Static charge coupling of intrinsic Josephson junctions

Physica C 362 (2001) 43±50

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Static charge coupling of intrinsic Josephson junctions Ch. Helm a,*, J. Keller b, Ch. Preis b, A. Sergeev c a

Los Alamos National Laboratory, Division T-11, M.S. B-262, Los Alamos, NM 87545, USA Institute of Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany c ECE Department, Wayne State University, Detroit, MI 48202, USA

b

Received 23 August 2000

Abstract A microscopic theory for the coupling of intrinsic Josephson oscillations due to charge ¯uctuations on the quasi-twodimensional superconducting layers is presented. Thereby in close analogy to the normal state the e€ect of the scalar potential on the transport current is taken into account consistently. The dispersion of collective modes is derived and an estimate of the coupling constant is given. It is shown that the correct treatment of the quasiparticle current is essential in order to get the correct position of Shapiro steps. In this case the in¯uence of the coupling on dc properties like the I±V curve is negligible. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.72.)h; 74.80.Fp; 74.50.‡r; 74.40.‡k Keywords: Layered superconductors; Intrinsic Josephson e€ect; SN junction; Shapiro steps

1. Introduction Since the discovery of the intrinsic Josephson e€ect not only the typical properties of conventional junctions were demonstrated [1,2], but also unique and surprising features like the coupling of Josephson oscillations to phonons [3] have been discovered. There has also been a considerable interest in the in¯uence of nonequilibrium e€ects on the I±V characteristic and collective modes [4± 14], as the quasi two-dimensional superconducting layers are expected to be more sensitive to external perturbations than bulk materials. Recent measurements of Shapiro steps on the resistive branch [15,16] suggest to study the role of nonequilibrium e€ects on these, as the in depth theoretical under*

Corresponding author. E-mail address: [email protected] (Ch. Helm).

standing of this problem is important for any highprecision applications of high temperature superconductors (HTSC) as a voltage standard. This paper is organized as follows: The microscopic theory for the electronic transport between the superconducting layers is developed and the close analogy of the static case to the normal state is pointed out. Then the consequences for the I±V curve, the dispersion of collective modes and the position of Shapiro steps are being discussed. Further technical details can be found in Refs. [10,11] and in a forthcoming publication [17].

2. Tunneling theory We consider a stack of N ‡ 1 (superconducting) layers l ˆ 0; 1; . . . ; N forming N intrinsic (Josephson) junctions in the homogeneous case. The

0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 6 4 5 - 1

44

Ch. Helm et al. / Physica C 362 (2001) 43±50

ql ˆ 0 1 …El;l‡1

El

1;l †;

…4†

lcl can be eliminated from Eq. (2):   …2† j ˆ jNN ˆ r 1 a D El;l‡1 ; l;l‡1 0 l;l‡1

…5†

with the discrete derivative D…2† fl :ˆ fl‡1 ‡ fl 1 2fl and the coupling constant 0  a0 ˆ 2  1: …6† 2e dN2d

Fig. 1. Schematic picture of the mesa of superconducting layers in a two-point measurement.

normal conducting electrodes attached at the top and bottom of the stack in a two-point measurement are denoted as l ˆ 1, N ‡ 1 (cf. Fig. 1). As a motivation for the following discussion let us ®rst recall the situation in the normal state. There the electrical current (density) jNN l;l‡1 ˆ

rl;l‡1 ec …ll‡1 ed

jNN l;l‡1 ˆ rl;l‡1 El;l‡1 ‡

lec l †; rl;l‡1 c …ll‡1 ed

…1† lcl †;

…2†

between the layers l and l ‡ 1 is given by the difc ference of the electrochemical potentials lec l ˆ ll eA0;l (Fermi energy) in neighbouring layers (d distance of layers, e ˆ jej). This can be separated in a di€usion term driven by the di€erence D…1† lcl :ˆ lcl‡1 lcl of the chemical potentials lcl and a ®eld term rl;l‡1 El;l‡1 ˆ rl;l‡1 …A0;l A0;l‡1 †. In turn, the (static) charge ¯uctuation ql;N ˆ

