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Superconductivity pairing induced by two level systems in amorphous metals
Volume 79A, number 4
PHYSICS LETFERS
13 October 1980
SUPERCONDUCTWITY PAIRING INDUCED BY TWO LEVEL SYSTEMS IN AMORPHOUS METALS J. RIESS and R. MAYNARD Centre de Recherches sur les TresBasses Temperatures, CNRS, 38042 Grenoble Cedex, France Received 10 July 1980
From the standard interaction between electrons and two level systems a superconductive pairing is envisaged. The solutions of the Eliashberg equations for the critical temperature as well as the zero temperature gap lead to expressions as exp(—l/..J~)instead of exp(—1/xo) in the BCS case, which enhances considerably the superconducting properties in the weak coupling case.
There seems to be sufficient experimental and theoretical evidence for the existence of tunneling levels
Here NILS is the total number of TLS in the sample, and w0 is the cutoff energy of the distribution (all
(two-level systems, TLS) in metallic glasses and their interaction with the conduction electrons. For a review see ref. [1]. A TLS model has also been used to describe interactions between electrons and quasiocal excitations in structurally unstable lattices of high-Ta superconductors [2,3]. In this context it has been shown recently [3] that TLS contribute an attractive part to the effective electron—electron interaction leading to an enhancementsame of the superconducting transition temperature. The result has been obtained earlier [4,5] for a BCS-superconductor perturbed by a random spatial
energies are given in units of kB). The TLS-electron system can be described by the following hamiltonian:
distribution of TLS-impurities. In refs. [2—5]all TLS were supposed to have the same splitting energy E. However, in metallic glasses at least, there is a distribution F(E) of the splitting energies, with F(E) being considerably different from zero for values of (where E ranging less than lkB 100kB kB from is themuch Boltzmann constant). up In to this lO—article we consider superconductivity induced by TLS-electron interaction alone, neglecting any other type of interaction, taking into account a distribution F(E) of the TLS energy splitting E. We assume that F(E) has the simple form F(E) = NTLS/wo =
E~
1
0
~ 0
334
H=
~~CkCk~Ck + 1
+
(1/Nlk~
~iqr
(viiS~+
ViS~)4+q acka
.
(2)
Here is the splitting for the TLS located 1 andE1described by theenergy pseudospin operator S~(S =at r1/2). We neglect all k-dependence of the coupling constants
and v
1. N is the total number of atoms in the sample and ek the single electron energy. Our first goal is to derive Eliashberg-type equations. To this end we follow closely the derivation for the case of electron—phonon interaction given in ref. [61. For the calculation we first write the pseudospin operatorsand S~,annihilation S~in (2) in operators. a representation of TLS-state creation This representation, together with the proper normalization (cf. [7,5 1) guarantees the applicability of the usual diagram rules in the Green’s function perturbation theory. Next we calculate the electronic self-energy. Here we simply have to replace the total phonon vertex in ref. [61(eq. (53)) by the total TLS-vertex, which in lowest order is equal to the free TLS-vertex. This approximation formally corresponds to the Migdal approximation for the phonon vertex. Averaging over the position of each
Volume 79A, number 4
PHYSICS LETTERS
site r1, summing over all the TLS and taking the frequency sums in the usual manner we obtain a result which consists of contributions from the diagonal and off-diagonal parts of the interaction matrix M = (u S~+ v 1S~)/N. (3) II
takes the form
2]th(w/2T)F(w)=S(w,fl. (5) S(w)= [N(0)u~/8N Here N(0) is the electronic density of states at the Fermi energy, and F(w) is the TLS-density of states with
Note the fact that now the Eliashberg function depends on temperature due to the factor th(w/2T), which reflects the statistics of two levels separated by the energy w. (In the case of phonons, where we have an infinite number of levels spaced by the energy the corresponding factor is equal to one.) According to eqs. (1) and (5) the Eliashberg function of the electron-TLS system has the form ~
=
_______
A(T) = 2 ~ S(w, T) dw .i w ~‘
2A~(1+ in (w0/2T)) w0~’2T.
10
Eq. (10) shows that McMillan’s coupling parameter A increases with decreasing temperature, and therefore the T~.loweringeffect due to small A 0 -values is partly cancelled. From eq. (9) one readily obtains T~=0.82w 1/2} (11) 0exp[—(0.25 + 1/2A0) In a second calculation we use a more rigorous approximation of the double integral in eq. (4) of ref. [8]. This leads in a long but straightforward calculation to .
1!2]
.
(12)
0exp[—(0.79 +0.67/A0)
NTLS.
