Energy gap and tunneling

Energy gap and tunneling

I.

13

Introduction

In Chapter 10, we introduced critical-state models such as the Bean Model, which gave us a chance not only to acquire simple understandings of some of the magnetic phenomena that had been discussed in Chapters 5, 8, and 9 but also to discuss critical currents and, thereby, introduce the area of transport properties, which is the subject of this chapter. Transport properties are of importance because the principal applications of superconductors are based upon taking advantage of their ability to carry electric current without any loss. The chapter begins with a discussion of tunneling and supercurrent flow in the absence of externally applied fields. After covering introductory material on tunneling, we will discuss the Josephson effect and macroscopic quantum phenomena. It will be shown how tunneling measurements provide energy gap values. Chapter 3 has already examined several transport processes that involve applied fields and thermal effects.

II. Phenomenon of tunneling Tunneling, or barrier penetration, is a process whereby an electron confined to a region of space by an energy barrier is nevertheless able to penetrate the barrier through a quantum mechanical process and emerge on the other side. The example shown in Fig. 13.1 involves electrons with kinetic energy EKE 5 1/2mυ2 confined to remain on the left side of a barrier by the potential Vb where 1/2mυ2 , eVb. We show in this figure an electron tunneling through the barrier to the right side where it ends up with the same kinetic energy. Such a phenomenon can occur because there is a quantum mechanical probability per unit time that the electron will penetrate the barrier and escape, as explained in standard quantum mechanics texts. Tunneling phenomena are rather common in physics. For example, radioactive decay of nuclei is explained by a barrier penetration model, with half-lives varying from less than nanoseconds to many centuries.

A. Conduction-electron energies The surface of a normal metal has a dipole charge density layer that produces a barrier potential Vb. The conduction electrons inside the metal move in a region

Superconductivity. DOI: http://dx.doi.org/10.1016/B978-0-12-409509-0.00013-5 © 2014 Elsevier Inc. All rights reserved.

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Superconductivity

Barrier

Figure 13.1 Tunneling of electrons through a barrier when the kinetic energy of the electrons is less than the barrier energy eVb.

Surface barrier

Energy

eVb eVw 0

⫺EF

Empty e levels Occupied e levels (conduction band)

Figure 13.2 Energy-level diagram of a conductor showing the levels occupied below the Fermi energy EF and the energy barrier eVb at the surface. The minimum energy (work function) eVw for extracting an electron is also indicated.

where they experience an attractive potential, so that their energies are negative with the value 2 jEFj, as indicated in Fig. 13.2. At absolute zero, the conductionband levels are filled up to the Fermi level EF and are empty above, corresponding to Fig. 1.4a. In some of the figures dark shading is used to indicate occupied levels. To remove an electron from the interior of a metal we must apply a potential equal to or greater than the work function potential Vw. The minimum energy eVw that can extract an electron is eVw 5 eVb 1 jEF j:

ð13:1Þ

Metals differ in their eVw, eVb, and EF values, so that a proper treatment of how an electron is transferred between two metals in contact through an insulating barrier should take these factors into account. However, to simplify the mathematics we will ignore these surface potential effects and assume that two metals in contact at the same potential have the same Fermi energy. Tunneling phenomena are sensitive to the degree of occupation of the relevant energy levels by electrons. Hence, in energy-level diagrams it is helpful to include

Energy gap and tunneling

503

f(E)

Energy

0

–EF

Figure 13.3 Energy-level diagram of a conductor with electron distribution f(E) at a finite temperature near the Fermi level EF plotted at the top. The shading in this and subsequent figures indicates electron occupancy.

information about the occupation of the levels involved in the tunneling. Figure 1.4b plots the temperature dependence of the FermiDirac distribution function, giving the fractional occupation of levels in the conduction band. We have replotted this function in Fig. 13.3 with energy as the ordinate, f(E) as the abscissa, and electron occupation is indicated by shading. Figure 13.4a presents a sketch of a conduction band that is filled at absolute zero and separated from an upper energy band by a gap. Figure 13.4b shows this same diagram at a finite temperature, combined with the distribution function plot of Fig. 13.3 to show the level populations.

B. Types of tunneling Tunneling can occur through an insulating layer, I, between two normal materials (NIN), such as a semiconductors, between a normal metal and a superconductor (NIS), or between two superconductors (SIS). Proximity junctions (SNS), in which the Cooper pair and quasiparticle transfer across the junction via the proximity effect, are discussed in Section VI.F. Junctions involving semiconductors, such as the S-Semicond and S-Semicond-S types, will not be discussed here (Furusaki et al., 1991, 1992; Kastalsky et al., 1991; Van Wees et al., 1991). The dc and ac Josephson effects involve particular types of tunneling phenomena across a barrier between two superconductors. In the next several sections, we will examine energy-level diagrams and then provide a qualitative picture of various tunneling processes, concluding with a more quantitative presentation.

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(a)

(b)

EF+Δ Excitations EF EF–Δ

T=0

T>0

Figure 13.4 Semiconductor representation of the energy-level occupancy of a superconductor (a) at T 5 0, and (b) at T . 0. The band gap 2Δ and level populations are shown for each case, using the convention of Fig. 13.3.

III.

Energy-level schemes

Before we discuss the NIN, NIS, and SIS types of tunneling it will be instructive to examine the energy-level systems that are involved in each. Two conventions for representing the energy levels will be introduced, called, respectively, the semiconductor representation and the Bose condensation representation.

A. Semiconductor representation A superconductor is considered to have an energy gap Eg 5 2Δ between a lower energy band, which is full of super electrons at absolute zero, and an upper energy band, which is empty at that temperature, as shown in Fig. 13.4a. At higher temperatures, some of the electrons are raised from the lower band to the upper band as illustrated in Fig. 13.4b; these excited electrons are often referred to as quasiparticles. They act like normal conduction electrons; that is, they behave approximately like free electrons moving at the Fermi velocity υF, as discussed in Chapter 1, Section II. When an electron jumps down from the bottom of the quasiparticle band to the top of the super electron band, its energy falls by the amount 2Δ. Following conventional semiconductor terminology, we assign the equivalent Fermi energy to the center of the gap. This single-electron picture, called the semiconductor representation of a superconductor, does not explicitly take into account the phenomenon of electron pairing.

B. Boson condensation representation Another way of representing a superconductor at T 5 0 is by a single level for the super electrons, as shown in Fig. 13.5a. This is justified by the argument that the

Energy gap and tunneling

505

(a)

(b)

Quasi particles Δ T=0

Cooper pairs T>0

Figure 13.5 Boson condensation representation of the energy-level occupancy of a superconductor (a) at T 5 0, and (b) at T . 0, showing the level populations in the quasiparticle band for each case. Figure 13.4 presents the corresponding semiconductor representation.

Cooper pairs that occupy this level are paired electrons with zero spin, and hence are boson particles that obey BoseEinstein statistics. For bosons there is no Pauli exclusion principle, so it is possible for all of them to have the same energy. Thus the transition to the superconducting state is an example of boson condensation, a phenomenon that is explained in quantum mechanics texts. The condensation takes place when the electrons drop into the single-superconducting level where they exist as Cooper pairs, as shown in Fig. 13.5a. This mode of presenting the energy-level diagram is called the boson condensation representation. The Cooper-pair binding energy Eg is shared by two electrons, so that Δ 5 1/2Eg is the binding energy per electron. In this representation, the Cooper-pair level is located at a distance Δ below the bottom of the conduction band, as shown in Fig. 13.5. At absolute zero all the conduction electrons are condensed at the Cooper-pair level. Above absolute zero, some of the pairs break up and the individual electrons are excited to the bottom of the conduction band, as shown in Fig. 13.5b. These electrons, which are produced by the breakup of Cooper pairs, are the quasiparticles mentioned earlier. We will find the boson condensation representation a little more convenient than the semiconductor representation for analysis of the tunneling of super electrons. Before proceeding, however, let us say a few words about tunneling in general, after which we will comment on normal electron tunneling.

IV.

Tunneling processes

We start with a brief qualitative description of the three types of tunneling processes, following it with a more detailed examination.

A. Conditions for tunneling Three conditions must be satisfied for tunneling to occur. First, there must be a barrier between the source and destination locations of the tunneling electrons

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preventing direct electron transport. Second, the total energy of the system must be conserved in the process, which is why single-electron tunneling occurs between levels that have the same energy on either side of the barrier. In two-electron tunneling, one electron gains as much energy as the other electron loses. Third, tunneling proceeds to energy states that are empty since otherwise the Pauli exclusion principle would be violated. A bias voltage that lowers the energy levels on the positive side relative to levels on the negative side is often applied. This can serve to align occupied energy levels on one side of the barrier with empty levels on the other so as to enable tunneling between the two sides. There are sign and direction rules that apply to the description of tunneling processes. When a metal (or superconductor) is positively biased relative to another metal, so that its potential is 1 V, its energy levels are lowered, as shown in Fig. 13.6b. The electron tunneling direction is toward the metal with the positive bias, but the tunneling current flows in the opposite direction, as indicated in the figure. This is because, by convention, current flow is expressed in terms of positive charges so that negatively charged electrons must flow in the opposite direction. Thus current flows toward the negative bias, as shown in Fig. 13.6c. In drawing energy-level diagrams, one of the two metals is normally grounded so that its Fermi level does not change when the bias is applied. The Fermi level falls for a metal that is biased positive relative to ground, and raised for one biased negative.

B. Normal metal tunneling Consider two normal metals grounded at absolute zero and separated by an insulating barrier. Their Fermi levels are aligned as shown in Fig. 13.6a, so no tunneling occurs. A positive-bias voltage is then applied to one of the metals, lowering its energy levels, as shown in Fig. 13.6b, so that the electrons are now able to tunnel from the top of the conduction band of the grounded metal to the empty continuum levels of the positively biased metal, as shown. The number of empty levels that can receive electrons is proportional to the bias, so that the current flow is also proportional to it, as shown in Fig. 13.6c. The magnitude of the tunneling current is, of course, small compared to the current that flows in the absence of the barrier. Such a process satisfies the three conditions for tunneling—namely, presence of a barrier, energy conservation, and empty target levels.

C. Normal metalsuperconductor tunneling Next, we consider the case of an insulating barrier between a superconductor and a normal metal. NIS tunneling occurs through the processes outlined in Fig. 13.7, where the top three diagrams are the semiconductor representation and the three middle diagrams sketch the boson condensation representation. In the unbiased cases of Fig. 13.7b and e no tunneling occurs because there is no way for energy to be conserved by electrons tunneling to empty target levels. This is also true for the range 2Δ/e , V , 1Δ/e of biases. For a positive bias, V $ Δ/e, electrons can tunnel from the conduction band of the normal metal to the empty states above the gap

Energy gap and tunneling

507

– + V

(b)

(a)

I EF2

EF1

EF1

e eV EF2

N

N

N

N

(c)

I

V

Figure 13.6 Normal metal tunneling showing (a) Fermi levels aligned for zero applied voltage, and hence zero tunneling current, (b) application of a positive-bias voltage V to metal 2, lowering its Fermi level by eV relative to metal 1 and causing electrons to tunnel from left to right (metal 1 to metal 2), corresponding to current flow opposite in direction to the electron flow, as indicated by the arrows, and (c) linear dependence of the tunneling current on the applied voltage. Sketch (b) is drawn with metal 1 grounded, so that when the bias is applied, the Fermi level of metal 1 remains fixed while that of metal 2 falls.

of the superconductor, as shown in Fig. 13.7c and f. The figures appear similar in both representations because Cooper pairs do not participate. For a negative bias, V # 2 Δ/e, the process must be considered more carefully since the explanation is different in the two representations. In the boson condensation picture shown in Fig. 13.7d tunneling involves the breakup of a Cooper pair, with one electron of the pair tunneling down to the top of the normal-metal conduction band and the other jumping upward to the quasiparticle energy band of the superconductor. Thus the paired electrons separate to create a quasiparticle in the

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Superconductivity

–V 0

0

0

0 +V

EF

2Δ

SC N

N

N

SC

(a)

(b)

V < – Δ/e

V=0

(c)

SC

V > Δ/e

Δ

(d)

(e) I

(f)

–Δ/e Δ/e

V

(g)

Figure 13.7 Normal metalsuperconductor tunneling. The semiconductor representation shows (a) super electron tunneling (SC!N) for V , 2Δ/e, (b) zero tunneling current for the V 5 0 case of the Fermi level in the gap (2Δ/e , V , Δ/e), and (c) normal electron tunneling (N!SC) for V . Δ/e, which are also shown in the boson condensation representation (d), (e), and (f), respectively. The arrows show the electron tunneling directions, which are opposite to the current flow directions; the currentvoltage characteristic is given in (g). The normal metal is grounded so that the superconductor bands shift downward when a positive bias is applied.

superconductor and transfer a conduction electron to the normal metal, with energy conserved in the process (Hu et al., 1990; Rajam et al., 1989; Van den Brink et al., (1991); Worsham et al., 1991). Both electrons of the Cooper pair are accounted for. In the semiconductor representation, only the electron that is transferred to the normal metal is taken into account, as shown in Fig. 13.7a. This electron leaves behind it a hole in an otherwise filled band; it is this hole that constitutes the quasiparticle. Figure 13.7g shows how the experimentally measured current flow between the metal and the superconductor depends on the bias.

Energy gap and tunneling

509

–V 0

0

0

0 +V 2Δ

SC SC

SC

SC

(a)

(b)

V < – 2Δ/e

V=0

(d)

SC (C)

SC

V > 2Δ/e

(e)

(f)

I –2Δ/e 2Δ/e

V

(g)

Figure 13.8 Superconductor-to-superconductor tunneling at absolute zero. The semiconductor representation shows (a) super electron tunneling for V , 22Δ/e, (b) zero tunneling current for bias voltages in the range 22Δ/e , V , 2Δ/e, and (c) opposite-direction super electron tunneling for 2Δ/e , V, which are also shown in the boson condensation representation (d), (e), and (f), respectively, where the tunneling arises from the breaking of a Cooper pair. The currentvoltage characteristic is given in (g). The arrows show the electron tunneling directions, which are opposite to the current flow directions. The sketches are drawn with the superconductor on the left grounded.

D. Superconductorsuperconductor tunneling Finally, let us consider the case of two identical superconductors. SIS tunneling occurs through the processes depicted in Fig. 13.8 for the two representations. Over the range of biases 22Δ/e ,V , 1 2Δ/e an electron in the semiconductor representation can tunnel from the superconducting state of one superconductor to become a quasiparticle in the normal state of the other, as shown in Fig. 13.8a and c.