2eN2d lcl

…3†

on the layer l is determined completely by the ®lling of the conduction band or the chemical potential lcl respectively. Using this and the Poisson equation (1 : background dielectric constant)

For a ®xed dc-bias current j Eq. (5) can be used to determine the electric ®eld El;l‡1 by applying the operator 1 ‡ a0 D…2† on Eq. (5). If all conductivities rl;l‡1 are equal, no charge ¯uctuations accumulate and El;l‡1 ˆ j=r, while in the case of only one barrier with a higher resistance (e.g. r0;1  rl;l‡1 ; l 6ˆ 0) the electric ®eld is not only localized at the highly resistive junctions, but is spread to neighbouring junctions E0;1 ˆ …1 E

1;0

2a0 †j=r0;1 ;

…7†

ˆ E1;2 ˆ a0 j=r0;1 :

…8†

In the superconducting state both the electric ®eld El;l‡1 and the di€erence (l ˆ 0; . . . ; N 1) Z 2e l‡1 cl;l‡1 ˆ vl vl‡1 Az …z; t† dz …9† h l of the phases vl of the superconducting order parameters Dl ˆ jDl jeivl in neighboring layers are gauge invariant quantities and have an independent physical meaning. Its time derivative directly leads to the general Josephson relation h …t† ˆ Vl;l‡1 ‡ ll‡1 c_ 2e l;l‡1

ll ;

l ˆ 0; . . . ; N

1; …10†

between the phase cl;l‡1 and the voltage Vl;l‡1 ˆ dl;l‡1 El;l‡1 across the junction, where the scalar potential ll is given by: ll ˆ A0;l

h v_ …t†; 2e l

l ˆ 0; 1; . . . ; N :

…11†

As can be seen from the Gorkov equation for the single particle Greens function, the static part c ldc l =e plays the role of the chemical potential ll in the superconducting state.

Ch. Helm et al. / Physica C 362 (2001) 43±50

In addition to this, a tunneling Hamiltonian X y HT ˆ tr cl‡1rr clrr lrr



Z

 exp

i…e=h†

l‡1 l

hr0 1 r1 D…1† ll c_ d 2ed l;l‡1 …r0 r1 † …1† D ll ; ˆ jsup l;l‡1 ‡ r0 El;l‡1 ‡ d

jl;l‡1 ˆ jc sin cl;l‡1 ‡

 Az …z; t†dz ‡ c:c: …12†

ql ˆ

between the (BCS-like) superconducting layers has to be speci®ed explicitly in order to calculate jl;l‡1 . In the following we will assume a hopping matrix 2 2 2 element tkk 0 ˆ ts ‡ td gk gk0 , which has both a constant s- and d-wave (i.e. gk ) part in order to get both a Josephson current and a ®nite quasiparticle conductivity at the same time. The general structure of the theory will be independent of this choice. A time dependent perturbation theory up to the second order in the hopping matrix element tkk0 is performed, but no assumption for the external electromagnetic potentials A0:l …t† and Az …t† is made. The resulting expressions for the tunnel current XZ t jl;l‡1 ˆ I dt0

ˆ



1

 P …t; t0 † exp‰…i=2†…cl;l‡1 …t†  cl;l‡1 …t0 ††Š …13† between the layers l and l ‡ 1 and the charge ¯uctuation ql ˆ

vqq ll ‡ Qt2 …t; cl;l‡1 †

…14†

on layer l are nonlinear in cl;l‡1 . Thereby vqq is the charge susceptibility of the superconducting layer, which is for x ˆ 0 connected with the twodimensional density of states N2d and independent of the temperature T: vqq …x ˆ 0; T † ˆ 2e2 N2d . The correlation functions P and the charge contribu2 tion Qt2 in O…tkk 0 † also depend on the scalar potential ll and have been expressed by Feynman diagrams in [10,11]. Using the general ansatz for the phase cl;l‡1 ˆ c0l ‡ Xl t ‡ dcl;l‡1 …t†

