signs of the order parameter
The Hamiltonian (1) can be transformed to a model are separated by antiphase domain walls into which the
where the on-site interaction is repulsive. In the case of excess spin is localized.
longer-rangeinteractions, Ising-like terms are generated.
On a bipartite lattice, this is achieved by a particle-hole
transformation on the down spins, IV. THE CONTINUUM MODEL
ψ (r) ψ (r),
↑ ↑ Although we are mostly interested in quasi-two-
→
ψ (r) ( 1)rψ†(r). dimensionalsuperconductors,itisusefultoconsiderone-
↓ → − ↓
dimensional systems where many ideas can be examined
In this transformation, the particle number operator analytically. Indeed, for quasi-one-dimensional systems,
transformstothez-componentofthespindensityopera- afruitfulconnectionbetweentheBCSandSSHHamilto-
tor: n(r) 2sz(r)+1,andvice versa. The Hamiltonian nianscanbemade. Thelatteronedescribes,forexample,
→
is mapped into conducting polymers where the order parameter ∆(x)
represents the lattice distortion [16]. In our case, such
H =−41W ψσ†(R+r)ψσ(R)+hz sz(r) systems can be organic superconductors or wires whose
X X
hRriσ r thickness is smaller than the coherence length ξ0. In the
weak-coupling limit, additional progress is achieved by
+2U s (r)s (r), (8)
z z
X considering a continuum field theory, which can be de-
r
rived because the coherence length is much longer than
wherehz =U−2µisaneffectivemagneticfieldalongthe the lattice spacing, ξ0 ≫a.
z-axis. Thus, for the Hubbard model, the particle-hole Since we are interested in low-energy and long-
transformationchangesthesignoftheon-siteinteraction wavelengthphenomena,theelectronicdegreesoffreedom
U. canbeexpressedbyslowlyvaryingfieldsψ (x)describ-
±σ
The superconductor has U(1) symmetry associated ing the left (+) and right ( ) moving electrons,
−
with the phase of the order parameter. Because the real
and imaginary parts of the order parameter are trans- ψσ(x)/√a=ψ+σ(x)eikFx+ψ−σ(x)e−ikFx, (9)
formed to the x and y components of the spin, a gauge
where k is the Fermi wave vector. Defining the four-
transformation corresponds a rotation of the spin in the F
component spinor as Ψ(x) = [Φ (x) Φ∗(x)]T, where
xy plane. ↑ ↓
Φ (x) = [ψ (x) ψ† (x)]T, the BCS Hamiltonian (2)
The particle-hole transformation establishes one-to- σ +σ −σ¯
one correspondencebetween the groundstates of the at- becomes
tractive and repulsive Hubbard models. For example,
H = dxΨ†(x)[v pˆ ∆(x)τˆ ]Ψ(x). (10)
consider the attractive Hubbard model away from half 0 Z F − 1
fillingsothattheaverageelectrondensity n <1andthe
h i The momentum operator is pˆ = i¯hτˆ ∂ , where v =
average spin density s = 0. The particle-hole trans- 3 x F
z −
formation maps it intoh thie half-filled, repulsive Hubbard (2ta/¯h)sinkFa is the Fermi velocity. Similarly, the gap
model with the effective magnetic field h . Its ground equation can be rewritten in the form [17]
z
statehasatransverseantiferromagneticorderbecausein
∆(x)= 1aU Ψ†(x)τˆ Ψ(x) . (11)
this way the system can lower its energy by generating −2 h 1 i
a small ferromagnetic component parallel to the z-axis. These equations are formally equivalent to those of the
TLM model [18], which is the continuum limit of the
3
SSH model. For example, at zero temperature, the su-
perconducting energy gap is ∆ = 2We−1/λ, where the
0
dimensionless interaction is λ = N U ; N is the den-
F F
| |
sity of states at the Fermi energy in the normal state.
Similarly, the coherence length is ξ =h¯v /∆ .
0 F 0
It is now straightforward to determine the nonequi-
librium properties of the quasi-one-dimensional s-wave
superconductor. In particular, it is obvious that inject-
ing spin-polarized electrons into the system, they form
solitons. They are topological excitations of the system,
acting as domain walls between two ground states that
differ by the sign of the order parameter ∆. The en-
ergy of the soliton is E = 2∆ /π and the order pa-
dw 0
rameter is ∆(x) =∆ tanh[(x x )/ξ ], where x is the
0 0 0 0
−
location of the center of the soliton. At low densities
of injected electrons, single quasiparticle bags may ap-
pear. They are counterparts of polarons; thus, also their
spatial form as well as their energy, E = √2E , is
qp dw
knownexactly. Whileininhomogeneoussuperconductors
individual quasiparticles may diffuse until they become
trapped into defects, they are not generic solutions, be-
cause at finite concentration of quasiparticle excitations
they “phase separate” forming domain walls.