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Superconductivity

This has its counterpart explanation in Fig. 13.8d and f where we see how a Cooper pair in the higher of the two boson condensation levels can break up, with one electron jumping up to become a quasiparticle in its own excited level and the other electron jumping down to become a quasiparticle in the other superconductor. As the bias voltage increases beyond the range 22Δ/e , V , 1 2Δ/e, the current increases abruptly in magnitude and then approaches its normal metal value, as indicated in Fig. 13.8g. The currentvoltage characteristic for two identical superconductors is, of course, antisymmetric about the point V 5 0. By antisymmetric we mean that when V! 2 V we will have I! 2 I. Note that the onset of tunneling for SIS junctions occurs at V 5 6 2Δ/e, which is twice the value for the NIS case. Figure 13.8 was drawn for the case T 5 0. For a finite temperature there will be some quasiparticles in each superconductor, so that a small tunneling current will flow for bias voltages below 2Δ/e, as shown in Fig. 13.9 in the boson condensation representation (a) and in the semiconducting representation (b). The current voltage characteristic is given in Fig. 13.9c.

(a)

(b)

Δ 2Δ

(c) I

2Δ e

V

Figure 13.9 Superconductor-to-superconductor tunneling at finite temperatures, T . 0. The semiconductor representation (a) and boson condensation representation (b) show finite tunneling between upper quasiparticle levels sparsely populated by thermal excitation. The currentvoltage characteristic (c) shows a small current flow for 0 , V , 2Δ/e, and the usual larger current flow for 2Δ/e , V.

Energy gap and tunneling

V.

511

Quantitative treatment of tunneling

The previous section discussed the different tunneling processes in terms of both the boson condensation and the semiconductor representations. The former seems to give a better physical picture of what is happening because it involves the breakup of Cooper pairs, whereas the latter provides a framework for carrying out quantitative calculations of the tunneling current as a function of temperature. We will now apply the Fermi statistics approach of Chapter 1, Section IX, to the semiconductor representation to derive quantitative expressions for the tunneling current.

A. Distribution function We explained in Chapter 1, Section IX, that the concentration of conduction electrons as a function of their energy is given by the product of the FermiDirac (FD) distribution function, f(E), and the density of states, D(E). We begin by expressing the former in a form that is convenient for treating tunneling problems, and then make use of the latter, which we write Dn(E) for the normal electrons involved in the tunneling. The super electrons have a different density of states, Ds(E), which was derived in Chapter 7, Section VI. In Chapter 1, Section IX, we expressed the energies of a conductor relative to the Fermi energy EF. In the present discussion, it is convenient to select the Fermi level as the zero of energy; that is, to set EF 5 0. With this in mind the FD distribution of Eq. (1.35) for electrons assumes the form f ðEÞ 5

1 ; exp ðE=kB TÞ 1 1

ð13:2Þ

which at absolute zero equals 1 for negative energies and 0 for positive energies. The corresponding distribution function for unoccupied states, sometimes called holes, is 1 2 f(E). If a bias voltage V is applied, the distribution function for electrons becomes f ðE 1 eVÞ 5

1 ; exp ½ðE 1 eVÞ=kB T 1 1

ð13:3Þ

and for holes is given by 1 2 f ðE 1 eVÞ 5

1 : 1 1 exp ½2ðE 1 eVÞ=kB T

ð13:4Þ

These distribution functions for T . 0 are plotted in Fig. 13.10 for zero, positive, and negative biases. Figure 1.4 shows the effect of temperature on the FD distribution.

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(d) 1 – f(E)

f(E)

(a) 1

0

E

(e) 1 – f(E + eV)

f(E + eV)

0

E

(b)

1

–eV 0

1

–eV 0

E

E

(f) 1 – f(E – eV)

(c) f(E – eV)

1

1

0

eV

E

1

0

eV

E

Figure 13.10 The dependence of the FermiDirac distribution function f(E) on the energy for zero bias (a), positive bias (b), and negative bias (c). The dependence of the distribution function in the case of holes, 1 2 f(E), on the energy for the same bias conditions (df).

B. Density of states Now that we have rewritten the FD distribution functions relative to the zero of energy set at the Fermi level Eq. (1.41) should similarly be rewritten for the density of states Dn(E) of normal electrons with this same zero of energy,
 EF 1E 1=2 Dn ðEÞ 5 Dn ð0Þ ; EF

ð13:5Þ

where Dn(0) is the density of states at the Fermi level (E 5 0). Plots of Dn(E)f(E) against the energy are shown in Fig. 1.7; we will make use of these plots with the zero of energy set at the Fermi level. Since E in Eq. (13.5) is usually very small compared with the Fermi energy jEj{EF, and since the energies of interest are generally limited by the maximum applied bias voltage Vmax, in tunneling calculations it is usually valid to write

Energy gap and tunneling

Dn ðEÞ  Dn ð0Þ

513

2eVmax , E , eVmax :

ð13:6Þ

The density of states in the superconducting state is given by the BardeenCooperSchrieffer (BCS) expression (7.65) 8 Dn ð0ÞjEj > > E , 2Δ ð13:7Þ > > > ðE2 2Δ2 Þ1=2 > > < 2 Δ , E , Δ ð13:8Þ Ds ðEÞ 5 0 > > > j j D ð0Þ E > n > Δ , E; ð13:9Þ > > : ðE2 2Δ2 Þ1=2 which is plotted in Fig. 7.4. Another property of the density of states that has important implications for superconductivity is the conservation of states in k-space that was mentioned in Chapter 1, Section X. This is reflected in the conservation of energy levels at the onset of superconductivity. When a material becomes superconducting, an energy gap forms, with some energy states shifting upward above the gap and some falling below it, with the total number of states remaining the same, in the manner illustrated in Fig. 1.9. Comparing Eqs. (13.7)(13.9) shows that the level spacing is the same just above and just below the gap. The area under the curve for D(E) versus E, which is numerically equal to the total number of energy levels, is unchanged during the passage through Tc, ð

ð Τ . Τc

Dn ðEÞ dE 5

Τ , Τc

Ds ðEÞ dE:

ð13:10Þ

We will use these expressions for f(E) and D(E) to write down an analytic expression for the tunneling current.

C. Tunneling current We begin by deducing a general expression for the tunneling current and then discuss the cases NIN, NIS, and SIS. Note, first, that in our qualitative discussion we were assuming that tunneling takes place in one direction only. A more careful analysis shows that it actually occurs in both directions, and that the net tunneling current is the difference between forward and backward tunneling processes. Consider the case of a bias voltage V, V 5 V1 2 V2 ;

ð13:11Þ

applied to a metalinsulatormetal junction, M1IM2, as shown in Fig. 13.11, where each M can be a normal metal or a superconductor. To simplify the

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Insulator

Metal 2

Metal 1

J12 J21 V2 = 0

V1 = V

I

–

+

I

V

Figure 13.11 Metal-to-metal tunnel junction in the form of a thin uniform insulating layer between two metals M1 and M2. Tunneling current densities J12 and J21 in both directions are shown, with J12 . J21 for the bias indicated.

mathematics we treat the special case of metal 1 at the potential V1 5 V and metal 2 at zero potential, V2 5 0. Electrons can tunnel to the right with the current density J12 and to the left with the current density J21, as shown. The tunneling current is, therefore, I 5 ðJ12 2 J21 ÞA;

ð13:12Þ

where A is the area of the junction. Most electrons that impinge on the barrier from the left are reflected, but a few penetrate it, as illustrated in Fig. 13.12, and contribute to the tunneling current. We recall from quantum mechanics that Fermi’s Golden Rule from time-dependent perturbation theory provides the probability per unit time W that an electron will undergo a transition from state 1 to state 2 in the energy range from E to E 1 ΔE, W1!2 5 ð2π=h ¯ Þjh2jHpert j1ij2 δ2 ðEÞ;

ð13:13Þ

where δ2 ðEÞ 5 D2 ðEÞ½1 2 f ðEÞ

ð13:14Þ

is the “target” density of empty states in the energy range into which the electron tunnels. We assume that the tunneling matrix element HT,

Energy gap and tunneling

515

J12 V=0

+V

Reflected electron Tunneling electron Metal 2

Metal 1 Barrier

Figure 13.12 One electron reflected from, and another tunneling through, the insulating barrier of a tunnel junction for the bias of Fig. 13.11. The tunneling of the electron is in the direction from the negative to the positive side of the junction, but the corresponding current flow J12 is from 1 to 2 because it is based on the convention of positive-charge carriers.

   HT 5 H12 5 H21 h2Hpert 1i

ð13:15Þ

of the perturbation Hamiltonian Hpert responsible for the penetration at the barrier can be evaluated. The tunneling current from metal 1 to metal 2 is related to the transition probability through the expression Ð J12 5 e W1!2 D1 ðE 2 eVÞf ðE 2 eVÞ dE ð 2πe jHT j2 D1 ðE 2 eVÞf ðE 2 eVÞ 5 ¯h

ð13:16aÞ

3 D2 ðEÞ½1 2 f ðEÞ dE; where W1!2 is given by Eq. (13.13). We note that the integrand is proportional to the overlap between the two densities of states. By the same reasoning, the tunneling current in the reverse direction, J21, is J21 5

ð 2πe jHT j2 D2 ðEÞf ðEÞ ¯h

ð13:16bÞ

3 D1 ðE 2 eVÞ½1 2 f ðE 2 eVÞ dE; where again the integrand contains the density of states overlap. Inserting Eqs. (13.16a) and (13.16b) in Eq. (13.12) gives ð 2πeA I5 jHT j2 D1 ðE 2 eVÞD2 ðEÞ ¯h ð13:17Þ 3 ½ f ðE 2 eVÞ 2 f ðEÞ dE for the total tunneling current I in the direction from metal 1 to metal 2, where it is assumed that the tunneling matrix element Ht is independent of the energy near E 5 0.

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Superconductivity

In this expression the density-of-states functions D1(E 2 eV) and D2(E) depend on the nature of the source and target states, whether they are normal or superconducting. The distribution function difference [f(E 2 eV) 2 f(E)], on the other hand, depends only on the potential V, being close to 1 for energies between 0 and eV and approaching zero rapidly outside this range. Therefore, strong tunneling can occur only where D1(E 2 eV) and D2(E) are both appreciable in magnitude in this energy range. Weak tunneling could occur in the tails of the function ½ f ðE 2 eVÞ 2 f ðEÞ just beyond this range. These characteristics will be illustrated in the next three sections for NIN, NIS, and SIS tunneling, respectively.

D. NIN tunneling current Normal metal-to-normal metal tunneling depends mainly on the difference in the distribution functions ½ f ðE 2 eVÞ 2 f ðEÞ; since there is very little difference in the two normal metal densities of states. The analysis of this case is left as an exercise (see Problem 1).

E. NIS tunneling current For tunneling between a normal metal and a superconductor, the normal-metal density of states Dn(E 2 eV) can be approximated by Dn(0) and factored out of the integral (13.17). NIS tunneling will then occur when the superconducting density of states Ds(E) overlaps with ½ f ðE 2 eVÞ 2 f ðEÞ: At absolute zero [f(E 2 eV) 2 f(E)] is 1 in the range 0 # E # eV and zero outside this range, so that for a positive bias, V . 0, the integrand of (13.17) becomes Dn ðE 2 eVÞDs ðEÞ½ f ðE 2 eVÞ 2 f ðEÞ  2Dn ð0ÞDs ðEÞ Δ , E , eV 5 0 otherwise:

ð13:18Þ

A similar reasoning shows that for a negative bias, Dn ðE 2 eVÞDs ðEÞ½f ðE 2 eVÞ 2 f ðEÞ  2Dn ð0ÞDs ðEÞ 2eV , E , 2Δ 5 0 otherwise:

ð13:19Þ

Energy gap and tunneling

517

With the aid of Eqs. (13.7) and (13.9) we see that Eq. (13.17) can be integrated in closed form, as shown in Problem 4, to give 8 2 0 12 31=2 > > > Δ Δ > > ,V Gn 4V 2 2 @ A 5 > > > e e > > > > < 2Δ Δ ,V , ð13:20Þ Ins 5 0 e e > > > > 0 12 31=2 2 > > > > > Δ > 4 2 @ΔA 5 > V ,2 > : 2Gn V 2 e e where Gn, which has the value Gn 5


 2πA jHT j2 Dn ð0Þ; ¯h

ð13:21Þ

is the normal metal electron tunneling conductance defined by Gn 5

1n V

ð13:22Þ

and plotted as on asymptotic (dashed) straight line in Figs. 13.7g, 13.8g, and 13.9c. The NIS tunneling current, Eq. (13.20), is shown plotted in Fig. 13.7g. We see that as the voltage increases, the current approaches the normal conductor value. Experimentally determined plots of this type can be used to evaluate the energy gap Δ from NIS tunneling measurements. The expression in Eq. (13.20) is valid for absolute zero. For T . 0 the tails of the distribution function difference, ½ f ðE 2 eVÞ 2 f ðEÞ; produce weak tunneling for potentials jV j in the gap close to the value Δ/e. The significance of the overlap conditions in producing NIS tunneling is illustrated in Fig. 13.13. Figure 13.13a shows the lack of overlap when jV j{Δ/e, so that no tunneling occurs. Figure 13.13b indicates the small overlap when V is in the gap near the edges and there is weak tunneling. Finally, Fig. 13.13c shows the strong overlap for V . Δ/e, which produces strong tunneling. These figures should be compared with the more qualitative representations sketched in Fig. 13.7.

F.

