The Interacting Impurity Josephson Junction: Variational Wavefunctions and Slave Boson Mean Field Theory

5 Pages·0.19 MB·English

by  A. V. Rozhkov

The Interacting Impurity Josephson Junction: Variational Wavefunctions and Slave
Boson Mean Field Theory
A. V. Rozhkov and Daniel P. Arovas
Department of Physics, University of California at San Diego, La Jolla CA 92093
(February 1, 2008)
0
We investigate the Josephson coupling between two superconductors mediated through an infinite
0
0 U Anderson impurity, adapting a variational wavefunction approach which has proved successful
2 for the Kondo model. Unlike the Kondo problem, however, a crossing of singlet and doublet state
energiesmaybeproducedbyvaryingtheratioofKondoenergytosuperconductinggap,inagreement
n
with recent work of Clerk and Ambegaokar. We construct the phase diagram for the junction and
a
J discuss properties of different phases. In addition, we find the singlet and doublet state energies
within a slave boson mean field approach. We find the slave boson mean field treatment is unable
6
to account for the level crossing.

]
l PACS numbers: 74.50.+r, 73.40.Gk
l
a
h
-
s
e I. INTRODUCTION singlet at energy scales below T W exp( π ε /2Γ),
K ≡ − | 0|
m where W is the half-bandwidth in the electrodes. In our
. The Ambegaokar-Baratoff formula, I = π∆/2eR, re- problem, then, one expects that the Kondo effect will
t c
a lates the critical current in a Josephson junction to the be mitigated whenever ∆>T . Roughly speaking, if
K
m superconducting gap ∆ and the normal state junction ∆ > TK, the ground state∼of the system is a Kramers
- resistance R. This result is perturbative in the elec- doublet, and breaks time reversal symmetry, whereas if
d
tron tunneling amplitude t; the normal state conduc- T > ∆, the ground state is a hybrid singlet formed
n K
tance is proportional to t2 when t is small. In certain from electrons onthe impurity and in the superconduct-
o | |
instances, however, it may be that the tunneling is me- ing electrodes [4].
c
[ diated through a magnetic impurity, rather than taking Recently,ClerkandAmbegaokar[7]appliedageneral-

placedirectlyfromonesuperconductortotheother. This izationofthe non-crossingapproximation(NCA), a par-
1
situation has been considered by a number of authors tial summation scheme, to attack this problem in the
v
2 [1–7]. The principal result is that magnetic impurity- U limit [7]. Adaptation of this method to the in-
→ ∞
6 mediated tunneling results in a negative contribution to teractingimpurityJosephsonjunctionallowsonetoseea
0 I . AsKulikoriginallyargued[1],themagneticimpurity transitionfrom0-junctionto π-junction whenthe super-
c
1 gives rise to an effective spin-flip hopping amplitude t conducting gap becomes comparable to the Kondo tem-
sf
0 between the superconductors. Since spin-flip tunneling perature.
0
results in a sign change of the Cooper pair singlet ( In this paper, we also explore the U limit, using
0 to ) [5], the critical current is I t2 t 2. Wh↑en↓ two approaches. The first is a variation→al∞wave function
/ c sf
t I <↓↑0, one has a π-junction, for wh∝ich| |th−e g|rou|nd state calculation, similar to that used by Varma and Yafet in
a c
m energy is minimized when the phase difference between the Kondo problem [8,9]. We generalize this wave func-
the superconductors is δ = π. Such π-junctions break tion in two respects. Firstly, there are two superconduc-
-
d time-reversalsymmetry( ),andaringcontainingasin- tors connected to the impurity. This is in contrast with
T
n gle π-junction will enclose trapped flux [3]. the usual setup of Kondo problem where we have only
o This analysis suffices in the perturbative limit where metallic electrode. Secondly, the spin-1 state must be
c 2
t is small. If the impurity level energy is ε and the considered as well. We find, in agreement with ref. [7],
: 0
v Coulomb integral is U, the condition for a magnetic that a first order transition occurs at T /∆ 1. We
i K ≈
X ground state (and a π-junction) is U > ε > 0 [5]. In also show how this transition may be precipitated by a
− 0
the nonperturbative regime, a new energy scale arises: change in the phase difference δ, as we found in ref. [6].