…15†

in resistive (Xl 6ˆ 0) and superconducting (Xl ˆ 0) junctions, jl;l‡1 and ql can be linearized in the (small) oscillations dcl;l‡1 and the scalar potential ll :

45

ir2 …2† hr1 …1† D ll D cl dx 2ed  iQ jl;l‡1 jQl 1;l ; vqq ll ‡ x vqq ll ‡

jQl;l‡1 ˆ r1 El;l‡1 ‡

…r1

r2 † d

ll‡1

…16†

1;l


ll :

…17† …18†

Here we restricted the discussion to a linear quasiparticle characteristic for simplicity. The conductivities ri …Xl ; x† will in general depend on both hXl =2e ˆ Vdc ‡ ldc ldc l‡1 l and the oscillation frequency x of dcl;l‡1 , which coincides with X e.g. on the autonomous ®rst R resistive branch. Products like …r0 El;l‡1 †…t† ˆ r0 …t t0 †El;l‡1 …t0 † dt0 are to be interpreted as a folding in time space. All conductivities ri are of the same order of magnitude as the experimental quasiparticle con1 ductivity rexp  2…kX cm† (Bi-2212) and consequently corrections ri =0 x  10 2 are small for frequencies x > xpl on the resistive branch. The conductivity r1 is presented for di€erent values of X in Fig. 2. There the transition to the conductivity rN ˆ 4e2 pN2d ts2 =h3 for x > 2eD0 (D0 : maximal dwave gap) and the slight di€erence between the value of r1 in a superconducting and a resistive junction can be seen. The negative part for small frequencies is not a violation of causality, but arises from the fact that r1 is not a conventional transport coecient. The static limit of r1 is subtle and deserves special attention. Formally the result for x ˆ 0 cannot be obtained in our formalism by expanding ql or jl;l‡1 in the static part ldc l of the scalar potential directly. Instead of this the static inhomogeneity has to be included in the equilibrium Greens function in zeroth order in tkk0 . For x ! 0 and T < eV  D0 one gets the ®nite value r1 …X; x ! 0†  rN

T eV ; D0 D0

…19†

while the strictly static function rdc 1 :ˆ r1 …x ˆ 0; T † ˆ 0 vanishes exactly for all temperatures. This

46

Ch. Helm et al. / Physica C 362 (2001) 43±50

out that this choice is strictly speaking model dependent and a ®nite value rdc 1 similar to our result for r1 …x ! 0† in Eq. (19) could be obtained, if additional scattering mechanisms are considered. The elimination of Vl;l‡1 , ll and dql from Eqs. (4), (10), (16) and (17) leads to a set of coupled equations of motion for the phases cl;l‡1 …x2pl :ˆ 2edjc =h0 , xc ˆ x2pl 0 =r0 , Da…2† :ˆ 1 D…2† a†: j 1 1 …2† ˆ Da…2† sin cl;l‡1 ‡ Dg…2† c_ l;l‡1 ‡ 2 Df cl;l‡1 jc xc xpl …20† The coupling constants are given as:   0  ir2 1‡ a…x† :ˆ ; dvqq 0 x  0  ir2 1‡ g…x† :ˆ dvqq 0 x f…x† :ˆ

Fig. 2. Real and imaginary parts of r1 in superconducting (X ˆ 0, top ®gure) or resistive (X 6ˆ 0, bottom ®gure) junctions at T ˆ 0.

discontinuity is due to the fact that our only relaxation mechanism is the incoherence of the hopping tkk0 between layers. Therefore in a tun2 neling process (which is already of order tkk 0 ) no additional relaxation within the layers is possible, 2 if we work strictly in O…tkk 0 †. Including strong impurity scattering in the layers, which is the main mechanism for charge imbalance relaxation in d-wave superconductors, leads to r1 …x ! 0† ˆ r1 …x ˆ 0† ˆ 0 like in Refs. [7,8], which will be considered mainly in the following. It is pointed