V. ANTIPHASE DOMAIN WALLS
While at zero temperature quasiparticle bags are not
generic excitations of the superconductor, they may ap-
pear as long-lived states because of defects. This may
happen if they are injected into the system at low rate
FIG.2. (a) The energy gap, (b) the spin density, and (c)
so that they can migrate without scattering from other
the density of states of a localized solution of spin-polarized
quasiparticle excitations long distances before they are
quasiparticles injected into an s-wave superconductor with
trapped to defects. Note that, in addition to magnetic
magnetic impurities. This configuration of well separated
impurities, a local order-parameter suppression caused quasiparticle bags is obtained self-consistently on a square
by nonmagnetic impurities leads to bound states in the lattice with thelattice spacing a, ∆0/W =0.1, πNFw=0.3,
superconductingenergygap,albeittheirbindingenergies andµ=0. Theconcentration ofquasiparticles andmagnetic
arenecessarilysmall[19]. Figure2 illustratesa situation impurities equals 1%.
which is obtained when the quasiparticle concentration
issmallandthere aremagneticimpuritiesinthe system.
a closed domain-wall loop is formed. Because the order
For simplicity, the magnetic impurities are assumed to
parameterchangessignacrossthedomainwall,thereare
be ferromagnetically orderedproducing a maximal trap-
midgapstates. Thefinitelengthofthedomainwallleads
ping potential. The bound quasiparticle states yield two
to the level repulsion yielding the density of states that
peaks in the density of states,
hasaminimumatzeroenergy. Asthesystemishalffilled
and either there are no impurities or their moments are
1
N(ω)=−2π ImGσσ(r,r;ω+i0+), (12) parallel to each other, the effective Hamiltonian Heff is
X
rσ invariantundertheparticle-holetransformation,Eq.(5).
Consequently, the density of states, depicted in Figs. 2
in the energygap. The oscillationsin (ω) for ω >∆
0
N | | and 3, is symmetric relative the zero energy.
are due to the finite size effects.
For a finite concentrationof quasiparticles,it becomes
With an increasing quasiparticle concentration, indi-
energeticallyfavorabletoformdomainwallswithinfinite
vidualquasiparticleexcitationsbecomeunstabletowards
length; see Fig. 4. This allows all the quasiparticles to
aspontaneousformationofantiphasedomainwalls. This
occupy the midgap states. Domain walls may become
tendencyisdepictedinFig.3,wherethenumberofquasi-
pinned to defects either because the defects have a mag-
particles is not large enough to form a domain wall that
netic moment or because the defects suppress the order
would extend all the way through the system. Instead,
4
FIG. 3. (a) The energy gap, (b) the spin density, and FIG. 4. (a) The energy gap, (b) the spin density, and
(c) the density of states when a finite number (40) of (c) the density of states in an s-wave superconductor with
spin-polarizedquasiparticlesisinjectedintoans-wavesuper- 5% spin-polarized quasiparticles. The antiphase-domain-wall
conductor. Thisconfigurationisobtainedself-consistentlyon configuration is obtained self-consistently on a square lattice
a square lattice with the lattice spacing a, ∆0/W =0.1, and with thelattice spacing a,∆0/W =0.1, µ=0, πNFV =0.3,
µ=0. Noimpurities are present. and nimp =2%.
parameter locally, and this local suppression then pins understood by considering the repulsive Hubbard model
a domain wall. In the case of extended defects, domain withafiniteeffectivemagneticfieldwhichinducesasmall
walls may find it preferable to wind through these de- longitudinal ferromagnetic component. In the supercon-
fects. ductor,thiscomponentisequivalenttoanon-zerocharge
All the quasiparticle and domain-wall textures are density. Incontrast,barequasiparticleexcitationsatthe
charge neutral when the system has particle-hole sym- Fermisurface (k=k )behave asspinons irrespectiveof
F
metry at the Fermi energy. In this regard, quasiparti- the energy spectrum in the normal state.
