Critical and non-critical coherence lengths in a two-band superconductor

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by  Teet Örd

Noname manuscript No.
(will be inserted by the editor)
Critical and non-critical coherence lengths in a two-band
superconductor
Teet O¨rd · Ku¨llike R¨ago · Artjom Vargunin
2
1
0
2

n
a
J
Received: date /Accepted: date
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] Abstract We study the peculiaritiesofcoherencyina condensates. In this situation the quantities, related
n
two-gapsuperconductor.Thebothintrabandcouplings, initially to the superconducting states of independent
o
inducing superconductivity in the independent bands, bands, acquire mixed character and they describe the
c
- andinterbandpair-transferinteractionhavebeentaken collectivefeaturesofthewholetwo-componentconden-
r
p intoaccount.OnthebasisoftheGinzburg-Landauequa- sate.An example of such quantities in a superconduct-
u tionsderivedfromtheBogoliubov-deGennesequations ing system with interband coupling are the coherence
s
and the relevant self-consistency conditions for a two- lengths which cannot be attributed to different bands
.
at gap system, we find critical and non-critical coherence involved [9], see also [10]-[12]. In the present contribu-
m lengths in the spatial behaviour of the fluctuations of tion we examine in details the properties of coherence
- orderparameters.Thecharacterofthetemperaturede- length scales in a two-band superconductor.
d
pendencies of these length scales is determined by the
n
relative contributions from intra- and interband inter-
o
c action channels. 2 System of Ginzburg-Landau equations for a
[ two-band superconductor
Keywords Two-gap superconductivity Intra- and
1 ·
interband interactions Coherence lengths
v · WestartwiththefollowingHamiltonianinthetermsof
8 Ψ-operatorsfor a two-bandsuperconductor with intra-
4 bandcouplingsandinterbandpair-transferinteraction:
1 Introduction
7
5 H =H +H , (1)
0 1
. Two-band models of superconductivity have been de-
1
0 veloped more than fifty years starting from the pa- where
2 pers [1]-[3]. For the present the number of discovered
1 multi-gap superconducting materials is quite large, in- H0 = dr Ψα+s(r)H0Ψαs(r) (2)
v: cluding cuprates, MgB2 and iron-arsenic compounds. Z Xα Xs
i Inthisfieldvarioustheoreticalschemeshavebeenelab- with
X
r orated, see [4]-[7] and references therein. In particular, ¯h2 ie 2
a the derivation of high-quality superconducting regions 0 = A µ+V(r), (3)
H −2m ∇− ¯hc −
(cid:18) (cid:19)
fromoxygenordering,observedrecentlyinLa CuO
2 4+y
and
[8], supports the multi-band theoretical scenario of su-
1
perconductivity in cuprates. H1 = Wαα′
In a two-gap superconductor with interband pair- 2α,α′
X
transfer interaction one has to deal with coupled band
TInesettitOu¨tredo·fKPu¨hlyliskiecsR,U¨agnoiv·eArsrittyjomof TVaarrtguunin ×Z drXs,s′ Ψα+s(r)Ψα+s′(r)Ψα′s′(r)Ψα′s(r) . (4)
T¨ahe 4, 51010 Tartu, Estonia In Eqs. (1)-(4), α=1,2 is the band index, s= , =
↑ ↓ ±
E-mail: [email protected] is the spin index, µ is the chemical potential, V(r)
2 Teet O¨rd etal.
is the periodic potential of a crystal, and Wαα′ are Uα(r)= Wαα Ψα+↑(r)Ψα↑(r)
theelectron-electroninteractionconstants.Wesuppose
D E
that Wαα < 0, i.e. the autonomous superconducting = Wαα Ψα+↓(r)Ψα↓(r)
phase transition takes place in the both bands if inter- D E
bandinteractionis absent.However,if interbandinter- = Wαα |uαk(r)|2f(Eα(k))
k
action is turned on, the superconducting states of the Xn
bands are not independent anymore. + vαk(r)2[1 f(Eα(k))] , (11)
| | −
The following derivation of the Ginzburg-Landau where f(E)=[1+exp(E/k T)]−o1.
equationsexploitstheschemefrom[13],generalizedfor B
InthespatiallyhomogeneouscaseEqs.(10)and(9)
two-band situation.
yield the system of superconductivity gap equations
First,oneintroducestheeffectiveone-electronHamil-
tonian in the self-consistent field approximation, ∆α′ Eα′(k)
H = dr Ψ+ (r) Ψ (r) ∆α =−Xα′ Wαα′Xk 2Eα′(k)tanh 2kBT (12)
eff Z ( α s αs H0 αs withE (k)= ε˜2 (k)+ ∆ 2 1/2, ε˜ (k)=ε (k) µ,
XX α α | α| α α −
+ Uαα′(r)Ψα+s(r)Ψα′s(r) where εα(k) ishthe normal-staite energy of an electron
αX,α′Xs in the αth band.
