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Superconducting StateMechanisms and Properties$

Vladimir Z. Kresin, Hans Morawitz, and Stuart A. Wolf

Print publication date: 2013

Print ISBN-13: 9780199652556

Published to Oxford Scholarship Online: January 2014

DOI: 10.1093/acprof:oso/9780199652556.001.0001

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Isotope effect

Isotope effect

(p.122) 9 Isotope effect
Superconducting State

Vladimir Z. Kresin

Hans Morawitz

Stuart A. Wolf

Oxford University Press

Abstract and Keywords

The isotope effect occupies a special place in the physics of superconductivity. In reality it is a complex effect controlled by many factors. Among them are intrinsic factors (anharmonicity, multi-component lattice, and so on) and also some extrinsic factors (such as pair-breaking) which are not related to the lattice dynamics. Doped materials, such as the cuprates, display the polaronic isotope effect.

Keywords:   coulomb pseudopotential, anharmonicity, pair-breaking, polaronic effect

9.1 General remarks

The isotope effect occupies a special place in the physics of superconductivity. Its discovery was the major step preceding the BCS theory by demonstrating that lattice is involved in the formation of the superconducting state. The effect was studied theoretically by Frohlich (1950) and was discovered by studying various isotopes of mercury (Maxwell, 1950; Reynolds, et al., 1950). As the mass M was varied between 199.5 and 203.4, the value of T c changed from 4.185K to 4.140K It was established that the following relation holds:

T c = ( const . ) M α ; α 0.5

The dependence (9.1) and value α ≈ 0.5 are in a total agreement with the BCS expression (3.13) for the critical temperature, since the vibrational frequency Ω ˜ M 1 / 2 .

At first sight, the isotope effect seems to be a rather straightforward phenomenon, but such an impression is misleading. In reality, it is a complex effect controlled by many factors. Indeed, in many cases one can observe large derivations from the canonical value α = 0.5. As a result, the presence of the isotopic dependence means that the lattice is involved in the pairing, but it is difficult to provide a quantitave estimation of the degree of this involvement.

In the following we discuss various factors affecting the value of the isotope coefficient.

9.2 Coulomb pseudopotential

The BCS expression (3.28) contains the Coulomb pseudopotential μ* defined by eqn. (3.27). One can see directly from eqn. (3.27) that the value of μ* depends on phonon frequency, and its change upon the isotope substitution also affects the value of T c and, correspondingly, the value of the isotope coefficient α.

Starting from the expression (9.1), defining the isotope coefficient, we can obtain the following general expression:

α = M / T c T c / M

Then we can write:

α = M / T c T c / Ω Ω / M

(p.123) Based on eqn. (3.27), (3.28), and (9.3), we obtain:

α = α | μ = 0 1 μ λ μ 2

One can see that the presence of the Coulomb factor leads to a decrease in the value of the isotope coefficient. For example, if λ = 0.5 and μ* = 0.15, we obtain α = 0.4. For λ = 0.4 and μ* = 0.15, the decrease is even more noticeable: α ≈ 0.3.

Eqn. (3.28) is valid for the case of weak coupling. One can see from eqns. (3.22) and (3.33) that for the intermediate and strong coupling (see also eqn. (3.35)), the presence of the Coulomb pseudopotential also leads to a noticeable impact on the isotope effect.

9.3 Multi-component lattice

As shown previously, the Coulomb pseudopotential noticeably affects the value of the isotope coefficient. If the lattice contains several atoms per unit cell, the picture becomes more complicated (Geilikman, 1976). Indeed, even for a two-atomic lattice we can obtain any value of α. This can be seen directly from the expression for the two phonon modes (Maradudin et al., 1963):

ω a c 2 = α = 1 3 γ α 1 M 1 + 1 M 2 α = 1 3 γ α 2 1 M 1 1 M 2 2 + 4 M 1 M 2 α = 1 3 γ α cos q 0 d 2 1 / 2 ω o p 2 = α = 1 3 γ α 1 M 1 + 1 M 2 + α = 1 3 γ α 2 1 M 1 1 M 2 2 + 4 M 1 M 2 α = 1 3 γ α cos q 0 d 2 1 / 2

The substitution M 1 M 1 can lead to any value of α, depending on M 2, force constants γ α , and so on.