r
a the bare impurity level width Γ πρt2, where t is The second calculation we describe is the slave boson
∼ | |
theelectrode-impurityhoppingmatrixelementandρthe mean field theory. While the this method does confirm
electrode density of states. The T =0 phase diagram as thatthe singletstate givesa0-junctionandthe Kramers
afunctionof ε /∆,U/∆,andΓ/∆wasinvestigatedby doublet a π-junction, within the static slavebosonmean
theauthors[6−]w0ithintheHartree-Fock(HF)approxima- fieldsolutionthesingletisalwayslowerinenergy(orthe
tion, where itwas foundthat π-junctionbehavioroccurs statesaredegenerate). TheNCA,whichgoesbeyondthe
forU > ε >0,providedΓissufficientlysmall(Γ<U). mean field level, is able to describe the transition.
However−, w0hen ∆ = 0, it is known that the HF ap∼prox-
imation is unable to describe the formation of a Kondo
1
II. VARIATIONAL WAVEFUNCTION
APPROACH
We start with the grand canonical Hamiltonian
K ≡
µN,
H−
= ξ ψ† ψ +ψ† ψ
K qα qα↑ qα↑ qα↓ qα↓
Xα Xq (cid:26) (cid:16) (cid:17)
+∆ eiδαψ† ψ† +e−iδαψ ψ
α qα↑ −qα↓ −qα↓ qα↑
1(cid:16) (cid:17)
t ψ† c +t∗ c†ψ
−√ α qασ σ α σ qασ
Nα σX=↑,↓(cid:16) (cid:17)(cid:27)
+ε c†c +c†c +Uc†c†c c , (1)
0 ↑ ↑ ↓ ↓ ↑ ↓ ↓ ↑
(cid:16) (cid:17)
where α labels the electrode, ξqα is the dispersion in the
α electrode relative to the chemical potential, ∆ is the
α
modulus of the superconducting gap and δ its phase,
α
is the number of unit cells in electrode α, t is the
α α
N
hoppingamplitudefromelectrodeαtotheimpurity,and FIG. 1. Singlet (solid) and doublet (dot-dash) energies
ε is the bare impurity energy. We set U , which versus phase difference for ε =−1.6 (upper left), ε =−1.8
le0aDdesfitnointhgetchoensatnragilnetθPqασc≡†σcσt≤an1−.1(∆α/ξ→qα)∞and the (riugphpt)e.rIrnigahltl),caεs0es=∆−=1.19 a(lno0dweΓrLl=eftΓ),Ra=nd0.ε30π.= −20.0 (lower
usual BCS coherence factors uqα = cos(21θqα), vqα =
sthine(B21θoqgαo)lieuxbpo(viδqαu),aswipeaerxtipcrleesospKer=atoKr0s:+K1 in terms of y-Da˜xαiαs)′.. WHeeren,exCtqα′s′qαet=toCzqαeqrα′o′,tCh˜eqα′′vqαar=iatC˜ioqαnqα′s′, and D˜qα′′qα =
qq′
−
K0 = q,αEqα(γq†α↑γqα↑+γq†α↓γqα↓)+ε0(c†↑c↑+c†↓c↓) δhΨ|K0+K1|Ψi−EδhΨ|Ψi=0 , (4)
X
+Uc†↑c†↓c↓c↑ wthheervear|iΨatiioinsa|lScoieoffirc|iDent↑si.,WtoeofibntdaintheaqtuAatiaonndstrheleatminag-
= t u (γ† c +γ† c +c†γ +c†γ ) trix Cqαqα′′ may be expressed in terms of the coefficients
K1 Xq,α qα(cid:26) qα qα↑ ↑ qα↓ ↓ ↑ qα↑ ↓ qα↓ Bqα, and similarly A˜ and the matrices C˜qαqα′′ and D˜qαqα′′
+v (γ† c† γ† c†)+v∗ (γ c γ c ) , may be expressed in terms of the coefficients B˜qα. We
qα qα↑ ↓− qα↓ ↑ qα qα↓ ↑− qα↑ ↓ then obtain the two eigenvalue equations for the singlet
(cid:27) and doublet energies E and E˜, respectively:
whWereenEoqwαt=woqvξaq2rαia+tio∆na2αl,manadnyt-qbαo≡dy|tsαta|/t√esNfoαr.theU = 2vqαvq∗′α′tqαtq′α′ + uqαuq′α′tqαtq′α′ B
∞ limit: a singlet, qX′α′(cid:20) E E−Eqα−Eq′α′ (cid:21) q′α′
u2 t2
1 q′α′ q′α′
|Si≡(cid:26)A+Xq,α √2Bqα(γq†α↑c†↓−γq†α↓c†↑) =(cid:20)E−Eqα−ε0−qX′,α′ E−Eqα−Eq′α′(cid:21)Bqα (5)
+ Cqαqα′′γq†α↑γq†′α′↓ |0i , (2) and
qXq′,,αα′ (cid:27) uqαuq′α′tqαtq′α′ + vqαvq∗′α′tqαtq′α′ B˜
and a doublet, qX′α′(cid:20) E˜−ε0 E˜−ε0−Eqα−Eq′α′(cid:21) q′α′
|D↑i≡(cid:26)A˜c†↑+Xq,αB˜qαγq†α↑+ qXq′,,αα′hC˜qαqα′′γq†α↑γq†′α′↓c†↑ =(cid:20)E˜−Eqα−qX′,α′ E˜−2ε|0v−q′αE′|q2αt2q−′αE′ q′α′(cid:21)B˜qα (6)
We solve these equations numerically for the symmet-
+√13D˜qαqα′′(γq†α↑γq†′α′↓c†↑−γq†α↑γq†′α′↑c†↓) |0i , (3) ric case ∆L = ∆R =∆, tL =tR =t, ΓL = ΓR = Γ. The
i(cid:27) normalstateofeachelectrodeisdescribedbyaflatband
where 0 isthefermionvacuum(theotherdoubletstate of width 2W; we use W/∆=10 in our calculations, but
| i
D is obtained by rotating the spins by π about the the generalfeatures are rather insensitive to the value of
| ↓i
2
∆=2W exp( π ε /2Γ)=2T . (9)
− | 0| K
ThispredictsastraightlineforΓversus ε withaslope
− 0
Γ/( ε ) = π/2ln(2W/∆). With W = 10∆, the slope is
0.52−4,0and the line would be bounded by the 0 0′ and
π π′ phase boundaries in fig. 2 (it is not show−nfor the
−
purposes of clarity).
III. SLAVE BOSON MEAN FIELD THEORY
We consider an extension of the model of eqn. (1) by
introducing a flavor -1 m which runs from 1 to N . The
f
Hamiltonian is then
Nf
= (ξ +µ Bσ)ψ† ψ +
K  qα B qαmσ qαmσ
Xα mX=1Xq σX=↑,↓
∆α eiδαψq†αm↑ψ−†qαm↓+e−iδαψ−qαm↓ψqαm↑
FIG. 2. Variational wavefunction phase diagram for the
(cid:16) (cid:17)
symmetric case ΓL =ΓR ≡Γ. 1 t ψ† c +t∗ c† ψ
−√ α qαmσ mσ α mσ qαmσ 
α
N σX=↑,↓(cid:16) (cid:17)
W. Energy versus phase difference, E(δ), is plotted in
Nf
figure 1 for ∆ = 1, Γ = 0.3π, for four different values of + (ε +µ Bσ)c† c , (10)
0 B mσ mσ
ε . When ε = 1.6, the singlet state is the ground
− 0 0 − mX=1σX=↑,↓
state for all δ, while for ε = 2.0, the ground state is
alwaysthedoublet. These0are0−-andπ-junctions,respec- where µBB is the Zeeman energy. The impurity level
occupancy satisfies the constraint c† c r.
tively. For ε0 = −1.8, the curves cross, and the ground We refer to this as model I. In ams,lσighmtlσymdσiffe≤rent
stateenergyhasakinkasafunctionofδ. Bothδ =0and
P
large-N extension (model II), we rescale the hopping
δ = π are local minima in E(δ), with δ = 0 the global f
minimum. Using the terminology of ref. [6], this is a 0′- amplitudes tα tα/√Nf and write the constraint as
junction. The final case of the π′-junction is reflected in m,σc†mσcmσ ≤→rNf. In both cases, 0≤r ≤2.