+ ∆ (r)Ψ+ (r)Ψ+ (r) In the non-homogeneous situation one can derive
α α↑ α↓
α from Eqs. (10) and (9) analogously with [13] the lin-
X
earized selt-condistency conditions
+ ∆∗ (r)Ψ (r)Ψ (r) . (5)
α α↓ α↑
α ) ∆α(r)= dr′Kαα′(r,r′)∆α′(r′) (13)
X
Inwhatfollowswesimplify thecalculationssupposing1 α′ Z
X
with
Uαα′(r)=Uα(r)δαα′. (6)
In this case the effective Hamiltonian (5) can be di- Kαα′(r,r′)= Wαα′ [1 2f(ε˜α′(k))]
− −
agonalized by means of the Bogoliubov-Valatin trans- kX,k′
formation (γ+ and γ are the operators of creation and Θ( ε˜α′(k)) Θ(ε˜α′(k))
− +
destruction of elementary excitations): ×(cid:26)|ε˜α′(k)|−ε˜α′(k′) |ε˜α′(k)|+ε˜α′(k′)(cid:27)
Ψαs(r)= uαk(r)γαks−sgn(s)vα∗k(r)γα+k−s ,(7) ×Φ∗α′k(r′)Φ∗α′k′(r′)Φα′k(r)Φα′k′(r) , (14)
Xk (cid:8) (cid:9) where Θ(x) is the Heaviside function and Φαk(r) is
where k is the wave vector. As a result thenormal-stateeigenfunctionofanelectronintheαth
Heff =Eg+ Eα(k)γα+ksγαks, (8) band, [H0+Uα(r)]Φαk(r)=ε˜α(k)Φαk(r).
α k s In the analogy with the single band case [13], we
XXX
where E is the ground state energy and E (k) is the obtain on the basis of Eqs. (12), (13) the system of
g α
energy of an elementary excitation. Using the commu- Ginzburg-Landauequationsforsuperconductivitygaps
tator [Ψ (r),H ] together with Eqs. (7) and (8) we in a two-band system (A=0),
αs eff
obtain the Bogoliubov-de Gennes equations for a two-
band superconductor ∆α(r) = wαα′ ρα′ g(T) ν ∆α′(r)2
− ρ − | |
Eα(k)(cid:18)uvααkk((rr))(cid:19) + 3Xαβ′α′i∇2ir∆αα′(r(cid:20)), (15)
= H0+Uα(r) ∆α(r) uαk(r) . (9) Xi=1 (cid:21)
(cid:18) ∆∗α(r) −H0∗−Uα(r)(cid:19)(cid:18)vαk(r)(cid:19) where the intra- and interband interactions are non-
Here the self-consistent potentials ∆α(r) and Uα(r) zero in the energy layer with the width 2¯hωD near the
have been determined as Fermi level, ̺α is the density of electronstates per one
spin direction in the αth band at the Fermi level, and
∆α(r)= Wαα′ Ψα′↓(r)Ψα′↑(r)
h i
Xα′ wαα′ =wα′α =Wαα′√ρα′ρα, (16)
= − α′ Wαα′ k vα∗′k(r)uα′k(r) g(T)=ln 1.13¯hωD , (17)
X X k T
[1 2f(Eα′(k))] , (10) (cid:18) B (cid:19)
× −
be1cTonhseidmeorerdegaesnaersaplesciitaulaptiroonblUemαα.′(r)6=0ifα6=α′ should βαi = 71ζ6((3π)kh¯2vTF2)α2i , (18)
B c
Critical and non-critical coherence lengths ina two-band superconductor 3
7ζ(3) 3 Critical and non-critical coherence lengths
ν = . (19)
8(πk T )2
B c
We will find now the coherence (correlation) lengths ξ
The equations (15) follow also from the minima condi-
whichcharacterizethe spatialvariationofsmallfluctu-
tions of the free energy functional suggested in [17].