SIS tunneling current

Superconductorsuperconductor tunneling is treated in a manner similar to the treatment we have just used for the NIS case. Unfortunately, in the SIS case

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Superconductivity

(a)

Ds (E)

Ds (E)

(c)

Ds (E)

(b)

Dn (0)

Dn (0)

Dn (0)

Δ

–Δ

0

–Δ

0 eV Δ E

–Δ

0

–Δ

0

E

Δ

–Δ

0

–Δ

0

E

Δ

f(E – eV ) – f(E)

f(E – eV ) – f(E)

f(E – eV ) – f(E)

E

Δ

E

eV

Δ

ev

E

Figure 13.13 Contribution of the occupancy (shaded) of the superconductor density of states Ds (E) to NIS tunneling for (a) small positive bias, 0 , V{Δ/e, and no tunneling current; (b) small bias, V 5 Δ/e, for which the tail of the distribution function difference [ f(E 2 eV) 2 f(E)] overlaps Ds(E), the occupancy of Ds(E) is small, and a weak tunneling current flows; and (c) more positive bias, V . Δ/e, producing a strong overlap so that the occupancy of Ds(E) is large near the gap and a strong tunneling current flows.

the tunneling current equation (13.17) cannot be integrated in closed form, and instead we present the more qualitative treatment that is outlined in Fig. 13.14. We select D1(E 2 eV) as a superconductor with a small gap Δ1 and D2(E) as a superconductor with a larger gap Δ2 . Δ1. As the bias voltage V is increased, the lower bands of D1 and D2 are made to coincide at a bias V 5 (Δ2 2 Δ1)/e, as indicated in Fig. 13.14b, and also in the semiconductor and boson condensation representation plots of Fig. 13.15. This coincidence and the overlap of the bands results in weak tunneling because of the very low concentration of electrons in each level, and a small peak appears in the current-versus-voltage plot of Fig. 13.16b. The current is less on either side of the peak because the amount of overlap of the bands is less. The decrease of I with increasing V beyond the peak at finite temperatures constitutes a negative resistance region of the I-versus-V characteristic of Fig. 13.16b. At absolute zero this current vanishes, as indicated in Fig. 13.16a, because the quasiparticle levels are all empty. We see from Figs. 13.13b and 13.14c that when the magnitude of the bias reaches the value V 5 (Δ1 1 Δ2)/e, the densities of states D1(E) and D2(E) begin to overlap at their infinity points, and there is a large jump in the tunneling current, as indicated in Fig. 13.16a for T 5 0 and in Fig. 13.16b for T . 0. The tunneling current is now large because it flows from a nearly full level to a nearly empty level. An evaluation of the integral (13.17) at T 5 0 and V 5 (Δ1 1 Δ2)/e gives for the jump in current as this bias

(a)

(b) D1(E)

519

D1(E)

Energy gap and tunneling

Dn

Dn Δ1

–Δ1 0

Δ1

D2(E)

D2(E)

–Δ1 0

Dn

Dn Δ2

–eV 0 f(E) – f(E + eV)

0

f(E) – f(E + eV)

–Δ2

0 V=0

–eV 0 Δ2–Δ1 V=– e

Figure 13.14 Densities of states for superconductor 1 (top) with the gap Δ1, for superconductor 2 (middle) with the gap Δ2, and distribution function difference (bottom) for (a) zero bias in which no tunneling current flows, (b) bias V 5 (Δ2 2 Δ1)/e producing weak tunneling current due to the overlap of the two quasiparticle bands, (c) bias V 5 (Δ2 1 Δ1)/e for onset of strong tunneling, and (d) bias Vc(Δ2 1 Δ1)/e producing strong tunneling current due to the large overlap between the occupied superconductor band of the first superconductor and the empty quasiparticle band of the second superconductor. Quasiparticle tunneling (b) arises from the tail of the distribution function difference f(E) 2 f(E 1 eV), hence is very weak and vanishes at absolute zero. Figure 13.15 presents energy-level diagrams for these four cases.

ΔIs 5

πGn ðΔ1 Δ2 Þ1=2 ; 2e

ð13:23Þ

where Gn is the normal tunneling conductance defined by Eq. (13.22). In Problem 5, we show that the ratio of the jump in current ΔIs to the normal tunneling current In at the bias V 5 2Δ/e is given by ΔIs =In 5

π ; 4

ð13:24Þ

which represents a jump of around 80%. Van Duzer and Turner (1981, p. 87) have shown that, for a finite temperature, this jump has the magnitude

520

Superconductivity

(c)

D1 (E)

D1 (E)

(d)

Dn Δ1

–eV

–Δ1 0 Δ1

D2 (E)

D2 (E)

–Δ1 0

Dn

0

f(E) – f(E + eV)

f(E) – f(E + eV)

–eV

–eV

0

V~ – Δ1 + Δ2 e

–eV

0 V<< – Δ1 + Δ2 e

Figure 13.14 (Continued)

ΔIs 5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi πGn Δ1 Δ2 4e 0

1 Δ 1 Δ 1 2A sinh@ 2kB T 1 0 1; 30 @cosh Δ1 A  @cosh Δ2 A 2kB T 2kB T

ð13:25Þ

which reduces to Eq. (13.23) for T 5 0. If the gaps are the same for the two superconductors, Δ1 5 Δ2 5 Δ, there will be no maximum in the weak quasiparticle tunneling current, but such a weak current does flow for V , 2Δ/e, as shown in Fig. 13.9. For T{Δ/kB, this current is given approximately by (Van Duzer and Turner, 1981, p. 87) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gn 2Δ ðΔ 1 eVÞ Is 5 2Δ 1 eV e 0 1 0 1 ; eV A @ eV A 2Δ=kB T 3 sinh@ K0 e 2kB T 2kB T

ð13:26Þ

Energy gap and tunneling

(a)

521

(b)

(c)

(d)

+ + + 2Δ2

2Δ1

V=0

V=

Δ2–Δ1 e

V=

Δ1+Δ2 e

V >>

Δ1 + Δ2 e

Figure 13.15 Semiconductor (top) and boson condensation (bottom) representations of the SIS tunneling cases of Fig. 13.14 for (a) zero bias, (b) quasiparticle band alignment and weak tunneling, (c) onset of Cooper-pair tunneling, and (d) strong tunneling of Cooper-pair electrons.

(a)

(b)

I

I

T=0

Δ2 – Δ1 e

Δ1 + Δ 2 e V

T>0

Δ2 – Δ1 e

Δ1 + Δ2 e V

Figure 13.16 Tunneling current versus bias voltage for SIS tunneling involving two superconductors with energy gaps Δ2 . Δ1 for (a) T 5 0, with no tunneling occurring until the bias V 5 (Δ2 1 Δ1)/e is reached, and (b) T . 0, with weak tunneling at the bias V 5 (Δ2 2 Δ1)/e and strong tunneling for V . (Δ1 1 Δ2)/e.

522

Superconductivity

where K0 is the zeroth-order modified Bessel function (cf. Chapter 9, Section III.B). In Eq. (13.22) we defined the normal metal electron tunneling conductance Gn as the asymptotic slope of the I-versus-V characteristic curve for very large V. We can also define the differential conductance, Gd 5

dI ; dV

ð13:27Þ

which is the slope at any point of the I-versus-V curve. Many workers report their tunneling measurements as plots of Gd versus V. This has the advantage of providing greater resolution, since structural features tend to be better resolved in plots of differential conductance than in plots of I versus V (see example in Section VI.C, especially Fig. 13.24).

G. Nonequilibrium quasiparticle tunneling So far we have assumed that the super electrons in the ground energy band and the quasiparticles in the excited band are in thermal equilibrium both between the bands and within each individual band. The tunneling process, of course, disturbs this equilibrium, but this effect is negligible. We now wish to treat so-called branch imbalance, in which the number of quasiparticles n1 with momentum in one direction, pi, is greater than the number of quasiparticles n2 with momentum in the opposite direction, 2 pi, in accordance with Fig. 13.17b. The imbalance [n1 2 n2] can be brought about by injecting quasiparticles across an NIS junction (Clarke, 1972). When a quasiparticle imbalance exists in the neighborhood of a tunnel junction, a current flows, I5e

d ½n1 2 n2 ; dt

ð13:28Þ

to reestablish balance between the positive and negative momentum states in the quasiparticle band. Equilibrium is restored in a time τ Q, called the branch imbalance relaxation time, and we can write ½n1 2 n2  5

Iτ Q : e

ð13:29Þ

For temperatures near Tc the relaxation time is predicted, assuming a spatially uniform case, to have the temperature dependence (Schmid, 1968)
 T 21=2 τ Q ðTÞ  τ Q ðTc Þ 12 : Tc

ð13:30Þ

Energy gap and tunneling

523

(a)

E/Δ 4 3 2

k (b)

E/Δ 4 3 2

k

Figure 13.17 Branch imbalance illustrated using a one-electron energy parabola (a) for the usual case of no imbalance where the number of electrons with positive momentum is equal to the number of electrons with negative momentum, and (b) for the branch-imbalance state (number of electrons with positive momentum greater than numbers of electrons with negative momentum).

This relation has also been found experimentally (Clarke and Paterson, 1974). A more extensive discussion of quasiparticle imbalance may be found in the works by Tinkham and Clarke (1972), and by Tilley and Tilley (1986).

H. Tunneling in unconventional superconductors a. Introduction Tunneling measurements have proved very useful in studying unconventional superconductors. Both Cooper-pair (Josephson) tunneling and single-electron (quasiparticle) tunneling measurements are used. In particular, the measurement of the tunneling current between a superconductor and a normal metal may be utilized for the direct determination of the superconducting gap. If a superconductor is unconventional then directional tunneling measurements help to clarify the superconducting gap structure. Furthermore, Josephson tunneling between two superconductors, where one of which is unconventional, can be used to probe the asymmetry of the superconducting phase, for example, by utilizing corner superconducting quantum interference device (SQUID) junctions. This is an example of so-called phasesensitive experiments, which are very important in distinguishing between a highly anisotropic s-wave gap and a d-wave gap. For the former case, the order

524

Superconductivity

parameter does not change sign anywhere on the Fermi surface, whereas in the latter case it does. In a simple tunneling experiment between a normal metal and a superconductor, the conductance is related to the density of states, and the bias voltage is related to the energy offset from the Fermi level. Since the quasiparticle energy spectrum is gapped with a superconducting gap Δ, an s-wave superconductor shows technically zero conductance up to V 5 Δ/e, where e is the electron charge. In unconventional superconductors with nodes the average over the Fermi surface of the quasiparticle energy spectrum is linearly proportional to the energy, and tunneling measurements do indeed reveal this structure. To refine the technique, we can use directional tunneling with small planar or point contacts. In this case, the experiments probe the gap structure along a particular direction with respect to crystallographic axes, and reveal the gap anisotropy. Many modifications of tunneling geometries and contacts are employed. Ramp junctions and single grain boundaries, as well as a variety of combinations of insulating layers, have been utilized during the last decade. As in any type of measurement, there are many complications in performing and interpreting the tunneling experiments. The main problems are the quality of the contacts, an incomplete knowledge of impurity distributions, distortions and stresses at the interface, and the uncertainty of the tunneling current distribution (so-called tunneling cone). As a result, there is still much controversy regarding the interpretation of various experiments. Many, sometimes conflicting, theories claim to describe the results. It will take some time and more experimental statistics on various superconducting systems for the situation to become clear.

b.

Zero-bias conductance peak

One of the signatures of a superconductor with nodes in tunneling measurements is the appearance of the zero-bias conductance peak arising from surface Andreev bound states (Hu, 1994). If quasiparticles scattered specularly off the interface can be Andreev-reflected by the pair potential then a hole will go back in the direction of the initial quasiparticle trajectory. Multiple scattering events at the interface lead to the formation of zero-energy (with respect to the Fermi level) Andreev bound states. Evidently, this interference with the pair potential is very sensitive to the phase of the superconducting wavefunction. In a dx22y2 superconductor the effect is maximal for a (110) orientation with respect to the surface. In this case, a quasiparticle reflected off the interface experiences a change in sign of the superconducting order parameter (maximum change of the pair potential), and the effect is maximal. As a result, the appearance of a large conductance via these available states at the gap center is possible, and this leads to the formation of the zero-bias conductance peak. The presence of Andreev bound states is not the only explanation for the experimentally observed conductance peak at zero bias. A trivial explanation is the presence of a mesoscopic superconducting short circuit, and before Andreev physics became established the dominant theory had been the AppelbaumAnderson model that invoked magnetic impurities at the interface.

Energy gap and tunneling

525

A single zero-bias conductance peak is formed only when time-reversal symmetry is preserved. Therefore, the application of a magnetic field splits the peak, as was observed experimentally. The problem is that a similar splitting occurs in the AppelbaumAnderson model as well. In that case, however, it seems that the required fields are much larger, and the field dependence of the splitting is quite different from that in Andreev bound-state physics. Impurities tend to smear the peak, which may be why this peak splitting had not been seen in some earlier experiments carried out with an otherwise “correct” geometry. Furthermore, in YBaCuO a spontaneous zero-bias conductance peak splitting has been observed below 8 K (Greene et al., 2002) This was interpreted as involving spontaneous time-reversal symmetry breaking and the creation of mesoscopic surface currents due to the bound states. Such a spontaneous change can be due to the crossover to a mixed symmetry state such as s 1 idxy or dxy 1 idx22y2.

c.

c-Axis tunneling

In a pure d-wave superconductor the order parameter has lobes of equal area, but opposite sign. Therefore, the total Cooper-pair tunneling current in the c-direction must be zero. However, if there is an s-wave admixture to the order parameter it must lead to the presence of a nonzero c-axis supercurrent. The experimental realization of this simple idea is not straightforward, and until now different groups have reported conflicting results. Various versions of this idea have been explored, including twisted junctions and grain boundaries.

VI.

Tunneling measurements

Let us now say a few words about the experimental arrangements used in carrying out tunneling experiments, followed by a discussion of representative experimental data that have appeared in the literature. Tunneling, like photoemission, represents a surface-probe sampling of a region of dimensions determined by the coherence length (Cucolo et al., 1991; Liu et al., 1991; Pierson and Vails 1992), which, for high-temperature superconductors, can be only one or two unit cell dimensions in magnitude. Lanping et al. (1989) constructed a histogram of the distribution of gap parameters Δ determined from tunneling measurements made at 600 different surface locations on the same YBa2Cu3O72δ sample, obtaining values ranging from 13 to 50 meV, with several data points outside this range.

A. Weak links If a superconducting rod is cut at some point and then joined through an intervening insulating section, either (1) the insulating region will turn out to be so thick that the two separated superconducting sections lose contact and have no interaction, or

526

Superconductivity

(2) the insulator layer will consist of a monolayer of foreign atoms, so that strong contact is maintained across it, or (3) the section will be intermediate in thickness, so that the superconductors are weakly coupled and electrons can tunnel. The third case, called a weak link, is the one that is most commonly used for tunneling studies and Josephson effect measurements (Furusaki and Tsukada, 1991).

B. Experimental arrangements for measuring tunneling The overall structure at the interface between two superconductors or between a normal metal and a superconductor is called a microbridge. The barrier region is the crucial part of the microbridge. A typical barrier thickness is a coherence length or less in magnitude, so that barriers must be much thinner for high-temperature superconductors (ξ  2 nm) than for an element like lead (ξ 5 80 nm). Beenakker and van Houten (1991) discussed weak links that are quantum point contacts. The original Zeller and Giaever technique (1969; Fulton et al., 1989; Giaever and Zeller, 1968) for making an AlAl2O3Al sandwich-type tunneling junction was to embed Sn particles in the aluminum oxide since Al oxidizes much faster than Sn, making it easier to form a thin insulating oxide layer on the tin. The preparation method shown in Fig. 13.18 consists of evaporation of a strip of aluminum film on a glass substrate, oxidizing the strip, evaporating tin on the film, and oxidizing once again. Then a second strip of aluminum at right angles to the first strip is evaporated on the substrate, as shown in the figure. An arrangement of the Sn particles for four different samples is shown in the electron micrographs of Fig. 13.19. The final asymmetric feed configuration arrangement is sketched in Fig. 13.20, with the details of the junction area indicated in Fig. 13.21 (Florjanczyk and Jaworski, 1989; Monaco, 1990a,b). To make a tunneling measurement, a bias

v

Glass slide with slightly oxidized AI films

Evaporation of Sn

Oxidation

Evaporation of AI top layer

Al Sn

Sn

Film Sn

Oxide Al

Film

Al

Film

Al

Film

Al

Film

Figure 13.18 Preparation of a tunnel junction containing tin particles. Aluminum oxidizes faster than tin, so that the oxide layer is thicker between the particles than on their surface, as indicated (Zeller and Giaever, 1969).