 2r1 ; r0

0  ir2 : dvqq 0 x

…21†

…22†

…23†

These are in the static case completely determined by the normal state value in Eq. (6): a…x ˆ 0† ˆ g…x ˆ 0† ˆ a0 and f…x ˆ 0† ˆ 0. The rough theoretical estimate for a0  0:28 (for  ˆ 10, d ˆ 1:5 nm) based on a two-dimensional electron gas with density of states N2d ˆ mel =2p2 h2 (not including spin) is to be considered as an upper bound, as the density of states at the Fermi surface could be enhanced by a factor 2±3 due to strong electronic correlations [18,19]. A more reliable estimate is possible from optical experiments and will be elaborated in detail elsewhere [20]. Linearizing the set of Eqs. (20) one obtains the 2 dispersion of collective modes (in order O…tkk 0 †, 2 2 2 1  ˆ xpl =c ): x0 :ˆ r0 =0  ˆ xpl =xc , x kz 1

 2pl x2  x

1 ‡ akz2 1 ‡ fkz2

 2pl 1 ‡ akz2 x2 ˆ x

ixx0


1 ‡ gkz2 ; 1 ‡ fkz2


ixx0 1 ‡ gkz2 :

…24† …25†

For frequencies xpl  x  D0 on the resistive branch these represent weakly damped plasma

Ch. Helm et al. / Physica C 362 (2001) 43±50

modes of the superconducting condensate. For the special case e ! 0 the dispersion x…kz † ! 0 for kz ! 0 of the Goldstone mode associated with the spontaneous breaking of the gauge symmetry in the superconducting state can be obtained explicitly. This reproduces in the Kuboformalism the results in Refs. [7±9], which were obtained by solving kinetic equations. Taking into account only the leading order 0 O…tkk 0 † in the hopping matrix element, a…x; T † has been calculated in [10] and is presented for T ˆ 0 in Fig. 3. This result is obtained for a constant background dielectric constant , while the frequency dependence of 1 …x† near phonon resonances can change the behaviour of a…x† considerably [3]. For the temperature dependence see Ref. [11]. We would like to point out the remarkable similarity of the static quasiparticle current in the superconducting and in the normal state. Due to r1 …x ˆ 0† ˆ 0, Eq. (16) reduces to: jdc ˆ jc hsin cl;l‡1 i ‡

Dr hr0 1 hc_ l;l‡1 i ‡ …l 2ed d l‡1

E ll † : …26†

In the last term of Eq. (26) the static part of l does not contribute due to r1 …X; x ˆ 0† ˆ 0, but a ®nite dc-contribution might arise from the combination of l…X† and Josephson-like terms in r1 . This term

Fig. 3. Real and imaginary parts of a…x† for d-wave pairing for 0 temperature T ˆ 0 in O…tkk 0 † (x in units D0 …T †, 1 ˆ constant).

47

4 is in intrinsic junctions negligible of order atkk 0  1 and vanishes exactly for SN junctions:   NN jdc ˆ hjsup i ‡ hj i ; …27† l;l‡1 l;l‡1

jdc ˆ jc hsin cl;l‡1 i ‡

hr0 hc_ i; 2ed l;l‡1

…28†

dc jdc ˆ jc hsin cl;l‡1 i ‡ r0 Da…2† El;l‡1 ; 0

…29†

where the quasiparticle current coincides with Eq. (2), if we express cl;l‡1 by El;l‡1 and ll via Eq. (10). The dc-equations SN j ˆ rSN 0 EN ;N ‡1 ‡ …r0 0 ˆ rSN 0 ……1 ‡ a0 †EN ;N ‡1 0 j ˆ rSN 0 ……1 ‡ a0 †E

1;0

…1† rSN 1 †D lN

a00 EN a00 E0;1 †;

1;N †;