0
The new aspect of eqs. (11) and (12) lies in the appearance of the square root in the exponential, which is not present in the T~expressions of phonon-induced superconductivity (BCS, McMillan, ...). This square root behaviour has its origin in the temperature dependence of the Eliashberg function of TLS-induced superconductivity (eq. (6)) and in the flat form of the distribution F(w) (eq. (1)), which extends to low frequencies of order w < T~.This is ifiustrated if we replace F(w) of eq. (1) by a s-function centered at wo~In this case eq. (11) reduces to
A 0 th(w/2T),
w ~ w0
,
(6)
where —
[8], we obtain for eq. (4) of ref. [8] the simple relation z~ ~ ‘ C’ (9) 1 + X(T~) 2 with
T~=0.35w
00
f F(w) dw
~ ~
First we calculate the superconducting transition temperature T~.In a first version of approximation we proceed as in ref. [8]. As a consequence of the special form (6) of S(w, T) and together with eq. (10) of ref.
_____
It turns out that the diagonal elements do not contribute to an electron pairing (cf. also [4,5]), since they do not introduce any retardation effect. Therefore only the contribution to the electronic self-energy due to the off-diagonal parts of M have to be considered. The gap equations which are obtained in this way are of the same form as in the case of electron—phonon interaction (eqs. (74), (75) of ref. [6]) with the only difference that now, in the case of electron-TLS interaction, the Eliashberg function 2(w) F(w) 5(w) (4) a
S(w, T)
13 October 1980
2~~O with A Tc = 0.5w0 eh/ 0 = 2A0 (13) i.e. to a BCS-type expression. As a consequence of the square root behaviour in eqs. (11) and (12) the transition temperature T~of a very weak coupling (2A0 ~ 1) TLS-induced superconductor is considerably larger than the BCS value. As a numerical illustration consider the value w0 = 100: here, for A~= 0.1, T~becomes 2.3 K (eq. (12)) instead of 0.34 K for the standard pairing (eq. (11)). For the lower value A0 = 0.01 the enhancement is still larger: ,
,
in’
2 ~r 10A72 W0 . V14!fTLS/QJV
~
~
In the following we are looking for approximate solutions of McMfflan type [8] of the Eliashberg equations, i.e., we choose the following trial function for the gap ~(w): = ~
0,
0 ~ ~ w0 w0
(8)
335
Volume 79A, number 4
PHYSICS LETTERS
T~= 0.01 K instead of 1020K from the usual formula, The parameter 2A 0 can roughly be compared to “N(0) V” of the BCS-theory, where N(0) V ~ 0.3. For metallic glasses the values of 2A0 are not known with great accuracy. From experimental values of PdSiCu (quoted in ref. [9]) an indication of the order of magnitude can be obtained forfor the(uvalue of the TLS-density 3 erg~),and ( 10~~ cm 1/N)N(0) (~0.2)corresponding to A0 ~ 2.3 X iO—~,which leads to a very low, non-observable critical temperature T~ 10—22 K(eq. (12)). Further the gap z~(T= 0) is calculated. Starting from the Eliashberg equations at zero temperature and evaluating the integrals in the gap equations by a similar type of approximation as before we obtain ~(T= 0) w exp[—(1/A 2)1/2] (14) —
.
0
Eq. (14) is valid for w0 ~ (and A0 ~ 0.15) and differs from the standard expression of ~(0) proportional to exp(—1/2A0). Here also the square root in the exponential leads to a very large enhancement of ~(T = 0) with respect to an expression of BCS type. Finally we have evaluated the gap ~.(T) in the region T ~ T~.Using our TLS distribution eq. (1) and similar approximations as before we find
13 October 1980
temperature and the gap at zero temperature are much higher than one would expect on the basis of the BCStheory. Moreover, the gap near T~increases more rap. idly with decreasing temperature than in the BCS-theory. From the presently known values of the parameters in metallic glasses (PdSiCu) the predicted critical temperature is too low to be observable. This is mainly a consequence of the low density of TLS. However, it would be sufficient to increase the number of TLS by two orders of magnitude in order to obtain an observable effect: A0 102 gives T~ 10 mK in the absence of any pairing induced by phonons. Finally we remark that near a structural transition the density of two-level systems is larger and the superconductivity due to electron pairing induced by TLS could be observable.
~‘
—
4’,J2 [T~(T~ T)] 1/2 —
,
T~
,
A0
‘~
1
.
(15)
The pre-factor is therefore distinct from the BCS case: 5.6 instead of 3.1. In conclusion we find that TLS-induced superconductivity (characterized by a flat TLS-distribution function) eq. (1)leads to expressions for T~,L~(T 0) and ~ near T~,which differ from the corresponding expressions of phonon-induced superconductivity. In particular for weak coupling (2A0 ‘~ I) the critical
336
The authors would like to thank Dr. 0. Béthoux for pertinent suggestions and discussions. References [1] J.L.