Energy gap and tunneling

527

5000 Å

Figure 13.19 Electron micrograph of tin particles on an oxidized aluminum film for four particle sizes (Zeller and Giaever, 1969). V

S I

S or N

V’

S

I

Figure 13.20 Sketch of tunnel junction on a glass substrate, showing current (I) and applied bias voltage (V, V0 ) leads.

voltage is applied across the junction using two of the leads, with the current monitored at the other two leads, as shown. Sandwich-tunnel junctions of this type have been made for many SIS and NIS cases using various combinations of superconducting and normal metals, but the use of embedded particles, such as Sn, to control the film thickness is not generally employed. Another common technique for tunneling measurements employs a scanning electron microscope (SEM). A probe ground to a point with a very small tip radius makes contact with the superconductor surface, as indicated in Fig. 13.22a (typical

528

Superconductivity

Applied bias voltage

Normal metal or superconductor

Insulator Ammeter

Superconductor

I

Figure 13.21 Details of the tunnel junction sketched in Fig. 13.20, showing the insulating layer between the metal strips. The ammeter measures the tunneling current produced by the bias voltage.

Probe

Tip Superconductor

(a)

SC

SC

Constriction (b)

Figure 13.22 Tunnel junction formed from (a) a probe tip in contact with, or almost in contact with, the superconducting surface, and (b) a constricted region in a superconductor.

probe materials are Au, Nb, PtRh, PtIr, and W). The tip touches the surface, or comes very close to it. If contact is made, tunneling probably takes place through a layer of inhomogeneous insulating or semiconducting material, such as the oxide coating the surface. Tunneling can also occur at a constriction, as shown in Fig. 13.22b.

Energy gap and tunneling

529

a.

20

Current (nA)

50

b.

1 20

10

c.

2

0 Voltage (mV)

Figure 13.23 Currentvoltage characteristics at different positions on the surface of aluminum-doped YBa2Cu3O6.51x determined by scanning tunneling microscopy using a Nb tip at 4.2 K. A jump in the current is observed at 95, 30, and 2.5 mV, respectively. Note the changes in scale for each curve (Gallagher et al., 1988).

Mooreland’s group developed what they call a break junction technique for tunneling measurements (Moreland et al., 1987a,b). A small piece of bulk material is electromechanically broken under liquid helium and the freshly fractured interfaces joined to form a tunneling barrier with the liquid helium acting as insulator. We will now present some typical experimental tunneling measurements that were made using these techniques.

C. NIS tunneling measurements Gallagher et al. (1988) made an NIS tunneling study of YBa2Cu3O7 using a scanning tunneling microscope operating in liquid helium. A coarse-adjust screw and fine-adjust piezoelectric transducer provided the desired tip-to-sample contact, where the tip is embedded in an insulating surface layer with a typical 1 MΩ resistance, which causes the junction to end up in the tunneling regime. Figure 13.23 shows the I-versus-V characteristic made with a Nb tip and Fig. 13.24 the dI/dVversus-V characteristic for a similar sample made using a W tip. Note the increased resolution of the differential curve. Ekino and Akimitsu (1989a,b) reported point-contact electron tunneling studies of BiSrCaCuO and BiSrCuO bulk, monocrystal, and sputtered film samples. Figure 13.25 presents the I versus V differential conductance plot for the 2:2:2:3 sample, and we again see that the differential data exhibit much more structure.

Superconductivity

dl/dV (arbitrary. units)

530

–8

0 Voltage (mV)

8

Figure 13.24 Recording of dI/dV obtained for YBa2Cu3O6.51x at 4.2 K using a tungsten tip (Gallagher et al., 1988).

T = 4.2 k

Bi-Sr-Ca-Cu-O

I

(a)

Bi-Sr-Ca-Cu-O T = 4.2 k 2Δ ≅ 92 meV (b) I2 (Arbitrary. units)

dI/dV (μS)

200

100

0 0

0 –150

–100

–50

0 Voltage (mV)

50

12 Δ

5

10 V2 (103 mV)

100

15

150

Figure 13.25 Currentvoltage and dI/dT characteristic curves of Bi2Sr2Ca2Cu3O10 determined by point-contact tunneling. The inset shows a plot of I2 versus V2 made from the I-versus-V curve (Ekino and Akimitsu, 1989a).

D. SIS tunneling measurements Figure 13.26 shows some experimental data on tunneling across the AlAl2O3Al junction formed from an oxide layer between two aluminum samples with Tc 5 1.25 K. We see from the figure that quasiparticle tunneling is negligible at T 5 0, becoming dominant just below T 5 Tc. The jump in current at V 5 2Δ/e and T 5 0 appears to be less than the expected 80%.

Energy gap and tunneling

531

AI-AI2O3-AI No. 102 Tc= 1.250 k 2Δ(o)= 379 μV

40 k

1.1

25 μamp

Current

2k k 25 241 8 k . 1 1. 2 k 2 1. 10 6 k 2 1. .18 1

1.077 k

0.998 k 0.890 k 0.756 k 0.331 k 0

100

200

300

400

500

Microvolts

Figure 13.26 Currentvoltage measurements on an AlAl2O3Al tunnel junction. Zero-current positions for each curve are staggered for clarity (Blackford and March, 1968).

E. Energy gap We saw in Sections IV.C and IV.D, respectively, that NIS tunneling occurs for biases with magnitudes greater than Δ/e, and that SIS tunneling occurs for biases exceeding 2Δ/e, as indicated in Figs. 13.7 and 13.8. The abrupt rise in current at these biases gives us the superconducting energy gap Δ. When the two superconductors that form an SIS junction have different gaps Δ1 and Δ2, a finite temperature tunneling measurement can give us the values of both gaps, as pointed out in Section V.F and indicated in Fig. 13.16. Thus a tunneling experiment provides a convenient way of measuring the energy gap. As an example of a gap determination, note that the peaks on the derivative NIS tunneling curve of Fig. 13.24 are separated by 5 meV, which gives 2Δ  5 meV. The inset of Fig. 13.25 shows the I2 versus V2 plot of Eq. (13.22), giving us the value of the energy gap Δ from the intercept at zero current. The temperature dependence of the energy gap, Δ(T), obtained by fitting the tunneling data to a broadened BCS density-of-states function,  DðEÞ 5 Re

E 2 iΓ ½ðE 2 iΓ Þ2 2Δ2 1=2

 ;

ð13:31Þ

where Γ is the gap broadening parameter, provided a good fit to the experimental data for two Bi superconductors, as shown in Fig. 13.27. However, the ratio 2Δ=kB Tc  10:5

532

Superconductivity

Bi-Sr-Ca-Cu-O

2Δ (meV)

120

80

40 BCS

0 0

20

40

60

80

100

T (K)

Figure 13.27 Temperature dependence of the energy gap 2Δ of Bi2Sr2CaCu2O8 (x) and Bi2Sr2Ca2Cu3O10 (’) obtained by point-contact tunneling. Some of the data points have vertical error bars. The solid lines are fits to a broadened BCS density of states (Ekino and Akimitsu, 1989a).

is about three times the BCS value of 3.53. Plots similar to Fig. 13.27 have been reported elsewhere (e.g., Ekino and Akimitsu, 1990; Escudero et al., 1989, 1990). Figure 13.26 shows SIS tunneling with the sharp rises occurring at the values 2Δ(T). We see from the figure that the gap 2Δ(T) decreases with temperature, as expected. The ratio 2Δ/kBTc 5 3.52 is almost precisely the BCS value. The ratio 2Δ(0)/kBTc has been reported in the range 210 for high-temperature superconductors (Mattis and Molina, 1991); the older superconductors usually had valves near the range 35 (Schlesinger et al., 1990), much closer to the BCS value of 3.54. Figure 13.28 shows the dependence of Δ(0) on Tc for several high-temperature superconductors, all with reported ratios 2Δ(0)/kBTc  6 (Takeuchi et al., 1989). The energy gaps of high-temperature superconductors are anisotropic (Bulaevskii and Zyskin, 1990; Mahan, 1989), being much larger in the a, b-plane than in the c-direction. Some reported values are 2Δab  6.2kBTc and 2Δc  2kBTc for (La12xSrx)2CuO4 (Kirtley, 1990a, 1990b), 2Δab  8kBTc and 2Δc  2.5kBTc for YBa2Cu3O72δ (Collins et al., 1989a,b,c), 2Δab  8kBTc for Bi2Sr2CaCu2O82δ, and 2Δab  8kBTc and 2Δc  4kBTc for Ba0.6K0.4BiO3 (Schlesinger et al., 1990; see also Kussmaul et al., 1990; Takada et al., 1989). The existence of these high anisotropies could account for much of the scatter in the reported gaps for the hightemperature superconductors.

F.

Proximity effect

We have been discussing the effect of an insulating layer between a normal metal and a superconductor or between two superconductors. If no intervening layer

Energy gap and tunneling

533

(2223)

Δ(O) (meV)

30

(2212) 20 TBCCO (bulk) YBCO (bulk) BSCCO (film// Cu–O) YBCO (film// Cu–O) with Tc(Δ=O)

10

0

50

100

150

Tc (K)

Figure 13.28 Correlation of the energy gap Δ(0) with the transition temperature Tc for bulk YBa2Cu3O7 (YBCO), two YBa2Cu3O7 films with tunneling in the CuO plane direction, Bi2Sr2CaCu2O8 (BSCCO) film with tunneling in the CuO plane direction, bulk Tl2Ba2CaCu2O8 (TBCCO, 2212), and bulk Tl2Ba2Ca2Cu3O10 (2223). The dashed line is drawn for the BCS slope 2Δ(0)/kBTc 5 3.53 (Takeuchi et al., 1989).

is present, another effect, called the proximity effect, comes into play. The direct contact at the junction and the overlap of wavefunctions causes the density ns of electron pairs to differ in the neighborhood of the surface from its value in the bulk. At an NS interface some electron pairs leak into the normal metal while some quasiparticles leak into the superconductor, thereby reducing the transition temperature of the superconductor. The proximity effect can cause two superconductors with different Tc that are in contact with each other to exhibit the same intermediate Tc. Theoretical treatments of this effect, such as the tunneling approach of McMillan (DiChiara et al., 1991, 1993; Kadin, 1990; McMillan, 1968; Maritato and Noce (1989); Stephan and Carbotte 1991) have been published. To elucidate this Tc reduction, an experimental study of composite films (Werthamer, 1963) was undertaken, each film consisting of a superconductor of thickness ds and a normal metal of thickness dn. Figure 13.29 shows a plot of the critical temperature T 0c of layered PbCu composite relative to Tc 5 7.2 K of bulk Pb versus the Cu layer thickness dn for various thickness ds of Pb. The reduction of the critical temperature is small for superconductor layer thicknesses greater than the coherence length ξ 5 80 nm irrespective of the normal layer thickness. The data show that there is a characteristic thickness Ln  40 nm of the normal layer beyond which (dn . Ln) there is no additional reduction of the transition temperature. Van Duzer and Turner (1981, p. 301) associate this effect with the diffusion constant D of the normal metal through the expression ¯ D=2πkB TÞ1=2 : Ln 5 ðh

ð13:32Þ

534

Superconductivity

ds 1.0

100 nm 50 nm

T’c/Tc

30 nm

0.5

15 nm

10 nm ds = 7 nm 0

20

40 60 dn (nm)

80

Figure 13.29 Proximity effect for a PbCu composite illustrating how the critical temperature T 0c of superconducting Pb in a copperlead composite relative to Tc of bulk lead depends on the Cu film thickness dn for several Pb film thickness ds from 7 to 100 nm. The vertical dashed line indicates the characteristic thickness Ln. Source: Adapted from Werthamer (1963).

The thinnest sample studied, ds 5 7 nm, went normal before this characteristic could be attained. Figure 13.30 shows how the limiting value of T 0c obtained from Fig. 13.29 for the condition dn . Ln depends on the superconducting layer thickness. Similar experiments have been carried out with layers of the superconductor YBa2Cu3O72δ containing Ny CuO layers (1/2Ny unit cells thick) adjacent to NPr CuO layers of the nonsuperconducting material PrBa2Cu3O72δ. The calculations by Wu et al. (1991), which are compared in Fig. 13.31 with the experimental data of Lowndes et al. (1990), provide results comparable with those presented in Fig. 13.29. Layered compounds are useful for studying other properties as well, such as resistivity (Minnhagen and Olsson 1992). Radousky (1992) reviewed the superconducting and normal-state properties of the Y12xPrxBa2Cu3O72δ system. Proximity junctions are (SNS) Josephson junctions in which the Cooper pair and quasiparticle transfer arises from the proximity effect (Agrait et al., 1992; Braginski, 1991; Claassen et al., (1991); Gijs et al., 1990; Han et al., 1990; Harris et al., 1991; Jung et al., 1990; Kleion and Aharony 1992; Yu et al., 1991). Studies have also been carried out on S-Semicond-S or S-Semicond junctions (Furusaki et al., 1991, 1992; Kastalsky et al., 1991; Kleiner et al., 1992; Van Wees et al., 1991) and arrays (Hebboul and Garland, 1991; Kwong et al., 1992; Lerch et al., 1990; Sohn et al., 1992).

Energy gap and tunneling

535

1.0 Tc’ Tc

sat

0.5

0

0

50 ds (nm)

100

Figure 13.30 Dependence of the limiting value of the relative transition temperature ðT 0c =Tc Þsat of the PbCu composite of Fig. 13.29 on the thickness ds of the superconducting component Pb. Source: Data from Fig. 13.29. (a)

(b) 1 8 6

0.8

8

16 N Pr

8

4

0.4

0.2

NY = 2

0

32

0.4

0.2

2

0.6

0.6

T ’c / Tc

T ’c / Tc

4

N

Pr =

4

0.8

1

24

32

0

6

8

10

NY

Figure 13.31 Normalized transition temperature T 0c =Tc of YBa2Cu3O72δ/PrBa2Cu3O72δ layers plotted (a) versus the number of CuO planes NPr in PrBa2Cu3O72δ, and (b) versus the number of CuO planes NY in YBa2Cu3O72δ. The calculated curves are drawn to fit the data points (Wu et al., 1991).