9.4 Anharmonicity

Anharmonicity of the lattice can strongly affect the value of the isotope coefficient. The well-known superconductor displaying such an effect is the Pd-H compound. The isotope substitution HD leads to an increase in T c , ; that is, the isotope coefficient α (see eqn. (9.1)) appears to be negative. The idea that such an unusual behavior of the isotopic dependence is caused by anharmonicity was proposed by Ganzuly (1973). A detailed analysis, based on the strong coupling theory (see Chapter 3) was carried out by Klein and Cohen (1992). They demonstrated that anharmonicity of the lattice leads to the dependence of vibrational frequency on mass which drastically different from the Ω ˜ M 1 2 law. The calculation for the Pd-H system performed up to sixth order of the ionic displacement show that the coupling constant λ (see eqn. (3.23)) increases upon the isotope substitution. This increase is caused by the effective decrease in the characteristic frequency; the Hopfield factor η = I 2 ν F , see eqn. (3.23), was assumed being constant, so that the increase in λ is caused by lattice dynamics only. The value (p.124) of T c was determined from eqn. (3.35). The value of μ * for the Pd-H system was chosen to be μ * = 0.115, since it provides the best agreement with the data (Schirber and Northrop, 1974). The value of the isotope coefficient obtained by Klein and Cohen (1992) appears to be negative, in agreement with experimental observation.

Therefore, anharmonicity can drastically change the isotopic dependence of the critical temperature; it may lead even to a negative sign of the isotope coefficient.

9.5 Isotope effect in proximity systems

In this and the following sections we consider the factors which are not related directly to the pairing mechanism, but nevertheless contribute to the isotope effect. Indeed, the value of the critical temperature could be affected by an external factor (T c;0T c; T c;0 is the intrinsic value of the critical temperature), which is not related to the lattice dynamics.

The most interesting case corresponds to the situation when the relation between Tc ;0 and T c is not linear. Let us be more specific. As an example, consider the isotope effect in a proximity system SN (where S and N are superconducting and normal films; see Chapter 8 and Fig. 8.1). It turns out that the value of α depends on the relative thicknesses of the films. Indeed, assume that the thickness L N ≪ ξN, where ξN = hv F;N/2πT is the coherence length for the N film. We can then use a well-known McMillan tunneling model (1968b). According to this model, the proximity effect is described by the parameter Γ = ΓSN + ΓNS, where Γ i k = T ˜ i k 2 ν k V k , T ˜ i k is the tunneling matrix element, υk is the density of states (per unit of volume V), i,k = {S,N}, ik. We assume that Γ Ω . One can see that, indeed, the relation between T c and T c0 is non-linear. The critical temperature of the whole system T c differs from T c0 (T c0 is the critical temperature of the isolated S film), and is described by eqn. (8.7).

If we make an isotope substitution MM * for the isolated S film, one can measure the shift in T c0 and determine the isotope coefficient α0 which is described by eqn. (9.2). The presence of the N film leads to a change in T c and in the value of the isotope coefficient. One can see directly from eqns. (9.2) and (8.7) that the shift in T c and the new value of the isotope coefficient α = −(MM)(ΔT c/T c) differ from ΔT c0/T c0 and the value of α 0. Indeed, the value of T c for the whole sandwich is determined by eqn. (8.7). Then the value of the shift ΔT c is determined not only by ΔT c0 but also by the value of the parameter ρ defined by eqn. (8.5b); it reflects the presence of the N film. More specifically, based on eqn. (8.7), we obtain:

Δ T c / T c = Δ T c 0 / T c 0 1 + ρ
As for the value of the isotope coefficient, we obtain
α = α o [ 1 + ( ν N L N / ν S L S ) ]

One can see that, indeed, the value of the isotope coefficient is modified by the proximity effect. Moreover, αα o. Therefore, a decrease in T c, which is a well-known feature of the proximity effect, and is described by eqn. (8.7), is accompanied by an increase in the isotope coefficient. It is interesting that one can modify the value of α by changing the thicknesses of the films. The increase in the thickness of the normal (p.125) film LN leads to decrease in T c, but the value of a increases. For example, if α 0 = 0.2, ν N/ν s = 0.8. and LN/Ls = 0.5, then α = 0.28. If we increase the thickness of the normal film, so that LN = Ls, then α = 0.36.

Note that the increase of the isotope coefficient discussed in this section is not related to lattice dynamics; as a result, the value of α can, in principle, exceed, the value of α 0;ph, max = 0.5.

9.6 Magnetic impurities and isotope effect

In this section we focus on another isotope effect which is also not related to lattice dynamics: we consider a superconductor which contains magnetic impurities.

The presence of magnetic impurities leads to decrease of the critical temperature, T c, relative to the intrinsic value T c0, because of the pair-breaking effect described in Ch. 8. This depression of T c is described by eqn. (8.12): ln ( T c o / T c ) = Ψ [ 0.5 + γ s ] Ψ ( 0.5 ) ; γ s = ( 2 π T c 0 τ s ) 1 ; Ψ is the digamma function. In this case also (c.f. eqn. (8.7)) we are dealing with a non-linear relation between T c0 and T c.

The isotope substitution MM * for the sample without magnetic impurities allows one to observe the shift in T c0 and measure the isotope coefficient. One can see directly from eqn. (8.12) that the shift in T c and the value of the isotope coefficient α in the presence of magnetic impurities differ from ΔT c0/T co and the value of α 0.

Based on eqns. (9.2) and (8.12), one can arrive at the following equation (Carbotte et al., 1991; Kresin et al., 1997):

α = α o [ 1 Ψ 0.5 + γ s γ s ] 1

Note that an increase in the concentration of magnetic impurities nM leads to increase in γs. Eqn. (9.7) is valid in a broad range except a very small region near ncr (then T c is close to T = 0 K and the condition ΔT c/T c ≪ 1 is not satisfied).

The presence of magnetic impurities leads to an increase of the isotope coefficient (α 〉 α0), since Ψʹ 〉 0; this can be seen directly from the expression

Ψ 0.5 + x = k = 0 k + 0.5 + x 2

One can study the dependence α on the concentration of magnetic impurities. For small γs (small values of nM), ΔαnM. Therefore, near T c0 the critical temperature displays a linear decrease with increasing nM, whereas the isotope coefficient increases linearly as a function of nM. In the region γs 〉〉 1 one can use an asymptotic expression for the digamma function, and we obtain Δ α n 2 M ; that is, the dependence becomes strongly non-linear.

We can see directly from eqns. (9.6) and (9.7) that the impact of the proximity effect, as well as the magnetic scattering on the value of the isotope coefficient, is caused by the non-linear relation between T c and T c0. Qualitatively, both factors provide the pair-breaking, and this leads to an effective increase in the normal electronic component.

(p.126) 9.7 Polaronic effect and isotope substitution

Some novel materials, including the high-T c cuprates, display dynamic polaronic effect (see Sections 2.6.2 and 3.9). These materials contain sub-groups of ions (for example, oxygen ions in the cuprates), and each of them is characterized not by one but two close equilibrium positions. In addition, the carriers in materials of interest (see Chapter 10) are provided by doping. Then one can observe a novel isotope effect (Kresin and Wolf, 1994). The point is that the doping in such materials represents a charge transfer process. In this case the isotope substitution affects the doping, and therefore the carrier concentration n. For doped materials the critical temperature depends strongly on n, and this leads to isotopic dependence of T c. In the following we will discuss all implications of such an isotope effect for the physics of high-T c cuprates (Chapter 10) and manganites (Chapter 12), but here we describe the basic scenario of the polaronic isotope effect.