Black, Low energy excitations in metallic glasses, in:
Metallic glasses, ed. H.J. Güntherodt (Springer, New York), to be published. [2] K.L. Ngai and T.L. Reinecke, Phys. Rev, B16 (1977) 1077. 131 G.M. Vuji~ié,V.L. Aksenov, N.M. Plakida and S. Stamenkovic, Phys. Lett. 73A (1979) 439. [4] U. Brandt, J. Low Temp. Phys. 2 (1970) 573. [5] P. Fulde, L.L. Hirst and A. Luther, Z. Phys. 230 (1970) 155. [6] D.J. Scalapino, in: Superconductivity, ed. R.D. Parks (Dekker, New York, 1969) Ch. 10. [7] A.A. Abrikosov, Physics 2(1965)5. [8] W.L. McMilan, Phys. Rev. 167 (1968) 331. [91 J.L. Black, B.L. Gyorffy, J. J~ickle,Phil. Mag. B40 (1979) 331.
PHYSICS LETFERS
13 October 1980
SUPERCONDUCTWITY PAIRING INDUCED BY TWO LEVEL SYSTEMS IN AMORPHOUS METALS J. RIESS and R. MAYNARD Centre de Recherches sur les TresBasses Temperatures, CNRS, 38042 Grenoble Cedex, France Received 10 July 1980
From the standard interaction between electrons and two level systems a superconductive pairing is envisaged. The solutions of the Eliashberg equations for the critical temperature as well as the zero temperature gap lead to expressions as exp(—l/..J~)instead of exp(—1/xo) in the BCS case, which enhances considerably the superconducting properties in the weak coupling case.
There seems to be sufficient experimental and theoretical evidence for the existence of tunneling levels
Here NILS is the total number of TLS in the sample, and w0 is the cutoff energy of the distribution (all
(two-level systems, TLS) in metallic glasses and their interaction with the conduction electrons. For a review see ref. [1]. A TLS model has also been used to describe interactions between electrons and quasiocal excitations in structurally unstable lattices of high-Ta superconductors [2,3]. In this context it has been shown recently [3] that TLS contribute an attractive part to the effective electron—electron interaction leading to an enhancementsame of the superconducting transition temperature. The result has been obtained earlier [4,5] for a BCS-superconductor perturbed by a random spatial
energies are given in units of kB). The TLS-electron system can be described by the following hamiltonian:
distribution of TLS-impurities. In refs. [2—5]all TLS were supposed to have the same splitting energy E. However, in metallic glasses at least, there is a distribution F(E) of the splitting energies, with F(E) being considerably different from zero for values of (where E ranging less than lkB 100kB kB from is themuch Boltzmann constant). up In to this lO—article we consider superconductivity induced by TLS-electron interaction alone, neglecting any other type of interaction, taking into account a distribution F(E) of the TLS energy splitting E. We assume that F(E) has the simple form F(E) = NTLS/wo =
E~
1
0
~ 0
334
H=
~~CkCk~Ck + 1
+
(1/Nlk~
~iqr
(viiS~+
ViS~)4+q acka
.
(2)
Here is the splitting for the TLS located 1 andE1described by theenergy pseudospin operator S~(S =at r1/2). We neglect all k-dependence of the coupling constants
and v
1. N is the total number of atoms in the sample and ek the single electron energy. Our first goal is to derive Eliashberg-type equations. To this end we follow closely the derivation for the case of electron—phonon interaction given in ref. [61. For the calculation we first write the pseudospin operatorsand S~,annihilation S~in (2) in operators. a representation of TLS-state creation This representation, together with the proper normalization (cf. [7,5 1) guarantees the applicability of the usual diagram rules in the Green’s function perturbation theory. Next we calculate the electronic self-energy. Here we simply have to replace the total phonon vertex in ref. [61(eq. (53)) by the total TLS-vertex, which in lowest order is equal to the free TLS-vertex. This approximation formally corresponds to the Migdal approximation for the phonon vertex. Averaging over the position of each
Volume 79A, number 4
PHYSICS LETTERS
site r1, summing over all the TLS and taking the frequency sums in the usual manner we obtain a result which consists of contributions from the diagonal and off-diagonal parts of the interaction matrix M = (u S~+ v 1S~)/N. (3) II
takes the form
2]th(w/2T)F(w)=S(w,fl. (5) S(w)= [N(0)u~/8N Here N(0) is the electronic density of states at the Fermi energy, and F(w) is the TLS-density of states with
Note the fact that now the Eliashberg function depends on temperature due to the factor th(w/2T), which reflects the statistics of two levels separated by the energy w. (In the case of phonons, where we have an infinite number of levels spaced by the energy the corresponding factor is equal to one.) According to eqs. (1) and (5) the Eliashberg function of the electron-TLS system has the form ~
=
_______
A(T) = 2 ~ S(w, T) dw .i w ~‘
2A~(1+ in (w0/2T)) w0~’2T.