G. Evenodd electron effect Measurements of single-electron tunneling through a small superconducting island of volume 3 3 106 containing 6 3 108 conduction electrons exhibited a 2e periodicity in the tunneling current for T , 0.2Tc. Such a parity effect arises from the electron pairing, whereby the free energy of the superconducting island depends on whether there is an even or odd number of electrons in the island (Tuominen et al., 1993).

536

VII.

Superconductivity

Josephson effect

Until now we have been discussing the participation of quasiparticles in tunneling. The SIS processes that have concerned us included the strong tunneling current that flows between an occupied super electron band and an empty quasiparticle band (S!Q), as well as the relatively weak tunneling between two quasiparticle bands (Q!Q). There is also a third case—tunneling between two occupied super electron bands at zero bias (S!S). In this process, there is transfer of Cooper pairs across the junction through an effect predicted by Josephson (1962) and observed experimentally shortly thereafter (Anderson and Rowell, 1963). Figure 13.32 compares these processes. In the following treatment, we assume that the Josephson junction is of the weak-link type referred to in Section VI.A.

A. Cooper-pair tunneling

Current I

When two superconductors are separated by a thin layer of insulating material, electron pairs are able to tunnel through the insulator from one superconductor

Ic

S

S

n

n

Q

S

Q

Q π 2Δ 4 e

2Δ e

Voltage V

Figure 13.32 Currentvoltage characteristic curves for tunneling via the S!S, Q!Q, and S!Q processes. All three processes follow the linear n!n tunneling slope at high voltages, above 2Δ/e.

Energy gap and tunneling

537

to the other. There are four modes of pair tunneling: (1) the dc Josephson effect, or flow of a dc current J 5 J0 sinφ across the junction in the absence of an applied electric or magnetic field, where φ is a phase factor and J0 the maximum zero-voltage current; (2) the ac Josephson effect, relating to the flow of a sinusoidal current, J 5 J0 sin[φ 2 (4πeVt/h)], across a junction with an applied voltage V, where v 5 2eV/h is the frequency of oscillation; (3) the inverse ac Josephson effect, whereby dc voltages are induced across an unbiased junction by incident radiation or an impressed rf current; and (4) macroscopic quantum interference effects, involving a tunneling current J with an oscillatory dependence on the applied magnetic flux sin(πΦ/Φ0), where Φ0 is the quantum of magnetic flux.

B. dc Josephson effect In deriving the basic equations for the dc Josephson effect we follow the classic approach of Feynman (1965). Consider two superconductors, 1 and 2, separated by an insulating barrier, as shown in Fig. 13.11. If the barrier is thick enough so that the superconductors are isolated from each other, the time-dependent Schro¨dinger equation for each side is ih ¯

dΨ 1 5 H1 Ψ 1 ; dt

ð13:33aÞ

ih ¯

dΨ 2 5 H2 Ψ 2 ; dt

ð13:33bÞ

where ψi and Hi are the wavefunctions and Hamiltonians on either side of the barrier. We assume that a voltage V is applied between the two superconductors. If the zero of potential is assumed to occur in the middle of the barrier between the two superconductors, superconductor 1 will be at the potential 2 1/2V with Cooper-pair potential energy 1 eV, while superconductor 2 will be at the potential 1 1/2V with Cooper-pair potential energy 2 eV. (The factor of 1/2 does not appear in the potential energy terms because the charge of each Cooper pair is 2e.) The presence of the insulating barrier couples together the two equations, ih ¯

dΨ 1 5 eVΨ 1 1 KΨ 2 ; dt

ð13:34aÞ

ih ¯

dΨ 2 5 2eVΨ 2 1 KΨ 1 ; dt

ð13:34bÞ

where K is the coupling constant for the wavefunctions across the barrier. Since the square of each wavefunction is the probability density that super electrons are present, the two wavefunctions can be written in the form Ψ 1 5 ðns1 Þ1=2 eiΘ1 ;

ð13:35aÞ

538

Superconductivity

Ψ 2 5 ðns2 Þ1=2 eiΘ2 ;

ð13:35bÞ

Φ1 5 Θ2 2 Θ1 ;

ð13:35cÞ

where ns1 and ns2 are the densities of super electrons in the two superconductors and φ is the phase difference across the barrier. If the two wavefunctions (13.35a) and (13.34b) are substituted in the coupled wave equations (13.34a) and (13.34b) and the results separated into real and imaginary parts, we obtain equations for the time dependence of the pair densities and the phase difference: ¯h

d ns1 5 2Kðns1 ns2 Þ1=2 sin φ; dt

ð13:36aÞ

¯h

d ns2 5 22Kðns1 ns2 Þ1=2 sin φ; dt

ð13:36bÞ

d 2e φ 5 V: dt ¯h

ð13:37Þ

We can specify the current density in terms of the difference between Eqs. (13.36a) and (13.36b) times e J 5e

d ðns1 2 ns2 Þ; dt

ð13:38Þ

which has the value J 5 Jc sin φ;

ð13:39Þ

where Jc 5

4eKðns1 ns2 Þ1=2 ; ¯h

ð13:40Þ

and the coupling constant K is an unknown quantity. Equations (13.37) and (13.39) are called Josephson relations; they are the basic equations for the tunneling behavior of Cooper pairs. Multiplying Eq. (13.39) by the area A of the junction gives the current I 5 JA, I 5 Ic sin φ

ð13:41Þ

where Ic 5 JcA is the critical current. Ambegaokar and Baratoff (1963a,b) showed that for Cooper-pair tunneling between two identical superconductors with temperature-dependent gaps, Δ(T), the critical current is given by

Energy gap and tunneling

539

  1 πGn 2ΔðTÞ=e tanh ΔðTÞ=2kB T ; 4

Ic ðTÞ 5

ð13:42Þ

where the normal tunneling conductance Gn is given by Eq. (13.22). In the respective limiting cases T!0 and T!Tc, this is Ic ð0Þ 5

 1 πGn 2Δð0Þ=e T  0 4

Ic ðTc Þ 5

ð13:43aÞ

 1 πGn Δ2 ðTÞ=ekB Tc T  Tc : 4

ð13:43bÞ

The voltage 14πð2Δ=eÞ on the right side of Eq. (13.43a) has the physical significance indicated in Fig. 13.32. Thus the maximum Josephson current Ic(0), which occurs for T 5 0 and φ 5 π/2, is equal to 14π or  80% of the normal-state current at the gap voltage V 5 2Δ/e. Figure 13.33 compares the temperature dependence of the maximum zero-voltage tunneling currents of an SnISn junction measured by Fiske (1964) with the values predicted by Eq. (13.42). The fit of the PbISn Josephson junction data to the same theory is also shown. Copper-oxide superconductors are often granular in texture with Josephson junctions forming at the intergranular boundaries and perhaps at defect centers as well. Current flows through and between the Josephson junctions, and sometimes the intrajunction phases are favorable for the formation of complete circuits, which can produce flux shielding (Jung et al., 1990; see Doyle and Doyle, 1993). There is a grain-decoupling (or phase-locking) temperature Tg below which the Josephson junction network exhibits coherent properties, as well as a grain depairing (or critical) temperature Tc below which individual grains superconduct, where Tg , Tc (Sergeenkov and Ausloos, 1993).

1.0

Ic (T ) / Ic (o)

0.8 0.6 Pb–I–Sn

Sn–I–Sn

0.4 0.2

Solid curves: calculation of Ambegaokar and Baratoff (1963 a, b) 0

1

2

3

4

T °K

Figure 13.33 Temperature dependence of the maximum zero-voltage current, showing fit of theoretical curves to the experimental tunneling data of PbISn (&) and SnISn (x) junctions. The normalized tunneling current Ic(T)/Ic(0), is plotted (Fiske, 1964).

540

Superconductivity

C. ac Josephson effect We have been discussing the dc effect, whereby a phase difference φ 5 Θ2 2 Θ1 between either side of a superconductor junction causes a dc current to spontaneously flow at zero voltage. Now let us examine what happens when a dc voltage is applied across the junction. From Eq. (13.37), we know that a rate of change of phase accompanies the presence of a voltage across a Josephson junction. Since the applied voltage is a constant, this equation can be integrated directly to give φðtÞ 5 φ0 1


 2e Vt; ¯h

ð13:44Þ

which provides a characteristic frequency vJ, known as the Josephson frequency v1 5

2eV V 5 h Φ0

ð13:45Þ

5 483:6 3 10 V Hz; 12

where Θ0 is the quantum of flux (Tsai et al., 1983). A more practical expression to remember is vJ 5 483:6 MHz=μV: V

ð13:46Þ

With the aid of these expressions and Eq. (13.39), the critical current density can be written in the form J 5 Jc sinðωJ t 1 φ0 Þ;

ð13:47Þ

where ωJ 5 2πvJ. It can be shown that the critical current density Jc depends on the frequency in terms of the voltage and that it reaches a maximum when the applied voltage is equal to the gap voltage, V 5 2Δ/e. This voltage dependence of Jc(V), which was confirmed experimentally by Hamilton (1972), is sketched in Fig. 13.34. The ac Josephson effect that we have been describing occurs when current flows across a junction at the frequency given by Eq. (13.45) when a dc voltage V is applied across it. There is also an inverse ac Josephson effect, whereby a dc voltage is induced across the junction when an ac current is caused to flow through it, or when an electromagnetic field is incident on it. This will be discussed in Section VII.E.

Energy gap and tunneling

541

V = 2Δ/e 2 Ic(fj) Ic’(O) 1

0

1

2 hfj / 4Δ

3

Figure 13.34 Dependence of the critical current on the Josephson frequency, showing the peak at the applied dc voltage V 5 2Δ/e (Van Duzer and Turner, 1981, p. 144).

I

DC current source

Icsin φ

C

G

I

Figure 13.35 Josephson junction represented by the parallel circuit on the right consisting of a junction current source Ic sinφ, a capacitor C, and a conductance G. The circuit is driven by the dc current source I shown on the left.

D. Driven junctions The Josephson relations (13.41) and (13.37) I 5 Ic sin φ

ð13:48Þ

d 2e φ 5 V; dt ¯h

ð13:49Þ

apply to an idealized case in which all the current is carried by electron pairs. In the more general case, there can be other types of current flowing, such as displacement current, quasiparticle tunneling current, and perhaps conduction current, if the barrier is not a perfect insulator. It will be instructive to analyze the junction in terms of the equivalent circuit shown in Fig. 13.35, which contains the current source Ic sinφ of the junction, a capacitor to represent the displacement current, and

542

Superconductivity

a conductance to account for the quasiparticle tunneling and capacitor leakage currents. We assume that the dc current source I 5 I(V), shown on the left, drives the junction circuit. The differential equation for the current flow I in the equivalent circuit is I 5 Ic sin φ 1 GV 1 C

dV ; dt

ð13:50Þ

where G is assumed to be constant, although in a more general analysis it can be taken as voltage dependent. Equation (13.49) can be used to eliminate the voltage and write the circuit equation in terms of the phase φ(t): I5

¯hC d2 φ ¯hG dφ   1 Ic sin φ: 1 2e dt2 2e dt

ð13:51Þ

With the aid of the Josephson angular frequency ωc 5 (2e/h
)Vc for the voltage Vc, Vc 5 Ic =G;

ð13:52Þ

obtained from Eq. (13.45), and a new dimensionless variable Θ, Θ 5 ωc t;

ð13:53Þ

the circuit equation assumes a simplified form, I d2 φ dφ 1 sin φ; 5 βc 1 2 Ic dΘ dΘ

ð13:54Þ

where β c is the admittance ratio, β c 5 ωc C=G:

ð13:55Þ

The solution to this second-order differential equation exhibit complex time variations of the current. We will not try to interpret these time dependences, and instead we will find the average value of the voltage, from Eq. (13.49)

 dφ ¯h ; V 5 hVi 5 dt 2e

ð13:56Þ

for simple cases. We readily see from the form of Eq. (13.54) that when I # Ic there is a solution corresponding to Eq. (13.48), I 5 Ic sin φ

I # Ic ;

ð13:57Þ

Energy gap and tunneling

543

with all the time derivatives equal to zero. In other words, this is the zero-voltage solution. At the other extreme, when IcIc, the term Icsinφ becomes negligible, and we can use Eq. (13.49) to obtain the constant-voltage solution I 5 GV

IcIc ;

ð13:58Þ

where dV/dt 5 0. Hence, from Eq. (13.49) we have d2φ/dt2 5 0 in this limit. The situation is more complex for driving currents that are near the critical current Ic. The case in which C  0, so that β c{1, which corresponds to I dφ 1 sin φ 5 Ic dΘ

ð13:59Þ

can be solved analytically (see Problem 9), and has the solution V 5 0 for I , Ic ; "
 #1=2 I 2 V 5 Vc 21 Ic V5

I G

for IcIc :

ð13:60aÞ

for I . Ic ;

ð13:60bÞ

ð13:60cÞ

This is plotted in Fig. 13.36a. Pairs of arrows pointing in opposite directions mean that there is no hysteresis. Figure 13.37 shows how the voltage oscillates with the average values from Eqs. (13.60b) and (13.60c), respectively, indicated by points A and B in Fig. 13.36a (for further details, see Orlando and Delin, 1991, pp. 458ff; Van Duzer and Turner, 1981, pp. 170ff). When β cc1, the two solutions (13.57) and (13.58) apply with φ 5 π/2 so that sinφ 5 1. The I-versus-V characteristic plotted in Fig. 13.36c shows that there is a hysteresis in which V remains pinned at the value V 5 0 as the current is initially increased from zero until the critical current is reached, at which point the voltage jumps to the value V 5 Ic/G and thereafter follows the diagonal line upward. Subsequent reduction of the voltage follows the diagonal line into the origin. For immediate values of β c, the I-versus-V characteristic follows the behavior illustrated in Fig. 13.36b for the case β c 5 4. Again there is hysteresis, with the initial rise of the current to Ic and its return to the value Imin at V 5 0. Figure 13.38 shows how Imin depends on the value of β c. The solutions that we have been discussing were average values (13.50) of the voltage V involving a time average hdφ/dti of the derivative of the phase. The voltage itself oscillates in time, and Fig. 13.37 shows two examples of these oscillations.