If the charge transfer occurs in the framework of the usual adiabatic picture (Chapter 2), so that only the carrier motion is involved, then the isotope substitution does not affect the forces and therefore does not change the charge transfer dynamics. However, the situation of strong non-adiabaticity (polaronic effect; see Section 2.6.2) is different, and does not allow the separation of electronic and nuclear motions. In this case, charge transfer appears as a more complex phenomenon, which involves nuclear motion, and this leads to a dependence of the doping on isotopic mass.

Let us consider the case when the lattice configuration—that is, the positions of some definite ions (such as axial oxygen for YBCO, see Chapter 10)—corresponds to a degenerate or near-degenerate state. This means that the degree of freedom describing its motion corresponds to electronic terms crossing (see Fig. 2.4). Then the ion has two close equilibrium positions.

The charge transfer in this picture is accompanied by the transition to another electronic term. Such a process is analogous to the Landau–Zener effect (see, for example, Landau and Lifshitz, 1977). The charge transfer corresponds to the transitions between the first and second terms.

The total wavefunction (see Section 2.6.2) can be written in the form

Ψ i ( r , R , t ) = a ( t ) Ψ 1 ( r , R ) + b ( t ) Ψ 2 ( r , R )


Ψ i ( r , R ) = Ψ i ( r , R ) Φ i ( R ) , i = { 1 , 2 }

Ψ i ( r , R ) , Φ i ( R ) are the electronic and vibrational wavefunctions that correspond to two different electronic terms (see Fig. 2.4).

Qualitatively, the charge transfer for such non-adiabaticity can be visualized as a multistep process. First, the carrier makes a transition from the reservoir site to the ion; then the ion transfers to another term, and this is finally followed by the transition of the carriers to the plane (for the cuprates). The second step is affected by the isotope substitution.

Let us separate the Z-coordinate of the ion (the axis Z has been chosen to be parallel to the ionic motion between the minima, so that Φi(R) = Φi(ρ i)φi(Z), R i (p.127) corresponds to the other degrees of freedom. We do not assume the electronic terms to be similar and, as a result, they differ on the energy level spacing. This leads to some splitting between the vibrational levels (see Fig. 2.4), and the transition of the ion is not a resonant one. In the harmonic approximation,

φ 1 Z = π a 1 1 / 2 exp ( z z 10 ) 2 2 a 1 2
φ 2 Z = π a 2 1 / 2 exp ( z z 20 ) 2 2 a 2 2 H ν [ ( z z 20 ) a 2 1 ]

Here, ai = (M Ωi)−1 are the vibrational amplitudes, Hν is the Hermite polynomial, M is the ionic mass, and Z i0 are the equilibrium positions. We assume that the ionic motion corresponds to the zero vibrational state of the first term, and ν th level for the second term. Assume also that b(0) = 0. The dependence b(t) describes the dynamics of the charge transfer (1 → 2).

As noted previously (Section 2.6.2), it is convenient to employ the diabatic representation, and for the average value, b ˜ 2 , we obtain the expression (2.40a). The asymmetry of the potential for the axial oxygen and correspondingly the inequality ɛ ≠ 0 are playing key roles. Assume a large asymmetry (see Fig. 2.4), so that | L 0 F 12 | ε . Then the carrier concentration

n p l b 2 = 2 L 0 2 F 12 2 ε 2

Since the Franck–Condon factor F 12 and the value of ɛ depends directly on the mass M, the carrier concentration also depends on M. Its presence reflects the fact that the electronic and vibrational states are not separate (polaronic effect).