ations of superconducting gaps. By introducing small
The superconductingtransitiontemperatureTc has deviations ηα(r) from the bulk values of gaps we write
been determined by the equation
∆ (r)=∆∞ +η (r). (25)
1,2 1,2 1,2
1+w11g(Tc) 1+w22g(Tc) −w122g2(Tc)=0. (20) In the normal phase ∆∞1,2 =0. One can take ∆1,2(r) to
bereal.Thenforthesmallη (r)weobtainonthebasis
(cid:0) (cid:1)(cid:0) (cid:1) α
In general, Eq. (20) has two solutions
of Eqs. (23) the following linearized equations:
kBTc± =1.13¯hωD ηα(r)= wαα′ ρα′ g˜α′(T)+βα′ 2 ηα′(r) (26)
− ρ ∇
w +w (w w )2+4w2 α′ r α (cid:20) (cid:21)
exp 11 22± 11− 22 12 . (21) X
×  2(wqw w2 )  with
11 22− 12
g˜ (T)=g(T) 3ν(∆∞(T))2 (27)
  α − α
The higher temperature T+ > T− increases as the in-
c c We seek for the solutions of equations (26) in the
terbandinteractionconstant w increases.Simultane-
| 12| form
ously the temperature T− decreases, and it disappears
c 3 x
apsertahteudreiffereregniocnewT11>w22T−c+wc122orarpespprooancdhsesto0+th.eThneotremma-l η1,2(r)∼exp(cid:18)−P√i=31ξ i(cid:19), (28)
phase. In the domain T+ > T > T− there exists a whereξrepresentsthelengthscales.Thecorresponding
c c
stable superconducting state. If T < T−, a metastable substitution into Eqs. (26) yields
c
phase (or at least the saddle points of the free energy
asa function ofnon-equilibriumgaporderparameters) 1+w11g˜1(T)+w11β1ξ−2 η1(r)
appears besides the stable superconducting phase [14]- (cid:18) (cid:19)
ρ ρ
[16]. Consequently, the phase transition into the stable + w 2g˜ (T)+w 2β ξ−2 η (r) =0
12 2 12 2 2
superconducting state takes place at T =Tc+ ≡Tc. (cid:18) rρ1 rρ1 (cid:19)
If the interband pairing is absent, w12 = 0, the w ρ1g˜ (T)+w ρ1β ξ−2 η (r)
quantities Tc± transform into the temperatures of au- (cid:18) 21rρ2 1 21rρ2 1 (cid:19) 1
tonomous superconducting phase transitions in the in-
+ 1+w g˜ (T)+w β ξ−2 η (r) =0. (29)
dependent bands: 22 2 22 2 2
(cid:18) (cid:19)
1 Thesystemofequations(29)hasthenon-zerosolutions
kBTc± =kBTc1,2 =1.13¯hωDexp w , (22) η1,2(r) if the determinant of this linear homogeneous
(cid:18) 11,22(cid:19) system equals to zero. The latter condition leads to a
with w <0 and w > w , i.e. T >T . bi-quadratic equation for the characteristic lengths ξ
αα 11 22 c1 c2
| | | |
In the isotropic situation (the case of the spherical K(T)ξ4 G(T)ξ2+γ =0 (30)
Fermisurface)thesquaredmodulusoftheFermiveloc- −
ity equals v2 = 3v2 , i = 1,2,3. Correspondingly, where
Fα Fαi
the system of equations (15) reads G(T)= w2 [g˜ (T)β +g˜ (T)β ]
12 1 2 2 1
[1+w g˜ (T)]w β
∆α(r) = − α′ wαα′rρραα′(cid:20)g(T)−ν|∆α′(r)|2 −− [1+w1212g˜12(T)]w2121β21, (31)
X
+ βα′ 2 ∆α′(r) (23) K(T)= [1+w11g˜1(T)][1+w22g˜2(T)]
∇
(cid:21) w2 g˜ (T)g˜ (T), (32)
− 12 1 2
with
γ =(w w w2 )β β . (33)
7ζ(3)h¯2v2 11 22− 12 1 2
β = Fα . (24) SolvingEq.(30)weobtaintheexpressionsfortwosquared
α 48(πk T )2
B c characteristic lengths
On the basis of Eqs. (23) we are going to consider the G(T) G2(T) 4K(T)γ
ξ2 (T)= ± − . (34)
peculiarities of coherency in the present model. s,r 2K(T)
p
4 Teet O¨rd etal.
OnecanobservefromEq.(34)thatthequantitiesξ (T)
s
(soft or critical coherence length) and ξ (T) (rigid or
r
non-critical coherence length) reveal substantially dif-
ferent temperature behavior near T 2. The character-
c
istic length ξ (T) behaves critically diverging at the
s
phase transition point: ξ (T ) = . At the same time
s c
∞
ξ (T)remainsfinite,ξ (T )= γ/G(T ),anditstem-
r r c c
perature dependence is weaker. The functions ξ (T)
p s,r
have been depicted in Figs. 1-3 with increasing inter-
bandinteractionconstant W .Inthetemperaturedo-
12
| |
mainT <T usedinFigs.1-3thedeviationofthesolu-
c
tionsofthehomogeneouspartofapproximateequations
Eq. (23) from the solutions of exactequations Eq.(12) Fig. 2 The coherence lengths ξs (solidline)and ξr (dashed
is very small, less than 5%. line) vs temperature. Parameters: W11 = 0.3eV cell,
− ·
W22 = 0.57eV cell, W12 = 0.009eV cell, ρ1 =
1(eV ce−ll)−1, ρ2·= 0.5|(eV | cell)−1, ¯hωD· = 0.07eV,
vF1 =·4 105m/s, vF2 =5 1·05m/s. Characteristic tem-
peratures×: Tc = Tc+ = 31.6K×, Tc1 = 30.4K, Tc− = 26.5K,
Tc2=27.6K.