544

Superconductivity

(a)

(b)

2Ic

Current I

Current I

Ic

B

βc << 1

2Ic

A

βc = 4

Ic

Imin

2Ic/G

Ic/G

Ic/G

2Ic/G

Voltage V

Voltage V (c)

Current I

2Ic

βc

Ic Imin

2Ic/G Ic/G Voltage V

Figure 13.36 Currentvoltage characteristics, I, versus V, for the Josephson junction circuit of Fig. 13.35 with: (a) negligible capacitance, β c{1; (b) appreciable capacitance, β c 5 4; and (c) dominating capacitance, β c!N, where β c 5 ωcC/G.

E. Inverse ac Josephson effect We have found that applying a dc voltage across a Josephson junction causes an ac current to flow. In the reverse ac Josephson experiment, dc voltages are induced across an unbiased junction by introducing an rf current into the junction. To explain this effect, we assume that the Josephson junction can be represented by the parallel equivalent circuit of Fig. 13.39. The circuit consists of the usual Josephson current Icsinφ in parallel with a conductance G. In addition, it has as inputs a dc source current I0 and an rf source current Is cos ωst, with the total source current I given by

Energy gap and tunneling

6.00

545

V (τ) / RI1

5.00

4.00 B

3.00

2.00 A 1.00

0 0

50.00

100.00

150.00

200.00

250.00

300.00

350.00

τ

400.00

Figure 13.37 Voltage oscillations across the Josephson junction of Fig. 13.35 for the negligible capacitance case β c{1, and small and large dc bias voltages as marked at points A and B, respectively, of Fig. 13.36a (Barone and Paterno, 1982, p. 128).

I Ic

min

1.0 0.8 0.6 0.4 0.2 0

1

10

102

103

βc

Figure 13.38 Dependence on β c of the minimum current Imin (indicated in Fig. 13.36b) in the circuit of Fig. 13.35 when the current is decreased from values above Ic (Van Duzer and Turner, 1981, p. 173).

546

Superconductivity

I IS

Io

Iscosωst

IG

Iosinφ

G

I DC current source

AC current source

Junction

Figure 13.39 Equivalent circuit of a Josephson junction represented by a junction current Ic sinφ in parallel with a conductance G 5 1/R irradiated with rf power. The junction is shown driven by a dc current source I0 in parallel with an rf current source Is cos ωst. I(mA)

Zero RF power

2.5 2.0 1.5

Nonzero RF power

1.0 0.5

0

0.1

0.2

0.3

0.4

V(mV)

Figure 13.40 Currentvoltage characteristics of a point-contact Josephson junction with applied rf power at 35 GHz, for the ac current source of Figure 13.39. The solid curves calculated for 100 K thermal noise fit the experimental data well. The dashed line is from calculations done without noise (Van Duzer and Turner, 1981, p. 184).

IðtÞ 5 I0 1 Is cos ωs t:

ð13:61Þ

I(t) inserted into Eq. (13.50) results in a nonlinear differential equation that is difficult to solve. A numerical solution provides the staircase I-versus-V characteristic presented in Fig. 13.40. Measurements carried out by Taur et al. (1974) with a 35-GHz source satisfying the condition (ωs 5 0.16) ωc compare well with the calculated curves shown in the figure, where the zero rf power curve (Is 5 0) is shown

Energy gap and tunneling

547

I Is

IR

Vs cos ωst Ic sin φ

R

Vo I

Figure 13.41 Josephson junction of Fig. 13.39 driven by a dc voltage V0 in series with an rf voltage Vs cos ωst.

for comparison. This staircase pattern, which is referred to as Shapiro steps (Eikmans and van Himbergen, 1991; Shapiro, 1963; Yu et al., 1992), has been reported by many observers (e.g., Kriza et al., 1991; Kvale and Hebboul, 1991; Larsen et al., 1991; Lee et al., 1991; Rzchowski et al., 1991; Sohn et al., 1991). It is mathematically easier to analyze this problem in terms of the circuit of Fig. 13.41 where the source is an applied voltage, VðtÞ 5 V0 1 Vs cos ωs t:

ð13:62Þ

It is shown in Van Duzer and Turner (1981; see also Orlando and Delin, 1991) that the current I(t) through the Josephson junction can be written as an infinite series of products of Bessel functions Jn and sine waves, IðTÞ 5 Ic

P

0

1 2eV s n @ A n ð21Þ Jn ¯hωs

ð13:63Þ

3 sin½ðωJ 2 nωs Þt 1 φ0 ; where φ0 is a constant of integration. Since the I-versus-V characteristic is drawn for the average current, I  hI(t)i, and since the sine term averages to zero unless ωJ 5 nωs, there are spikes appearing on this characteristic for voltages equal to V5

nh ¯ ωs ; 2e

ð13:64Þ

with the maximum amplitude,

Imax 5 Jc Jn


 2eVs ; ¯hωs

ð13:65Þ

548

Superconductivity

occurring for the phase φ0 5 π/2. Figure 13.42 shows these spikes at intervals proportional to the source frequency and indicates their maximum amplitude range. The current can be anywhere along a particular spike, depending on the initial phase (see Orlando and Delin, 1991). Este`ve et al. (1987) reported that an LaSrCuO sample with the current voltage characteristic shown in Fig. 13.43 exhibited the IV characteristic of Fig. 13.43b when irradiated with x-band (9.4 GHz) microwaves. The microwaves



Vo / (hωs / 2e)

Figure 13.42 IV characteristic for the dc component of the current versus applied dc voltage V0 of the equivalent circuit of Fig. 13.41. The value of the current can be anywhere along a particular current spike, depending on the initial phase. The dashed line is for hI(t)i 5 V0G.

d 20 μV

f = 9.4 GHz c

b 1μA f a

e

Vs= 19 μV

(a)

(b)

Figure 13.43 Oscilloscope presentation of current-versus-voltage characteristics of a tunnel junction at 4.2 K formed by an Al tip on a (La0.925Sr0.075)2CuO4 sample (Este`ve et al., 1987). (a) Trace obtained in the absence of rf power, the letters af giving the sense of the trace and the dashed lines indicating switching between branches; (b) steps induced by incident microwave radiation at 9.4 GHz (Este`ve et al., 1987).

Energy gap and tunneling

549

produced the spike-step pattern we have already described. The researchers attributed the results to the beating of the oscillating Josephson supercurrent with the microwaves. The separation in voltage between these steps is proportional to the microwave frequency, and their amplitude is Bessel-like. The Josephson junction characteristics were observed even when the point-contact metal tip was itself superconducting, which indicates that the junction was inside the material underneath the tip.

F.

Analogues of Josephson junctions

Josephson tunneling involves a quantum phenomenon that is difficult to grasp intuitively. This is especially true when we try to picture how the total current flowing through a Josephson junction depends on the phase difference of the electron pairs on either side of the junction. The differential equation for this phase difference φ happens to be the same as the differential equation for the rotational motion of a driven pendulum. We will describe this motion and then relate it to the Josephson junction. Consider a simple pendulum consisting of a mass M attached to a pivot by a massless rod of length R. If a constant torque τ is applied by a motor, it will move the mass through an angle φ, as shown in Fig. 13.44. We know from our study of

Applied torque τ

∅ R

m mgR sin∅ restoring torque

Figure 13.44 Pendulum model of a Josephson junction showing the counterclockwise restoring torque mgR sinφ arising from the presence of a clockwise applied torque τ.

550

Superconductivity

(a)

τ

(b)

τc

(c)

∅ ∅

1

τ=0 ∅=0

τ = 2 mgR ∅ = 30°

τ = mgR = τc ∅ = 90°

Figure 13.45 Pendulum (a) with no applied torque, τ 5 0, (b) with the torque τ 5 1/2 mgR, and (c) with the critical torque applied, τ c 5 mgR.

mechanics that the force of gravity acting on the mass m produces a restoring torque mgR sinφ. For a relatively small applied torque the pendulum assumes an equilibrium position at the angle given by τ 5 mgR sin φ


 dφ 50 ; dt

ð13:66Þ

as indicated in Fig. 13.45b. The greater the torque, the larger the angle φ. There is a critical torque τ c indicated in Fig. 13.45c for the angle φ 5 π/2, τ c 5 mgR:

ð13:67Þ

If the applied torque exceeds this critical value, the pendulum will continue its motion beyond the angle φ 5 π/2 and rotate continuously as long as the applied torque τ . τ c operates. The motion is fast at the bottom and slow at the top, corresponding to a large angular velocity ω 5 dφ/dt at the bottom and a small ω at the top. For a large torque, τcmgR, the average angular velocity of the motion hωi increases linearly with the torque, reaching a limit determined by retarding drag forces coming from, for example, the viscosity η of the air or mechanical friction. The drag force is assumed to be proportional to the angular velocity ω, and is written as ηω. The dependence of the average angular velocity on the applied torque is shown in Fig. 13.46. We see from the figure that ω remains zero as the torque τ is increased until the critical value τ c 5 mgR of Eq. (13.67) is reached. Beyond this point, ω jumps to a finite value and continues to rise in the way we have already described. If the torque is now decreased down from a large

Energy gap and tunneling

551

2τc

τc τc′

<ω>

Figure 13.46 Relationship between the average angular velocity of the pendulum hωi and the applied torque τ. For low applied torques the pendulum oscillates and the average velocity is zero, whereas at high torques, τ . τ c, motion is continuous with hωi proportional to τ. Note the hysteresis for increasing and decreasing torques.

magnitude, once it passes the critical value of Eq. (13.67), the pendulum will have sufficient kinetic energy to keep it rotating for torques below τ c, as indicated in the figure. The torque must be reduced much further, down to the value τ 0c ; before friction begins to dominate and motion stops, as indicated in the figure. Thus we have hysteresis of motion for low applied torques, and no hysteresis for high torques. If we compare Fig. 13.46 with Fig. 13.36b, we see that the torqueangular velocity characteristic curve of the driven pendulum has the same shape as the currentvoltage characteristic of the Josephson junction. The correspondence between the driven pendulum and a Josephson junction can be demonstrated by writing down a differential equation that governs the motion of the pendulum, setting the applied torque τ equal to the rate of change of the angular momentum L, d dω L 5 mR2 ; dt dt

ð13:68Þ

and then adding the restoring and damping torques, τ 5 mR2

d2 φ dφ 1 mgR sin φ; 1η dt2 dt

ð13:69Þ

552

Superconductivity

where mR2 is the moment of inertia; here we have made use of the expression ω 5 dφ/dt. This equation is mathematically equivalent to its Josephson counterpart (13.51), so we can make the following identifications: Applied current Average voltage term Phase difference Capacitance term Conductance term Critical current

2 2 2 2 2 2

I V(2e/h
) 5 dφ/dt φ h
C/2e h
G/2e Ic

τ ω 5 dφ/dt φ mR2 η mgR

Applied torque Average angular velocity Angular displacement Moment of inertia Viscosity Critical torque

This analogue has been found useful in the study of the behavior of Josephson junctions. Another mechanical device that illustrates Josephson junction-type behavior is the washboard analogue sketched in Fig. 13.47, in which a particle of mass m moves down a sloped sinusoidal path in a viscous fluid, passing through regularly spaced minima and maxima along the way. Electrical analogues have been proposed (Bak and Pedersen, 1973; Hamilton, 1972; Hu and Tinkham, 1989; cf. Goodrich and Srivastava, 1992; Goodrich et al., 1991) that do not give as much insight into Josephson junction behavior as the mechanical analogues, however, they are useful for studying the behavior of Josephson junctions when the parameters are varied.

Max

Viscous fluid

Potential energy

m Max V

Min 2π Displacement

Figure 13.47 Washboard analogue of the Josephson junction showing a particle of mass m descending along a sloped wavy path in a viscous fluid.

Energy gap and tunneling

553

VIII. Magnetic field and size effects Until now we have assumed that no magnetic fields are applied, and that the currents circulating in the Josephson junctions produce a negligible amount of magnetic flux. The next few sections examine the effect of applying a magnetic field parallel to the plane of a single Josephson junction as well as perpendicular to a loop containing two such junctions. To determine how the presence of the field affects the phase Θ, we make use of Eq. (6.33): I rΘ  dI 5

2π Φ0

I A  dI:

ð13:70Þ

This shows that rΘ and the vector potential A play similar roles in determining the phase. In writing out this expression, we have assumed that the line integration is performed over regions of the superconductor where the current density is either zero or makes no contribution to the integral, so that the term J  dI of Eq. (6.33) is omitted. This expression will be applied to several cases. We begin with a discussion of a short Josephson junction in which the magnetic fields produced by the currents are negligible compared with the externally applied field, and then will treat long junctions where this is not the case. We first examine two-junction loops and arrays of many junctions, followed by ultrasmall junctions in which single-electron tunneling is observable. We will conclude with a brief section on SQUIDS.

A. Short Josephson junction Consider a weak-link tunnel junction of the type sketched in Fig. 13.11 with a magnetic field B0 k^ applied along the vertical z direction, as shown in Fig. 13.48. z Bz(y) a

Ax(y)

JTr y c

SC2

SC1

d x L

Figure 13.48 Application of a magnetic field B0 (y) transverse to the Josephson junction of Fig. 13.11 with a transport current of density JTr flowing to the left. The vector potential Ax(y) of the applied field is indicated.

554

Superconductivity

The junction is of thickness d normal to the y-axis with cross-sectional dimensions a and c along x and z, respectively. It is small enough so that the applied magnetic field is larger than the field produced by the currents. One superconductor SC1 is to the right of the insulating barrier and the other SC2 to the left, as indicated in the figure. Because of symmetry, the magnetic field Bz(y) has no x or z dependence, but does vary with distance along y into the superconductors, ^ B 5 Bz ðyÞk:

ð13:71Þ

This applied field is derived from the vector potential B 5 r 3 A, A 5 Ax ðyÞ^i;

ð13:72Þ

which has the value (cf. Eq. (6.41))   1 A 5 2 yB0^i y # d; 2

ð13:73Þ

in the barrier layer where the material is normal and Bz 5 B0, as sketched in Fig. 13.49. We assume that the magnetic field decays exponentially into the

A1∞ Ax(y) SC1

SC2

y

–A2∞ (a)

B0 Bz(y)

0

y

(b)

Figure 13.49 Variation of (a) vector potential Ax(y) and (b) magnetic field Bz(y) in the neighborhood of the junction of Fig. 13.48.