Introducing a slave boson b and a Lagrangemultiplier
thecurvesforε = 1.9,whereδ =0isalocalminimum
0 − Pλtoimposetheconstraint,weevaluatetheimpuritycon-
and δ =π the global minimum of the energy.
tribution to the free energy at the mean field level, as-
The phase diagram is displayed in fig. 2. In the HF
treatment [6], one also finds a transition from π to π′ suming both b and λ to be static. The impurity free
to 0′ to 0-junction phases as Γ is increased from zero. energy per flavor is then
However, the critical value of Γ there turns out to be
F /N =ε µ B+p(ε ε )(b2 r)
inversely proportional to lnU. There are two reasons imp f − B − 0 | | −
∞
for this: first, the HF calculations assumed an infinite dω
bandwidth,andsecond,HFisunabletodescribethefor- + π f(ω+µBB)Im lnH(ω+i0+)
mation of a Kondo singlet. Here, the transition from −Z∞
0-junction to π-junction occurs roughly for ∆ T , in 2
∼ K 2b2Γ ∆sin(1δ)
athgirseermelaentitonwictahnrbefe. d[7e]r.ivIenddeweidt,hiwnhpeenrt0ur<ba−tiεo0n,Γth≪eor∆y. H(ω)=ω2−ε2−(2|b|2Γa)2+ h | | ∆g2 ω22 i
−
The perturbative value of energy of the singlet state is 4b2Γ ω2
+ | | a (11)
√∆2 ω2
4Γ 2W
E ln , (7) −
0
≈− π ∆
where Γ = 1, Γ = Γ Γ with Γ = πρ t 2 (ρ
while the energy of the doublet is a 2 g L R α α| α| α
is the bare density ofqstates per unit cell in electrode
α), δ = δ δ is the phase difference between the two
2Γ 2W L− R
E↑ ≈ε0− π ln ∆ . (8) saunpdefrc(oωn)d=uc[teixnpg(ωel/eTct)ro+d1e]s−(1wies tahsesuFmerem∆i1fu=nc∆tio2n≡. T∆h)e,
Comparing these two we find that the symmetry of the Lagrange multiplier λ is absorbed into the renormalized
ground state changes when impurity level energy, ε ε +λ. The factor p in the
≡ 0
3
second term is 1/N in model I, while p = 1 in model
f
II. Hence it is model II which generates a true large-N
f
expansion,u with all terms in the impurity free energy
F proportional to N .
imp f
From H(0) < 0, H(∆−) = + , and dH/dω > 0, we
∞
conclude that there is a unique solution to the equation
H(ω) = 0 on the interval ω [0,∆]. Call this root Ω.
∈
We use the external field B as a Lagrange multiplier so
thatwemayfixthetotalvalueofSz. MakingaLegendre
transformation to G (Sz) = F (B) 2µ BSz, with
imp imp − B
Sz = 1,0,+1, we set ∂F /∂(µ B) = 0, and obtain,
−2 2 imp B
at T =0 (assuming 0<µ B <∆),
B
G0 /N =ε+p(ε ε )(b2 r)+ (12)
imp f − 0 | | − A
1 ω◦
= dωIm lnH(ω i0+) ΩδSz,0 , (13)
A π − −
Z∆
where ω =√W2+∆2; W is the half-bandwidth in the
◦
electrodes. Themeanfieldequations,obtainedbysetting
∂G0imp/∂|b|2 =0 and ∂G0imp/∂ε=0 are Γ F=IG0..36.50, ΓMe=an0.fi6e3l0d, Bpar=am2e5t,erδs =ver0s,uasndε0.p =∆r== 11,.
a g
∂ Solid curveis Sz =0, dot-dash is Sz =±1.
1+p(b2 q)+ A =0 (14) 2
| | − ∂ε
∂
p(ε ε )+ A =0 , (15)
− 0 ∂ b2
| |
and the Josephson current is
2e∂
I = A . (16)
¯h ∂δ
Thus, for 0 µ B < Ω, the ground state has Sz = 0,
B
while for Ω ≤< µ B < ∆, the ground state has Sz =
B
1sgn(B). | |
−2
For p = r = 1, we can show that the ground state
energiessatisfy ∆G0 G0 G0 >0, whichmeans
imp ≡ imp− imp
that the mean field slave boson theory cannot describe
the π or π′ phases. The energy difference ∆G0 is min-
imp
imized at δ = π. The value of ∆G0 is an increasing
imp
function of the bare impurity level energy ε . However,
0
there are no solutions to the mean field equations for
tεh0e<enεm0dpino=int−s4oπlu−t1ioΓnacεo=sh−b12(ω=◦/0∆h)o−ld2sΓ. aTδShzu,s0,.tRheatbheesrt,
| | FIG. 4. Mean field parameters and Josephson current
we can do is ∆G0imp = 0, but in this case G0imp = ε0, versus δ = 0 for ε0 = −2, ∆ = 1, Γa = 0.650, Γg = 0.630,
independent of δ, and there is no Josephson current. B =25, δ =0, p=r =1. Solid curve is Sz =0, dot-dash is
Sz =±1.