…30† …31†

for the SN junctions are slightly di€erent, as the change of the chemical potential lc 1 ˆ lcN ‡1 ˆ 0 in bulk materials is negligible due to the large density dc of states. Thereby a00 ˆ a0 …1 rdc 1;SN =r0;SN †, which is in the model with strong impurity scattering in the layers given as: a00 ˆ a0 . If we neglect the supercurrent for a moment, the inversion of the Da…2† shows that the total 0 Poperator N dc dc voltage V ˆ iˆ 1 Vl;l‡1 is exactly given like in a stack of independent junctions, i.e. the coupling constant a0 does not enter in the I±V curve. The dc-component of the supercurrent might add a correction to this due to the interaction a at the frequency X, but this contribution is suppressed 2 by a factor tkk 0 a0  1 and therefore no signi®cant e€ect on dc-properties is expected. It is pointed out that this general result cannot be obtained by assuming an ohmic quasiparticle current jqp l;l‡1 ˆ rEl;l‡1 as in Ref. [4]. An additional indication that the correct choice of the quasiparticle current is essential will come from the discussion of Shapiro steps in the next section. This might also be the reason that up to now no clear experimental evidence for the coupling a in the I±V characteristic, like small deviations in the additivity of resistive branches [11], could be found. Nevertheless, the distribution of the electric dc ®elds El;l‡1 within the stack is a€ected by the

48

Ch. Helm et al. / Physica C 362 (2001) 43±50

coupling a0 like in Eqs. (7) and (8) in the normal state. For a single resistive junction the electric ®eld is not contained in this junction, but leaks into neighbouring junctions in the same way as around a junction with much lower resistance in a stack of normal junctions. It is stressed that the above discussion makes use of the fact that rdc 1 ˆ 0 in the presence of strong impurity scattering in the layers. In a more general model this might not be the case and dc coupling terms a0 rdc 1 =r0 appear. The coupling via charge ¯uctuations might also have a considerable e€ect at frequencies larger than the charge imbalance relaxation rates (e.g. for phase locking at microwave frequencies), where the r1 =r0 in Eq. (22) can be of order O…1† in any model.

A Shapiro step is generated by microwave irradiation of frequency X, which creates an oscillating component of the c-axis bias current Iac ˆ Irf sin…Xt†. Thereby the dynamics of the phase in a resistive junction is given as …32†

Therefore the supercurrent (Jk : Bessel functions of ®rst kind) Isupra ˆ Ic sin…cn;n‡1 …t†† 1 X ˆ Ic Jk …cn1 † sin…cn0 ‡ …1 ‡ k†Xt†

…33†

Edc1;0 ˆ ENdc;N ‡1 ˆ …1 dc ˆ 0; El;l‡1

dc Isupra ˆ Ic J 1 …c1n † sin…c0n †;

V0dc ˆ

which creates the (®rst) Shapiro step in the I±V curve. The nontrivial question in the following will be, what dc-voltage Vstep is connected with a given X, if the generalized Josephson relation Eq. (10) is taken into account. Using Eqs. (4) and (17) this can be presented as:

‡ El‡1;l‡2 †; …35†

a00 †j=r0;SN ;

l ˆ 1; . . . ; N 1;N

ˆ

…36†

2;

…37†

a00 j=rdc 0;SN

…38†

N X lˆ 1

dc Vl;l‡1 ˆ

2jd : rdc 0;SN

…39†

This linear branch re¯ects the contact resistances at both SN junctions and is not a€ected by the coupling a00 , although the voltages Vl;l‡1 in the intrinsic junctions in general do not vanish. For a single resistive junction in the top layer l ˆ N 1 …hc_ N 1;N i ˆ X; hc_ n;n‡1 i ˆ 0; n else† one obtains analogously: EN ;N ‡1 ˆ …1