Energy gap and tunneling

555

superconductors on either side of the barrier, as indicated in Fig. 13.49b. If we proceed far enough inside where Bz drops to zero A will become constant, as seen from the expression B 5 r 3 A. We assign it the value A1N far inside SC1 and the value 2 A2N far inside SC2, as indicated in Fig. 13.49a. We start by calculating the value for the phase Θ1(x) at an arbitrary point x along the interface between the barrier and the superconductor SC1, as shown in Fig. 13.50. This phase will be found relative to the phase Θ10 of Θ1 (x) at a reference point x0 on the interface, as indicated. The phase difference Θ1(x) 2 Θ10 may be determined by integrating rΘ  dl along the path A!B!C!D in Fig. 13.50, but it is easier to make use of Eq. (13.70) and carry out the equivalent integration of A  dl along this same path. Since A is a vector in the x direction, it is perpendicular to the vertical paths A!B and C!D, so that the line integral vanishes for y B

A

A1∞ (x)

C

A dl = 0

A1θ10

D θ1 (x)

SC1 Barrier

x0 A′ θ20

A dl = 0

x D′ θ2 (x)

SC2

A2∞ (x) B′

C′

a

Figure 13.50 Path of integration around the junction of Fig. 13.48 for determining the phase difference φ(x) 5 Θ2(x) 2 Θ1(x) across the junction at a position x relative to the phase difference φ0 5 Θ20 2 Θ10 at the position x0 on the left.

556

Superconductivity

these two segments of the path. We already mentioned that the vector potential has the constant value A1N along the path B!C, so that integration gives Θ1 ðxÞ 5 Θ10 1 ð2π=Φ0 ÞA1N ðx 2 x0 Þ:

ð13:74Þ

An analogous expression can be obtained for the other superconductor SC2 using the path A0 !B0 !C0 !D0 . Hence we can write for the phase difference φ(x) φðxÞ 5 Θ2 ðxÞ 2 Θ1 ðxÞ 5 φ0 1

2π ½A1N 1 A2N x; Φ0

ð13:75Þ ð13:76Þ

where φ0 5 Θ2(x0) 2 Θ1(x0) 5 Θ20 2 Θ10 at the reference point x0. The quantity [A1N 1 A2N] is evaluated by integrating along a similar closed path that has been enlarged to enclose the entire barrier and extend deep into both superconductors. The resulting line integral equals the total flux through the barrier, ð Φ5

A  ds

ð13:77Þ

Again, the integrand vanishes for the two vertical paths along which the vectors A and ds are antiparallel. The contributions from the top and bottom paths far inside the two superconductors add to give for the total enclosed flux Φ 5 a½A1N 1 A2N :

ð13:78Þ

This total flux is approximately equal to the applied magnetic field strength B0 times the effective area of the junction, Φ 5 aðd 1 2λÞB0 :

ð13:79Þ

The quantity d 1 2λ constitutes the effective thickness of the junction, deff 5 d 1 2λ:

ð13:80Þ

Inserting Eq. (13.78) in (13.76) gives φðxÞ 5 φ0 1

2πΦ x  : Φ0 a

ð13:81Þ

If this is substituted in Eq. (13.39) and integrated over the area A 5 ac of the junction (see Problem 12), ð I 5 Jc sin½φðxÞdx dz;

ð13:82Þ

Energy gap and tunneling

557

Imax/Ic

0

1

2

3

4

5

Φ Φο

Figure 13.51 Josephson Fraunhofer diffraction pattern showing the maximum normalized zero-voltage current Imax/Ic versus Φ/Φ0 through the parallel junction of Fig. 13.48 when the current density is uniform across the x, z-plane of the junction. The values of Imax/Ic at the peaks of the curve, from the center outward, are 1, 2/3π, 2/5π, 2/7π, . . . Source: From Van Duzer and Turner (1981), p. 135.

we obtain I 5 Ic sinφ0

sinðπΦ=Φ0 ; πΦ=Φ0

ð13:83Þ

where Ic 5 AJc is the critical current. This has a maximum for the phase difference φ0 5 π/2, Imax 5 Ic

sinðπΦ=Φ0 ; πΦ=Φ0

ð13:84Þ

We call this the Josephson junction diffraction equation. The plot of Eq. (13.84) sketched in Fig. 13.51, which has been seen for many samples (e.g., Rosenthal et al., 1991; Seidel et al., 1991), illustrates how the tunneling current varies with increasing magnetic flux Φ through the junction. Figure 13.52 presents four special cases. When there is no flux, Φ 5 0, the current in the junction is uniform, as shown in Fig. 13.52a, and has the critical value Ic. When half a flux quantum is present, Φ 5 1/2Φ0, as in Fig. 13.52b, the average value of the current is the average of a sine wave over a half-cycle, namely (2/π)Ic. For the next maximum, Φ 5 3Φ0/2, two of the half-cycles cancel to give the current I 5 (2/3π)Ic, which is one-third of the half-cycle case. By induction, the nth maximum of the current Ic/[π(n 1 1/2)] occurs at the flux value Φ 5 (n 1 1/2)Φ0. We also deduce from Fig. 13.52c that the current cancels for even cycles, where Φ 5 nΦ0. For the case of Fig. 13.52c, in which the total phase change across the length of the junction is 2π, one flux quantum fits in it. We see from the directions of

558

Superconductivity

(a)

Magnetic field = 0 (c)

Magnetic field = Ho

(b)

Magnetic field = Ho/2 (d)

Magnetic field = 3Ho/2

Figure 13.52 Effect of an applied magnetic field on the tunneling current oscillations across a uniform Josephson junction, where the field μ0H0 corresponds to one flux quantum Φ0 in the junction (Langenberg et al., 1966).

the arrows on the figure that the supercurrent flows down across the junction on the left and up on the right. To complete the circuit it flows horizontally within a penetration depth λ inside the superconductor to form closed loops, as illustrated in Fig. 13.53a. These current loops encircle flux, and the resulting configuration is known as a Josephson vortex (Miller et al., 1985). There is no core because none is needed; the supercurrent density is already zero in the center. When the phase change across the junction is 2πn, where n is an integer, there will be n Josephson vortices side by side in the junction, each containing one flux quantum and each having a horizontal length 1/n times the length for the singleflux quantum case. Figure 13.53b sketches two fluxons in the gap for n 5 2, with total phase change of 4π. Equation (13.83) is mathematically equivalent to the well-known expression for single-slit Fraunhofer diffraction in optics. Peterson and Ekin (1989, 1990; cf. Barone and Paterno, 1982) suggest that an Airy diffraction pattern, in which the quantity sin(πΦ/Φ0) in Eq. (13.83) is replaced by twice a first-order Bessel function, 2J1(πΦ/Φ0), more properly characterizes superconductors with grain-boundary barriers in bulk materials. They also give a figure in which the Airy and Fraunhofer diffraction patterns are compared.

Energy gap and tunneling

559

(a)

λ d

d λ

π 2

0

π

3π 2

2π

Phase (b)

λ d

d

λ

0

π

2π

3π

4π

Phase

Figure 13.53 Current distribution around Josephson vortices in the junction of Fig. 13.52 for (a) single-vortex case of Fig. 13.52c when the magnetic flux in the junction is Φ0, and (b) the double-vortex case when the magnetic flux is 2Φ0.

B. Long Josephson junction We will now examine what are called long Josephson junctions, leaving discussion of the criterion for longness until the next section, VIII.C. We begin by taking the derivative of Eq. (13.81), dφ 2πΦ 5 dx Φ0 a

ð13:85Þ

560

Superconductivity

5

2πB d: Φ0

ð13:86Þ

Although we will not prove it, this expression is more general than (13.81), which may be obtained from it by assuming that Φ arises from a constant magnetic field and then integrating. If we make use of the Maxwell relation r 3 B 5 μ0J, which for the present case is given explicitly by dBz ðxÞ 5 μ0 Jy ðxÞ; dx

ð13:87Þ

we obtain d2 φ 2πμ0 Jy ðxÞd 5 : dx2 Φ0

ð13:88Þ

Using Eq. (13.39), this becomes the pendulum equation (which might also be called the stationary sine Gordon equation; Fehrenbacher et al., 1992). d2 φ sin φðxÞ 5 ; dx2 λ2J

ð13:89Þ

where λJ 5 (φ0/2πμ0Jcd)1/2 called the Josephson penetration depth, is the natural length scale for the junction. A long junction is one whose length a is greater than λJ, while for a short junction a{λJ. If the time dependence is taken into account, it can be shown that the sine Gordon equation is obtained (Orlando and Delin, 1991, p. 437), d2 φ 1 d2 φ sin φðxÞ 2  5 ; dx2 u2p dt2 λ2J

ð13:90Þ

where up, given by up 5

1 d ;  ðμ0 AÞ1=2 d 1 aλ

ð13:91Þ

is the velocity of a transverse electromagnetic mode (TEM) wave in the junction region. This equation has two types of solitary wave (soliton) solutions. The first, called kinks or topological solitons, are able to propagate and have the property that φ(x) increases monotonically from 0 to 2π as x increases from 2 N to N. There are also propagating antikink solutions for which φ(x) decreases monotonically from 2π to 0 as x increases from 2 N to N. Kinks represent magnetic flux quanta Φ0 in superconductivity (Holst et al., 1990; Kivshar and Soboleva, 1990), and

Energy gap and tunneling

561

domain walls in the theory of two-dimensional magnetism. The second type of solution, called a breather, is a nontopological variety of soliton that is stationary (i.e., does not travel; Dodd et al., 1982; Drazin and Johnson, 1989; Kivshar et al., 1991). Sometimes, perturbation terms for dissipation and energy (current) input are added to the sine Gordon equation (Grønbech-Jensen et al., 1991; Holst et al., 1990; Malomed, 1989, 1990; Malomed and Nepomnyashchy, 1992; Olsen and Samuelsen, 1991; Pagano et al., 1991; Petras and Nordman, 1989; Ustinov et al., 1992). Phase locking can also occur, in which the fluxon motion in the long junction follows the frequency of the external field, or two such junctions can be phase locked to each other (Fernandez et al., 1990; Grønbech-Jensen, 1992; GronbechJensen et al., (1990); Pedersen and Davidson, 1990). Frequency locking to the external field produces an ordered state and can lead to the appearance of Shapiro steps. The absence of phase locking can produce a disordered state and a condition of chaos (Chi and Vanneste, 1990).

C. Josephson penetration depth Equation (13.89) was obtained from Eq. (13.88) by defining the Josephson penetration depth λJ, λJ 5 ðΦ0 =2πμ0 Jc deff Þ1=2 ;

ð13:92Þ

which is the length criterion that distinguishes short from long junctions. To obtain a physical significance for this characteristic length, let us compare the energies associated with the stored fields and with the current flow through the junction, which is sketched in Fig. 13.48. For a constant magnetic field, the stored magnetic energy UB is UB 5

Ð B2 dx dy dz 2μ0

ð13:93Þ

5 ðB20 =2μ0 Þacdeff 5

Φ20 c  ; 2μ0 adeff

ð13:94Þ

where we have assumed one Josephson vortex present in the junction, as in Figs. 13.52c and 13.53a, with B0 5 Φ0/ad. The energy UJ associated with the current flow is ð UJ 5

JV dx dy dt;

and, using Eqs. (13.39) and (13.49), this becomes

ð13:95Þ

562

Superconductivity

UJ 

ð Φ0 Jc ac sin φ dφ: 2π

ð13:96Þ

If we equate the magnetic and current energies, UB 5 UJ, we obtain

1=2 a 5 λJ 2π = sin φ dφ ; ð

ð13:97Þ

2

where the factor in the square brackets is close to but larger than unity. Thus the two energies become comparable when the junction length a approaches the Josephson penetration depth λJ. A short junction is one for which a{λJ, UJ{UB, the magnetic fields arising from the current flow are much less than the applied field, and the field B is effectively constant over the junction region. A long junction is one for which a . λJ, UJ . UB, and so on.

D. Two-junction loop In Section A, we derived the diffraction equation (13.84) for a short Josephson junction in the presence of an applied magnetic field. In a typical case, applied fields in the millitesla range (see Problem 13) are used to see the current pattern. We will now consider the case of a superconducting loop containing two weak links (short junctions) in parallel, as shown in Fig. 13.54. For this arrangement flux quantization occurs in the area of the loop, which we are assuming to be considerably larger than the area of either junction, so that the system is sensitive to much smaller changes in applied flux.

B0 Iα θα2

I

θα1

SC1

SC2 d θβ2

Iβ

I

θβ1

d

Figure 13.54 Superconducting loop containing two weak links α and β of thickness d showing the phases Θij at the junctions and the direction of current flow I. The dashed line indicates the path of integration around the loop.

Energy gap and tunneling

563

In analyzing the two-junction loop we are assuming that the individual junction areas are small enough so as to be negligible. Integration of Eq. (13.70) around the dashed path shown in Fig. 13.54 gives


 2πΦ ðΘα2 2 Θα1 Þ 2 ðΘβ2 2 Θβ1 Þ 5 ; Φ0

ð13:98Þ

where again we are neglecting current flow effects on the phases. Using the phase difference notation of Eq. (13.75), this becomes φα 5 φβ 1 2π Φ=Φ0 :

ð13:99Þ

The total current I flowing through this parallel arrangement of weak links is the sum of the individual currents Iα and Iβ in the two arms, I 5 Iα 1 Iβ

ð13:100Þ

and each current satisfies its own individual Josephson equation (13.40), to give I 5 Icα sin φα 1 Icβ sin φβ

ð13:101Þ


Φ 5 Icα sin φα 1 Icβ sin φα 2 2π Φ0

 ;

ð13:102Þ

where we have used Eq. (13.99). For equal individual currents, Icα 5 Icβ 5 Ic ;

ð13:103Þ

the total current I is maximized by the choice of phase φα 5

1 π 1 πΦ=Φ0 ; 2

ð13:104aÞ

φβ 5

1 π 2 πΦ=Φ0 ; 2

ð13:104bÞ

to give for the magnitude of the maximum current   Imax 5 2Ic cosðπΦ=Φ0 Þ;

ð13:105Þ

an expression that we call the Josephson loop interference equation. It has its optical analogue in Young’s experiment for detecting the interference of light from two identical slits. The phase φα is an unknown function of the flux in the ring, and adjusts itself to maximize the current. The dependence of Imax on the applied flux Φ given by Eq. (13.105) for this equal-current case is plotted in Fig. 13.55.

564

Superconductivity

3Ic Imax 2Ic

Ic

–2

–1

0

1

2 φ/φ0

Figure 13.55 Dependence of the current maximum Imax of Eq. 13.105 of a balanced Josephson junction loop (Icα 5 Icβ) on the applied flux Φ normalized relative to the quantum of flux Φ0.