2
IV. LARGE ∆ LIMIT
c (ω )
Ψ(ω )= ↑ m
m
Thecase∆→∞(withU finite)maybesolvedexactly. c¯↓(−ωm)!
We begin with the Hamiltonian of eqn. (1), integrating
out the fermion degrees of freedom in the superconduc- (ω )= Γα iωm ∆eiδα . (17)
tors [6]. This generates an induced action, G m α ωm2 +∆2α (cid:18)∆e−iδα iωm (cid:19)
X
p
Sind = Ψ¯i(ωm)[σz (ωm)σz]ijΨj(ωm) If the dynamics all occur on frequency scales ω ≪∆, we
G mayignorethe Matsubarafrequenciesω incomparison
Xωm with ∆. Adding the induced action tomthe bare action
4
for the impurity, we find the resultant action is that for VI. BIBLIOGRAPHY
a Hamiltonian
=ε (c†c +c†c ) χc†c† χ∗c c +Uc†c†c c ,
Heff 0 ↑ ↑ ↓ ↓ − ↑ ↓− ↓ ↑ ↑ ↓ ↓ ↑
(18)
where
χ=Γ eiδL +Γ eiδR . (19) [1] I. O.Kulik, Sov.Physics JETP, 22, 841 (1966).
L R [2] H. Shibaand T. Soda, Prog. Theor. Phys. 41, 25 (1969).
The ground state energy is E0 = ε for the Kramers [3] L.N.Bulaevskii, V.V.Kuzii, andA.A.Sobyanin, JETP
D 0 Lett. 25, 290 (1977)
doublet, and
[4] L.I.GlazmanandK.A.Matveev,Pis’maZh.Eksp.Teor.
Fiz. 49, 570 (1989) [JETP Lett. 49, 659 (1989)].
1 1
E0 =ε + U (ε + U)2+ χ2 (20) [5] B. I. Spivak and S. A. Kivelson, Phys. Rev. B 43, 3740
S 0 2 −r 0 2 | | (1991).
[6] A. V. Rozhkov and D. P. Arovas, Phys. Rev. Lett. 82,
for the singlet. Thus, for U <U , where
c 2788 (1999).
[7] A.ClerkandV.Ambegaokar,preprintcond-mat 9910201.
1
U =4 Γ2 Γ2sin2( δ) , (21) [8] C.M.VarmaandY.Yafet,Phys.Rev.B13,2950(1976).
c r a− g 2 [9] O. Gunnarsson and K. Schonhammer, Phys. Rev. B 28,
4315 (1983).
the Coulomb repulsion is too weak to overcome hy-
bridization effects, and the ground state is a singlet for
all ε . For U > U , the ground state will be a doublet
0 c
provided
U U2 U2 <ε < U + U2 U2 . (22)
− − − c 0 − − c
p p
Thisallowsthepossibilityofalevelcrossingasafunction
ofδ ifE0 <E0 <E0. However,thedoubletenergyisin-
s d s
dependentofδ inthismodel,hencethereisnoJosephson
coupling in the doublet ground state.
V. CONCLUSIONS
Using a generalization of the variational wavefunc-
tions applied in the study of the Kondo effect [8,9], we
havedemonstratedthat the U = interactingimpurity
∞
Josephsonjunctionexhibitsafirstorderphasetransition
for ∆ T . When ∆<T , the ground state of the sys-
tem is≈a sKinglet, and∼E(δK) is minimized at δ = 0 and
maximized at δ = π. When ∆>T , the ground state is
K
aKramersdoublet,thestability∼ofδ =0andδ =π isre-
versed, and the system forms a π-junction. For ∆ T ,
≃ K
thesingletanddoubletenergycurvesmaycross,inwhich
case the ground state energy has a kink as a function of
δ [6] and is strongly nonsinusoidal.
We also solved for the junction’s properties within a
slave boson mean field theory. Unfortunately, this ap-
proach is unable to identify a phase transition, and the
singlet state always is lower in energy than the doublet.
One must go beyond the static mean field approach, as
hasrecentlybeenaccomplishedinref.[7],toseethetran-
sition.
5