Vstep ˆ …34†

1;l

and the total dc-voltage

kˆ 1

has a dc-component

a…El

for l ˆ 0; . . . ; N 1. This set of N equations is not sucient to determine all N ‡ 2 voltages Vl;l‡1 and has to be complemented by the current equations (30) and (31) in the SN junctions. Note that for the total voltage Vstep across the stack the transport coecients in the intrinsic Josephson junctions are irrelevant. In the ``superconducting'' state, which is de®ned by hc_ l;l‡1 i ˆ 0 for all l ˆ 0; . . . ; N 1, one easily obtains …in O…a00 ††

dc ˆ ENdc E0;1

3. Shapiro steps

cn;n‡1 …t† ˆ c0n ‡ Xt ‡ c1n sin…Xt†:

h ˆ …1 ‡ 2a†El;l‡1 c_ 2ed l;l‡1

V0dc

0 a00 †j=rdc 0;SN ‡ a0

h ‡ X 1 2e

Vstep ˆ V0dc ‡

h X: 2e

hX ‡ O…a02 2 †; 2ed

rdc 1;SN a0 dc r0;SN

!

rdc ˆ0 1;SN

ˆ

…40†

…41†

The last result is correct in all orders of a00 . Here the dc-conductivity rdc 1;SN ˆ 0 has been kept explicitly in Eq. (40) in order to demonstrate that the position of the step in general depends on the value dc of rdc 1;SN . In the model considered here r1;SN ˆ 0 and the Shapiro step is where expected, which

Ch. Helm et al. / Physica C 362 (2001) 43±50

4. Conclusions

I

V

Normal

Normal N

µ=0

CuO2 CuO 2

N-1

1- - - - - - - - - - - - - - - -

µ>0

0+ + + + + + + + + + + +

µ>0

Normal

µ=0 Normal V

I Fig. 4. Schematic four-point geometry in Ref. [21]

would not be obtained by using the ohmic quasidc particle current alone …rdc 0;SN ˆ r1;SN † like in Ref. [4]. Nevertheless this result opens up the principal possibility that there is a shift in the position of the Shapirostep in a more general theory (e.g. including electron±phonon scattering) due to nonequilibrium e€ects, where rdc 1 might not vanish. Taking our result Eq. (19) seriously the relative shift would be small 10 5 , but detectable. This result is also valid in a four-point measurement geometry, as sketched in Fig. 4 and used in Ref. [21]. Although the total voltage int Vstep ˆ

N 1 X lˆ0

dc Vl;l‡1 ˆ

49

X h …1 2e

a0 † ‡

2ja0 rdc 0;SN

…42†

across the intrinsic contacts depends on a0 , this deviation of the expected position of the Shapiro step cannot be detected at the contacts. The electrochemical potential is constant along the superconducting layers, but the potential ll …x† and the electric ®eld Ek are not and will compensate the additional contribution in Eq. (42). Finally, note that all the above considerations only apply to an experimental situation where the electric ®eld Edc is measured directly to determine the dc-voltage V dc . This might not be operational with conventional voltmeters, which actually detect a current through a circuit with high resistance, which is driven by the di€erence of the electrochemical potentials lec l rather than by the voltage V dc [22].

The microscopic theory of tunnelling in layered superconductors has been studied including the e€ect of the scalar potential on the layers and an estimate of the coupling constant a  0:2 has been given. The static quasiparticle current includes both ®eld and di€usion terms as in the normal state, which turns out to be crucial for all dc properties. In the model with strong impurity scattering in the layers, Shapiro steps are exactly at the expected position Vstep ˆ V0dc ‡ hX=2e. In this case the e€ect of charging on the total I±V 2 curve is suppressed by a factor a0 tkk 0 and therefore negligible, but it a€ects the distribution of the electric ®eld within the stack. This results motivate the study of more general microscopic models including di€erent relaxation mechanisms like electron±phonon scattering, as a ®nite rdc 1 would modify the position Vstep in a characteristic way with important consequences for potential applications as a voltage standard. Also precision experiments on Shapiro steps could then be used as a sharp test for microscopic theories.

Acknowledgements The authors would like to thank S. Rother, R. Kleiner and P. M uller for discussing their unpublished experimental data and L.N. Bulaevskii for his interest in this work. Financial support by DFG, FORSUPRA and DOE under contract W7405-ENG-36 (C.H.) is gratefully acknowledged.

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Ch. Helm et al. / Physica C 362 (2001) 43±50

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