When the two currents are not the same (Saito and Oshiyama, 1991), it is more complicated to calculate the phase that maximizes the total current, Eq. (13.101), subject to the condition in Eq. (13.99). Van Duzer and Turner (1981) give
 1=2 2 2 πΦ Imax 5 ðIcα 2Icβ Þ 14Icα Icβ cos : Φ0

ð13:106Þ

This has the minimum and maximum values minimum 5 Icα 2 Icβ

1 Φ 5 ðn 1 ÞΦ0 ; 2

ð13:107aÞ

maximum 5 Icα 1 Icβ

Φ 5 nΦ0 ;

ð13:107bÞ

where we have assumed that Icα . Icβ. The dependence of Imax on the applied flux given by Eq. (13.106) for the case Icα 5 2Icβ is plotted in Fig. 13.56 to the same scale as in Fig. 13.55 with the ordinate scale labeled with the limits of Eq. (13.107a,b). Equation (13.106) reduces to Eq. (13.105) for the equal-current case Icα 5 Icβ. Equation (13.106) corresponds to the analogue of light interference from two nonidentical slits, but this is rarely studied in optics because it is so easy to make matching slits. Identical Josephson junctions are not so easy to fabricate, so the case of Eq. (13.106) is of interest in superconductivity.

E. Self-induced flux In the previous two sections, we tacitly assumed that the flux Φ in the circuits is the applied flux, which meant neglecting the contribution of the currents. We noted in

Energy gap and tunneling

565

Imax

Icα + Icβ

Icα

Icα – Icβ

–2

–1

0

1

2 φ/φ0

Figure 13.56 Dependence of the current maximum Imax of an unbalanced Josephson junction loop (Icα 6¼ Icβ ) on the applied flux Φ normalized relative to the quantum of flux Φ0. The plot is made for the case of setting Icα 5 2Icβ in Eq. (13.106). φα Iα

I = Iα + Iβ

I Icirc =

1 I -I 2 α β

Iβ φβ

Figure 13.57 Circulating current Icirc 5 1/2(Iα 2 Iβ) in an unbalanced Josephson junction loop, Iα 6¼ Iβ , in the presence of an applied current I 5 Iα 1 Iβ.

Chapter 2, Section X that the current Icirc circulating in a loop can contribute the amount IcircL to the flux, where L is the inductance of the loop. The currents Iα and Iβ in the two arms of the loop flow in the same direction, as indicated in Figs. 13.54 and 13.57, producing magnetic fields (cf. Fig. 2.35) pointed in opposite

566

Superconductivity

directions through the loop; for Iα 5 Iβ these fields cancel each other. However, when the two currents are not equal, we can decompose them into a symmetrical component 1/2(Iα 1 Iβ ), which flows in the same direction in each arm of the loop and does not contribute to the flux, and an antisymmetrical circulating component, Icirc 5

1 ðIα 2 Iβ Þ; 2

ð13:108Þ

as indicated in Fig. 13.57, which contributes to the flux. The total flux Φ is then the sum of the applied flux Φapp and the self-induced flux arising from the circulating current, Φ 5 Φapp 1

1 LðIα 2 Iβ Þ: 2

ð13:109Þ

This self-induced flux should be taken into account for a proper treatment of Josephson junctions.

F.

Junction loop of finite size

As a final example of tunneling we examine a loop in which the two identical junctions are large enough in area so as to contribute to the observed oscillatory current pattern. For this case, we can combine the diffraction equation (13.84) for the junction and the interference equation (13.105) for the loop,    sinðπΦJ =Φ0 Þ  ; I 5 2Ic cosðπΦL =Φ0 Þ  πΦ =Φ  J

ð13:110Þ

0

where ΦJ 5 BappAJ and ΦL 5 BappAL are the amounts of flux in the junctions of area AJ and in the loop of area AL, respectively, for a particular applied field Bapp. We can define the critical applied fields BJ and BL for which one flux quantum Φ0 is present in the junction and in the loop in terms of their respective areas: BJ 5 Φ0 =AJ ;

ð13:111aÞ

BL 5 Φ0 =AL :

ð13:111bÞ

These expressions permit us to write Eq. (13.110) in terms of the applied field,    sinðπBapp =BJ Þ : I 5 2Ic cosðπBapp =BL Þ πBapp =BJ 

ð13:112Þ

We call this the Josephson loop diffraction equation. Since ALcAJ, we have BL{BJ, and the expected current pattern is sketched in Fig. 13.58 for the case

Energy gap and tunneling

567

I

–2

–1

0

1

Bapp/BJ

2

Figure 13.58 Josephson loop diffraction pattern showing the dependence of the supercurrent I flowing through a two-element junction loop of finite size on the magnetic field Bapp, producing the flux Φ passing through the plane of the loop. The figure is drawn for the case of setting BJ 5 3BL in Eq. (13.112).

Josephson current (Arbitrary units

(a)

(b)

–500 –400–300–200 –100 0

100 200 300 400 500

Magntic field (mG)

Figure 13.59 Experimentally measured dependence of the Josephson current in a twoelement loop on the applied magnetic field. (a) The case BL  4 mT and BJ  50 mT with negligible self-induced flux (Iα  Iβ), so that the oscillations all return to the baseline, as in Figure 13.58. (b) The case BL  1.5 mT and  35 mT, so that the self-induced flux is appreciable (IαiIβ) and the rapid oscillations do not return to the baseline (Jaklevic et al., 1965).

BJ 5 3BL. We see from the figure that the slower individual junction variations constitute an envelope for the more rapid loop oscillations. Figure 13.59a shows some experimental results for a small loop in which the self-induced flux is negligible, so that a pattern similar to the pattern in the center of Fig. 13.58, but with more oscillations, is obtained. Figure 13.59b shows a large loop result in which the selfinduced flux is appreciable, so that the minima in the oscillations do not reach zero, as in the pattern of Fig. 13.56. Because this second loop is larger, it has more rapid oscillations, as shown. On each side of Fig. 13.59b we can see traces of the next set

568

Superconductivity

of oscillations arising from the second cycles of Eq. (13.112) for the field range BJ , Bapp , 2BJ, and these are also shown in the pattern of Fig. 13.58. Equation (13.112) corresponds to the optics analogue of Fraunhofer diffraction from two identical wide slits.

G. Ultrasmall Josephson junction When a Josephson junction becomes much smaller than a typical weak link or short junction, new phenomena can appear. As an example, consider an ultrasmall junction or nanobridge with area A of 0.01 μm2, thickness d of 0.1 nm, and capacitance estimated from the expression C 5 A0A/d of about 10213 F (Kuzmin and Haviland, 1991; Kuzmin et al., 1991). The change in voltage ΔV brought about by the tunneling of one electron across the junction barrier is given by ΔV 5 e/C 5 0.16 mV, which is an appreciable fraction of a typical junction voltage. This can be enough to impede the tunneling of the next electron. Blocking of current flow has been termed Coulomb blockade (Furusaki and Ueda 1992; Tagliacozzo et al., 1989). Note that it is only in recent decades that techniques such as electron-beam lithography have developed to the point where nanobridges with capacitances in the range 10213 to 10216 F can be fabricated (Rails et al., 1989). Single-electron tunneling manifests itself by the appearance of fluctuations, called a Coulomb staircase, on an I-versus-V or dI/dt-versus-V characteristic, as illustrated in Fig. 13.60 (McGreer et al., 1989). The Coulomb blockade is a quantum effect that represents the quantization of charge q transferred across a junction. The Hamiltonian, considered as a function of 50

600 500 400

0

300

dI nA dV V

Current (nA)

25

200 –25 100 –50 –0.3

–0.2

–0.1

0

0.1

0.2

0 0.3

Bias voltage (V)

Figure 13.60 Coulomb staircase structure on plots of I versus V and dI/dT versus V of tunneling between a granular lead film and the tip of a scanning tunneling microscope (McGreer et al., 1989).

Energy gap and tunneling

569

the two conjugate variables q and the phase φ, may be written as the sum of a capacitor charging energy, a current bias term, and a Josephson coupling energy. Hðq; φÞ 5 q2 =2C 2 ðh ¯ =2eÞIφ 2 ðπh ¯ =4e2 ÞðΔ=Rn Þcos φ;

ð13:113Þ

where I is the bias current, Δ is the energy gap, C is the junction capacitance, and Rn is the normal resistance (lansiti et al., 1989; Shimshoni and Ben-Jacobs, 1991). For conventional junctions, the coupling energy, (h/8e2)(Δ/Rn) cos φ, is dominant and for ultrasmall junctions the charging energy q2/2C predominates. There is an uncertainty relationship between the two conjugate variables Δ φ Δq $ e (Graham et al., 1991; Kuzmin et al., 1991). In observing the Coulomb blockade, q 5 e and the phase (or voltage) fluctuations are large. In the usual dc Josephson effect, which we described in Section VII.B, the phase is well defined and the fluctuations occur in the charge. In an ultrasmall Josephson junction biased by a dc current, correlated tunneling of Cooper pairs can lead to what are called Bloch oscillations at the frequency vB 5 I=2e:

ð13:114Þ

One-dimensional N-junction arrays have been used to observe single-electron tunneling (Delsing et al., 1989; Kuzmin et al., 1989). A current of 0.01 μA 5 1028C/s corresponds to a frequency  3 3 1010 Hz, which is in the microwave region. Singularities of the observed microwave resistance at the current values corresponding to Eq. (13.114) have provided evidence for the presence of Bloch oscillations (Furusaki and Ueda, 1992; Geerligs et al., 1989; Hu and O’Connell, 1993; Kuzmin and Haviland, 1991; Shimshoni et al., 1989). Another type of quantum oscillation is resonant tunneling in the tilted cosine potential illustrated in Fig. 13.61. The figure shows a microwave photon emission accompanying resonant tunneling between levels aligned in adjacent wells (Hatakenaka et al., 1990; Schmidt et al., 1991). In an ultrasmall junction, the effect of a bias current is taken into account by the term 2 h
Iφ/2e in the Hamiltonian, Eq. (13.113). When the bias current energy h
Ic/2e becomes comparable with the charging energy e2/2C, we have IcC  3.9 3 10223 AF, where Ic is the ideal critical current (Kautz and Martinis, 1990). This relation is satisfied by representative current and capacitance values of some ultrasmall junctions.

H. Arrays and models for granular superconductors The discussion until now has concerned single Josephson junctions, although we have also examined a pair of junctions. There is an extensive literature on chains, arrays, and layers of Josephson junctions that are coupled together in various ways,

570

Superconductivity

φ1

φ2

φ3

hωp

Figure 13.61 Macroscopic resonant tunneling of phase in tilted cosine potential showing the phases φi at the turning points and the microwave photon emission h
ω accompanying the tunneling (Hatakenaka et al., 1990).

and sometimes coupled to an applied current, magnetic field, or radiation (e.g., Eckern and Sonin, 1993; Van der Zant et al., 1993). High-temperature superconductors can be modeled by arrays of superconducting grains coupled together by Josephson junctions at their interfaces (Babic et al., 1991; Cai and Welch, 1992; Deutscher and Chaudhari, 1991; Fishman, 1988, 1989; Majhofer et al., 1990; Saslow, 1989; Sugano et al., 1992). The junctions in the arrays are often phase locked to each other (cf. Section VIII.B).

I.

Superconducting quantum interference device

Flux changes in a loop with two weak lines were shown to produce oscillatory variations in the supercurrent through the loop. A SQUID is a practical circuit that measures these current variations to quantitatively determine the strength of the applied field. It is more accurate to call such a device a dc SQUID since it measures a slowly changing applied field. A dc SQUID can detect much smaller changes in field than is possible by nonsuperconducting technology. One example of a dc SQUID is described in Chapter 5, Section VIII (see Fig. 5.12), and another version is sketched in Fig. 13.62. In the latter arrangement, the current change through the weak links is detected as a voltage change across the pair of weak links and then amplified by a step-up transformer followed by further amplification and measurement. Another type of SQUID, called an rf SQUID, is shown in Fig. 13.63. It consists of a loop with one weak link W coupled to an LC-tuned circuit driven by an rf current source. A change in flux in the loop produces a change in the loading of the tuned circuit, and this is detected by measuring the change in rf voltage across the circuit. More information on SQUIDs may be obtained from the texts by Van Duzer and Turner (1981) and by Orlando and Delin (1991). SQUIDs have been fabricated from high-temperature superconductors (Gross et al., 1990; Siegel et al., 1991; Vasiliev, 1991). Miller et al. (1991) discussed a superconducting quantum interference grating (SQUIG), an interferometer consisting of several Josephson junctions in parallel.

Energy gap and tunneling

571

V SQUID

V Out

Figure 13.62 Diagram of a SQUID consisting of two weak links, on the left, showing the voltage VSQUID across them coupled via the transformer to produce the output voltage Vout. rf Current source

Output Weak line

L

C

Figure 13.63 An rf SQUID consisting of one weak link coupled to an LC-tuned circuit driven by an rf current source. The rf output voltage is a measure of the change in loading of the tuned circuit produced by a change in flux through the loop.

Since a SQUID easily detects a change in one quantum of flux in an area with dimensions in the centimeter range, it is said to measure a macroscopic quantum phenomenon. Orlando and Delin (1991) build on Fritz London’s observation that superconductivity is inherently a quantum mechanical phenomenon with macroscopic manifestations in their utilization of a macroscopic quantum model to describe superconductivity.

Problems 1. Describe NIN tunneling in the manner that NIS and SIS are described in Sections V.E and V.F, respectively. Calculate D1 and D2. 2. Derive and justify Eqs. (13.13), (13.16a), and (13.16b). 3. Show how to derive the expressions (13.18) and (13.19) for the integrand of Eq. (13.17).

572

Superconductivity

4. Show how to obtain the NIS expression (13.20) for Ins from the general tunneling current equation (13.17). 5. Show that in SIS tunneling the ratio of the jump in current ΔIs to the normal tunneling current In at the bias V 5 2Δ/e is given by ΔIs =In 5 π=4; which is about an 80% jump. 6. Show how Eq. (13.25) for ΔIs reduces to the simple expression (13.23) in the limit T!0. 7. Plot Δ(T) versus T using the data of Fig. 13.26, and compare the plot with Fig. 13.27. 8. Derive Eq. (13.96) from Eq. (13.95). 9. Show how to solve Eq. (13.59): I dφ 5 1 sin φ: L dΘ 10. Explain how the washboard analogue sketched in Fig. 13.47 mimics the behavior of a Josephson junction. 11. Justify and derive Eq. (13.70). What is the physical significance of each term? 12. Carry out the integration (13.82) to obtain I 5 Ic sin φ0

sinðπΦ=Φ0 Þ : πΦ=Φ0

ð13:83Þ

13. Consider a barrier junction that is 35 μm long and has a 45-nm oxide layer. What applied magnetic field will put one flux quantum in the junction if the superconductors are Nb on one side of the oxide layer and Sn on the other? At what value of the applied field will the first maximum following the principal center maximum appear in the “diffraction pattern”?

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