cle bags can be described as spinons [10], because they Finally, consider the stability of domain-wallsolutions
carry spin but no charge. On a square lattice with the against a formation of isolated quasiparticle bags. Their
nearest-neighbor hopping, this happens exactly at half energies per particle can be computed numerically. In
filling (µ = 0). However, if particle-hole symmetry at two dimensions, the energy of a vertical domain wall
the Fermi energy is broken, self-consistently determined per particle is estimated as E 0.66∆ and the en-
dw 0
≃
quasiparticle configurations usually acquire charge, be- ergy of a quasiparticle bag as E 0.86∆ . These es-
qp 0
≃
cause they are a linear combination of plane-wave states timates are in agreement with those computed in the
with an energy spread ∆ǫ ¯hvF/ξ0 about the Fermi antiferromagnetic system for vertical domain walls [13]
∼
energy. Similarly, away from half filling, the domain and spin polarons [20]. Thus, approximatelyat the tem-
walls become charged, although their total charge per perature T ∆ /5 a considerable fraction of domain
∗ 0
∼
unit length can be quite small. This feature is naturally walls begins to evaporate forming isolated quasiparticle
5
bags. It is interesting to compare this temperature with
the critical temperature of the superconductor, which is
T ∆ /2. Thus, there is a sizable temperature regime
c 0
∼
below T where most of the excitations appear as iso-
c
lated quasiparticle bags. At low enough temperatures,
T < T /3, domain-wall textures are thermodynamically
c
fav∼oredovernon-topologicalquasiparticleconfigurations.
VI. OPTICAL ABSORPTION
Theopticalabsorptionprovidesaspecificprobetovar-
ious inhomogeneous states of nonequilibrium supercon-
ductors. The optical absorption is the real part of the
complex conductivity, σa′b(ω) = Reσab(ω) (a,b = x,y), FIG.5. Optical absorption σx′x(ω) in an s-wave supercon-
where ductor in the presence of quasiparticle excitations forming
a domain-wall lattice and localized quasiparticle bags. The
1 ne2 concentration of quasiparticles in both cases is 1%. These
σ (ω)= Λ (q=0,ω+i0+)+ δ . (13)
ab −iω ab m∗ ab configurationsareobtainedself-consistentlyinonedimension
(cid:2) (cid:3)
for ∆0/W = 0.1, µ = 0, πNFV = 0.3, and nimp = 5%.
Theratiobetweenthedensityofchargecarriersandtheir The dashed line denotes the optical absorption (Drude-like)
effective mass is defined as n/m∗ = −a2hHkini/2, where obtained in thenormal state (∆0 =0).
H is the kinetic-energy part of the Hamiltonian H.
kin
The current-current correlation function is given by the 2n
τ−1 = imp sin2δ, (19)
formula
πN
F
Λab(q,t)= Tja(q,t)jb( q,0) , (14) where δ is the phase shift for s-wavescattering and nimp
−h − i
is the impurity concentration. For point-like impurities,
and its Fourier transform is the scattering phase shift is obtained from the equation
1 ∞ cotδ = c, where c = (πNFV)−1, for nonmagnetic im-
Λ (q,ω)= dteiωtΛ (q,t). (15) purities, and c = (πN w)−1, for magnetic impurities.
ab N Z ab F
0 Below, as a reference, the numerically determined opti-
cal conductivity in the presence of randomly distributed
The current operator in Heisenberg picture is defined as
j (q,t)=eiHtj (q)e−iHt, where impurities inthe normalstate (∆0 =0)is alsoshown. It
a a
is well described [21] by the Drude form, Eq. (18).
j (q)= ieaW Ψ†(R+r )τˆ Ψ(R)e−q·r. (16) In the superconductor, the zero-temperature optical
a 4 a 0
hXRri conductivity typically has a threshold of 2∆0 due to the
superconductingenergygapintheelectronicspectrumat
It is useful note that optical absorption obeys the sum the Fermi energy. Quasiparticle bags and antiphase do-
rule: main walls introduce states within this energy gap that
can be used as a characteristic signature of them. Fig-
∞ πne2
dωσ′ (ω)= . (17) ure5illustratesthe opticalabsorptionwhenthe injected
Z aa 2 m∗
0 quasiparticles form either isolated bag states pinned to
nonmagnetic impurities or domain walls. In the former
It is a quantity describing any state that is linearly per-
case, there is a very large peak in the absorption that
turbed by the electric field. Typically, it is associated
comes from the excitation processes from the localized
with the equilibrium state. For non-equilibria states,
bag states to states just below the energy gap. Because
suchasthedomainwallsandquasiparticlebags,thesum
the order-parameter suppression occurs on the length
rule must be modified.