Let us focus now on the isotope coefficient α = (−M/T c)(∂T c/∂M). We should distinguish two contributions, so that α = α 0 + αna, where α 0 = −(M/T c)(dT c/dΩ) (dΩ/dM) describes the usual isotope effect caused by change in the phonon spectrum (Ω is a characteristic phonon frequency for that mass). If the corresponding mode in the polyatomic system does not contribute noticeably to the pairing, then the coefficient α 0 is small.

The term ana corresponds to a different isotope effect which is due to the dependence n(M) described previously, and also to the dependence Tc(n). Namely:

α n a = M T c T c n n M

According to eqns. (9.10) and (9.11), the dependence n(M) is determined mainly by the Franck–Condon factor. Based on eqns. (9.9)–(9.11), and the dependence ɛ2 ∝ M−1 (therefore n M e s M 0.5 , s = k 0.5 d 2 2 , k 0.5 = k 1 0.5 + k 2 0.5 , where ki are the elastic constants and d is the distance between the minima) we obtain the following expression for the isotope coefficient:

α n a = γ n T c T c n

(p.128) where γ ≅ constant (it has weak logarithmic dependence on M). Expression (9.12) can be compared directly with experimental data (see Chapter 10).

Equation (9.12) contains the derivative T c n . Therefore, the isotope coefficient αna is equal to zero if T c  = T c,max. For the cuprates, T c has a maximum at some value n = nmax. Therefore, the described mechanism leads to the situation that the maximum value of T c corresponds to a minimum in the value of the isotope coefficient. This phenomenon, indeed, has been observed experimentally (see Chapter 10).

Let us make several comments. Continuous doping leads not only to an increase in concentration n, but to a change in electronic terms; the structure of the terms becomes less asymmetric. For example, for the cuprates (see Chapter 10) this is reflected in the movement of the average position of the axial oxygen toward the plane. This motion has been observed experimentally (Jorgensen et al., 1990). Such an evolution leads to an additional decrease in the value of α.

The second comment is related to the overdoped region. Our analysis is applicable to a single-phase system, that is, for the underdoped region of the phase diagram. Structural transitions can drastically modify this picture. If the overdoped region corresponds to the same phase as the underdoped region (this is probably the case for BiPbCaSrCuO), then in the region nnmax, the isotope coefficient should become negative; such an effect has indeed been observed by Bornemann et al. (1991). However, if the region nnmax is a multiphase compound (as is the case for La2–xSrxCuO4; see Bozin et al., 2000), then the picture described above can be used for nnmax only. Finally, one should note that according to the described model the values of α are not limited by a maximum of 0.5, and may exceed this value.

Therefore, a strong non-adiabaticity of the ions in the doped superconductors, such as the high-T c oxides, leads to the dependence of the carrier concentration on M, and this factor affects the value of T c.

9.8 Penetration depth: isotopic dependence

The polaronic isotope effect described previously is a consequence of the dependence of the carrier concentration on the ionic mass M : n = n (M). Since not only T c but other properties are also affected by the value of the carrier concentration, one can expect the isotopic dependence of various quantities. As an important example, let us discuss the isotopic dependence of the penetration depth, δ. This dependence was introduced theoretically by Kresin and Wolf (1995) and then analyzed in detail by Bill et al. (1998). The effect has been observed experimentally for the high-T c oxides, and we will discuss it in Chapter 10.

By analogy with T c, let us define a new isotope coefficient β by the dependence

δ M β

As a result, β is determined by the expression (see eqn. (9.2))

β = ( M / δ ) ( δ / M ) .

describing the change in the value of the penetration depth induced by the isotope substitution: MM * = M + ΔM.