Fig. 1 Thecoherence lengths ξs (solidline)and ξr (dashed
line) vs temperature. Note that the minimum of ξs and the
maximumofξrareactuallyseparatedbyasmallgap.Param-
eters: W11 = 0.3eV cell, W22 = 0.57eV cell, W12 =
0.0001eV cel−l, ρ1 = 1·(eV cell)−1−, ρ2 = 0.5·(eV |cell)−| 1,
C¯hωhDara=cte0r.i·0st7iceVt,emvFpe1ra=tur4e×s·: 1T0c5m=/sT,c+vF2 =Tc51×=·10350.m4/Ks,. Flinige). 3vsTtheemcpoehrearteunrcee. lPenagrathmseξtesrs(:soWlid11lin=e)a−nd0.3ξreV(d·ascheelld,
Tc−≈Tc2=27.6K. ≈ Wcel2l2)−=1,−ρ02.57=eV0.·5c(eelVl, |Wce1l2l)|−=1,0.¯h0ω8DeV=·ce0l.l0,7ρe1V=, v1F(1eV=·
4 105m/s, vF2 = 5· 105m/s. Characteristic tempera-
tu×res: Tc = Tc+ = 50.9×K, Tc1 = 30.4K, Tc− = 12.6K,
In Fig. 1, the maximum of ξ (T) near T− T re- Tc2=27.6K.
s c ≈ c2
flectsthe”memory”aboutthelowerautonomousphase
transition which takes place if interband interaction is
One can find by using the expansion in powers of
absent.Theincreaseof W suppressesthismaximum
12
| | (T T )/T thatthe followingexpressionsapproximate
and finally ξs(T) decreases monotonically as tempera- − c c
the temperature dependence of the critical coherence
ture T < T decreases, cf. Figs. 1 and 2. At the same
c
length near the phase transition point:
timethetemperaturedependenceofξ becomesweaker
r
as one canobservefrom Figs. 1-3. The further increase ξ0 Tc , T >T
s T−Tc c
of interband coupling does not introduce any qualita-
ξs(T)= q , (35)
tive changes into the temperature behaviour of coher- ξ0 Tc ,T <T
ence lengths. s 2(Tc−T) c
The length scales found are related to the criti- where  q
calandnon-criticalfluctuations whichappear as linear G(T )
ctiovmitbyinoardtieornpsaorfatmheeteflrusc,tsueaetaiolsnos[o1f5b],a[n1d2].superconduc- ξs0 =s−w11−w22−2(w11cw22−w122)g(Tc). (36)
The coefficient ξ0 coincides with the length found in
s
2 Note thatK(Tc)=0according to Eq.(20). [18] as a single coherence length in the rigorously lim-
Critical and non-critical coherence lengths ina two-band superconductor 5
ited Ginzburg-Landauscheme in a two-bandsupercon- 5 Conclusions
ductor.
Toconclude, wedetermined twocoherencelengths in a
two-gapsuperconductorwithintra-andinterbandcou-
4 Coherence length-scales and microscopic plings. These length scales are not related to the con-
length-scales crete bands involved in the formation of the supercon-
ductingorderinginasystemwithinterbandpairtrans-
Thegradientexpansionanditscut-offintheGinzburg- fer interaction. One of these lengths as a function of
Landau equations (23) is justified if temperature deverges near the phase transition point,
ξ2 (T) the other one is non-critical. The non-monotonic tem-
s,r
1, (37) perature dependence of coherence lengths appears in
β ≫
1,2
the superconducting phase if the interband coupling is
i.e.thelengthscalesξs,r(T)arelargeenoughcompared sufficientlyweak.Wesuggestthattheappearanceofthe
to the microscopic lengths β1,2. non-criticalcoherence length is a substantial feature of
It is relativelyeasyto satisfy the inequality (37)for two-gapsuperconductivity.
p
ξ (T).However,thesituationisdifferentinthecasethe
s
non-critical coherence length ξ (T). In Fig. 4 the com-
r Acknowledgements This research was supported by the
parison of the dependencies of ξr(Tc) and √β2 on in- European Union through the European Regional Develop-
terband interaction constant is demonstrated (we have mentFund(CentreofExcellence”Mesosystems:Theoryand
chosenv >v ,i.e.β >β ).Itisseenthatinthere- Applications”, TK114). We acknowledge the support by the
F2 F1 2 1
EstonianScience Foundation,Grant No7296.
gionof weak interbandcoupling the condition (37) can
be satisfied for ξ . At that the smaller the difference
r
between the temperatures of autonomous phase tran-
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