scaledeterminedbythecoherencelength,itactsasanat-
In the normal state, the optical conductivity has the
tractivepotentialwithafiniterangethatcanbindstates
Drude form
justbelowthe energygap. Thus,inadditiontothestate
ne2τ 1 occupied by the quasiparticle, the order-parameter re-
σ′ (ω)= , (18)
aa m∗ (τω)2+1 laxation may admit additional discrete states below the
energygap. Transitionsbetweenthesestateshaveavery
whereτ−1 isthescatteringrateduetotheimpurities. In largeoscillatorstrength. Onecanalsoseeapeakat2∆ ,
0
the limit of dilute concentration of impurities, it can be which is due to the pair-breaking processes across the
approximated as energy gap. In the case of domain walls, the optical ab-
6
This will require the use ofspin-polarizedquasiparticles,
whichmaynotalwaysbefeasible. Aqualitativelysimilar
situation may be created by maintaining a steady state
of unpolarized quasiparticles by continuously pumping
quasiparticles into excited states. Even though a gen-
uinelymetastablestatemaynotdevelopbecauseτ τ ,
∗ s
∼
the fact that τ can be many ordersof magnitude longer
∗
than the time scale associated with the superconducting
energygap,h¯/∆ , suggeststhatsome ofthe features ex-
0
ploredheremayactuallyberelevantforsuchstates,too.
Time resolvedtechniquesareanidealtoolto probetheir
properties.
While in gapless superconductors, such as in d-wave
superconductors, it is no longer clear that quasiparticle
′
FIG. 6. Optical absorption σxx(ω) in an s-wave super- excitations will lead to antiphase domain walls, various
conductor with 5% scalar (πNFV = 0.6) and magnetic external defects that suppress the order parameter may
(πNFw = 0.6) impurities in the absence of quasiparti- locally favor a phase shift. Such textures may appear in
cle excitations. The ground-state configuration is obtained
magnetic superconductors with static spin-density-wave
self-consistentlyinonedimensionfor∆0/W =0.1andµ=0. ordering where the phases of the magnetic and super-
The dashed line denotes the optical absorption (Drude-like)
conducting order parameters intertwine to form a new
obtained in the normal state (∆0 =0).
collective state with midgap quasiparticle states.
sorptionbeginsat∆ duetothemidgapstates. Onecan
0
therefore clearly distinguish between these two nonequi- ACKNOWLEDGMENTS
librium states based on the optical absorption.
Impurities yield a quite different absorption spectrum We would like to thank S. Kivelson for the interest-
in the absence of quasiparticle excitations; see Fig. 6. ing suggestion that a ferromagnetic metal could be used
Nonmagnetic impurities produce a spectrum that has a to create and inject spin-polarized quasiparticles into a
clear threshold near 2∆0 and a peak, whereas magnetic superconductor. This work was supported by the NSF
impurities yield a relatively smooth absorption profile under Grant Nos. DMR-9527035 and DMR-9629987, by
which may extend deep below 2∆0, depending on the theU.S.DepartmentofEnergyundertheGrantNo.DE-
coupling strength between electrons and impurity mo- FG05-94ER45518, and by the Many-Body-Theory Pro-
ments. For a high enough concentration of magnetic gram at Los Alamos.
impurities, the superconductor becomes gapless and the
absorption will begin at zero energy.
VII. FINAL REMARKS
[1] L.R.Testardi, Phys.Rev.B4, 2189 (1971).
Based on both analytical and numerical approaches,
[2] G.A.Sai-Halasz et al., Phys.Rev.Lett. 33, 215 (1974).
we have demonstrated that s-wave superconductors
[3] P.Hu,R.C.Dynes,andV.Narayanamurti,Phys.Rev.B10,
driven away from equilibrium exhibit interesting topo-
2786 (1974).
logical textures. They develop as quasiparticles in node- [4] I.Iguchi, Phys.Rev.B16, 1954 (1977).
lesssuperconductorssegregateformingantiphasedomain [5] J.Fuchs et al.,Phys. Rev.Lett. 38, 919 (1977).
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ing magnetization. Moreover, optical absorption pro-
(1990).
vides another unambiguous tool for exploring these tex-
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7
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[17] For simplicity, in this section, τˆ1 denotes τˆ1⊗11, which is
a block-diagonal matrix equalto τˆ10 .
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[21] In one dimension, the actual conductivity does deviate
from the Drude form at low energies because of localiza-
tion.However,sinceinourexamplesdescribings-wavesu-
perconductors the optical conductivity vanishes below a
smallenergyscale,thisdeviationfromtheDrudebehavior
is not shown.
8