(p.129) Let us consider the London limit. Then the penetration depth is given by the well-known relation:

δ 2 = m c 2 4 π n s e 2 = m c 2 4 π n φ ( T T c ) e 2 ,

where m is the effective mass. n s is the superconducting density of charge carriers, related to the normal density n s through n s = (T/T c). The function φ = (T/T c) is a universal function of (T/T c). For example, for conventional superconductors, φ ≃ 1 − (T/T c)4 near T c, whereas φ ≃ 1 near T = 0. We can now determine the isotope coefficient β of the penetration depth from the relation which follows from eqns. (9.14) and (9.15):

β M δ δ n S n S M = M 2 n S n S M .

Because of the relation n s = (T), one has to distinguish two contributions to β. There is the usual (BCS) contribution, β ph, arising from the fact that φ(T/T c) depends on ionic mass through the dependency of T c on the characteristic phonon frequency. Indeed, isotopic substitution leads to a shift in T c and thus in β, which might be noticeable near T c.

The presence of the factor n (see eqn. (9.15)) leads to a peculiar isotope effect for the penetration depth with ββ na (β na corresponds to the non-adiabatic contribution.)

Let us consider the region near T = 0 K. Then

β n a M 2 n . d n d M

Comparing eqns. (9.11), (9.12), and (9.17), one infers that β na = −γ/2 and thus establishes a relation between the non-adiabatic isotope coefficient of T c, α na, and β na:

α n a = 2 β n a n T c T c n .

The equation contains only measurable quantities, and can thus be verified experimentally. It is interesting to note that β na and α na have opposite signs when T c / n 0 (which corresponds to the underdoped region in high-T c cuprates). In Chapter 10 we will discuss the isotopic dependence of T c and δ for the cuprates.

Let us consider another case when the penetration depth is affected by isotopic substitution, and again the dependence is not related directly to the lattice dynamics. Namely, consider the SN proximity system (Bill et al., 1998), where S is a superconductor and N is a metal or a semiconductor (see Section 8.5.1). Assume that δ 〈 L N 〈〈 ξN (this is certainly satisfied in the low-temperature regime, since ξN = ħνF/2πT), where L N and ξN are the thickness and the coherence length of the normal film, respectively. One can show that in this case the penetration depth is described by the expression:

δ 3 = a N Φ

(p.130) where a N = constant; its value depends only on the material properties of the normal film (it is independent of the ionic mass), and

Φ = π T n 0 1 x n 2 p n 2 + 1 , p n = 1 + t x n 2 + 1

x n = ωn/ɛS(T), ωn = (2n + 1)πT, and ɛS(T) is the superconducting energy gap of S film. In the weak-coupling limit considered here, ɛS(0) = rT c0, with r = r BCS = 1.72, t is the dimensionless parameter, t = r(LN/LS)(Tc;S0) ( Γ 0 Γ α β L S 1 is the McMillan parameter; see Chapter 8), and L N and L S are the thicknesses of the normal and superconducting films.

Since δ depends non-linearly on T c0 (through the proximity parameter t), the penetration depth will display an isotope shift due to the proximity effect. From eqn. (9.19) we can calculate the isotope coefficient β prox of the penetration depth due to the proximity effect:

β p r o x = 2 α 0 3 Φ n 0 x n 2 p n 2 x n 2 p n 2 + 1 1 t p n x n 2 + 1 ,

where α 0 is the isotope coefficient of T c0 for the superconducting film S alone.

Note that the isotope coefficients β prax and α 0 generally have opposite signs. In addition, the isotope coefficient depends on the proximity parameter t; that is, on the thickness ratio l = L N/L S of the normal and superconducting films, and on the MacMillan tunneling parameter Γ0. Note also that the isotopic shift of the penetration depth is temperature dependent: | β p r o x | increases with increasing temperature.

Therefore, the presence of a normal layer on a superconductor induces an isotope shift of the penetration depth.

Note that, similarly to the proximity effect, the isotopic dependence of the penetration depth is also affected by another pair-breaking effects: by magnetic scattering (Bill et al., 1998). A detailed description of this effect, however, is beyond the scope of this book.