10
Eq. (10) shows that McMillan’s coupling parameter A increases with decreasing temperature, and therefore the T~.loweringeffect due to small A 0 -values is partly cancelled. From eq. (9) one readily obtains T~=0.82w 1/2} (11) 0exp[—(0.25 + 1/2A0) In a second calculation we use a more rigorous approximation of the double integral in eq. (4) of ref. [8]. This leads in a long but straightforward calculation to .
1!2]
.
(12)
0exp[—(0.79 +0.67/A0)
NTLS.
0
The new aspect of eqs. (11) and (12) lies in the appearance of the square root in the exponential, which is not present in the T~expressions of phonon-induced superconductivity (BCS, McMillan, ...). This square root behaviour has its origin in the temperature dependence of the Eliashberg function of TLS-induced superconductivity (eq. (6)) and in the flat form of the distribution F(w) (eq. (1)), which extends to low frequencies of order w < T~.This is ifiustrated if we replace F(w) of eq. (1) by a s-function centered at wo~In this case eq. (11) reduces to
A 0 th(w/2T),
w ~ w0
,
(6)
where —
[8], we obtain for eq. (4) of ref. [8] the simple relation z~ ~ ‘ C’ (9) 1 + X(T~) 2 with
T~=0.35w
00
f F(w) dw
~ ~
First we calculate the superconducting transition temperature T~.In a first version of approximation we proceed as in ref. [8]. As a consequence of the special form (6) of S(w, T) and together with eq. (10) of ref.
_____
It turns out that the diagonal elements do not contribute to an electron pairing (cf. also [4,5]), since they do not introduce any retardation effect. Therefore only the contribution to the electronic self-energy due to the off-diagonal parts of M have to be considered. The gap equations which are obtained in this way are of the same form as in the case of electron—phonon interaction (eqs. (74), (75) of ref. [6]) with the only difference that now, in the case of electron-TLS interaction, the Eliashberg function 2(w) F(w) 5(w) (4) a
S(w, T)
13 October 1980
2~~O with A Tc = 0.5w0 eh/ 0 = 2A0 (13) i.e. to a BCS-type expression. As a consequence of the square root behaviour in eqs. (11) and (12) the transition temperature T~of a very weak coupling (2A0 ~ 1) TLS-induced superconductor is considerably larger than the BCS value. As a numerical illustration consider the value w0 = 100: here, for A~= 0.1, T~becomes 2.3 K (eq. (12)) instead of 0.34 K for the standard pairing (eq. (11)). For the lower value A0 = 0.01 the enhancement is still larger: ,
,
in’
2 ~r 10A72 W0 . V14!fTLS/QJV
~
~
In the following we are looking for approximate solutions of McMfflan type [8] of the Eliashberg equations, i.e., we choose the following trial function for the gap ~(w): = ~
0,
0 ~ ~ w0 w0
(8)
335
Volume 79A, number 4
PHYSICS LETTERS
T~= 0.01 K instead of 1020K from the usual formula, The parameter 2A 0 can roughly be compared to “N(0) V” of the BCS-theory, where N(0) V ~ 0.3. For metallic glasses the values of 2A0 are not known with great accuracy. From experimental values of PdSiCu (quoted in ref. [9]) an indication of the order of magnitude can be obtained forfor the(uvalue of the TLS-density 3 erg~),and ( 10~~ cm 1/N)N(0) (~0.2)corresponding to A0 ~ 2.3 X iO—~,which leads to a very low, non-observable critical temperature T~ 10—22 K(eq. (12)). Further the gap z~(T= 0) is calculated. Starting from the Eliashberg equations at zero temperature and evaluating the integrals in the gap equations by a similar type of approximation as before we obtain ~(T= 0) w exp[—(1/A 2)1/2] (14) —
.
0
Eq. (14) is valid for w0 ~ (and A0 ~ 0.15) and differs from the standard expression of ~(0) proportional to exp(—1/2A0). Here also the square root in the exponential leads to a very large enhancement of ~(T = 0) with respect to an expression of BCS type. Finally we have evaluated the gap ~.(T) in the region T ~ T~.Using our TLS distribution eq. (1) and similar approximations as before we find
13 October 1980
temperature and the gap at zero temperature are much higher than one would expect on the basis of the BCStheory. Moreover, the gap near T~increases more rap. idly with decreasing temperature than in the BCS-theory. From the presently known values of the parameters in metallic glasses (PdSiCu) the predicted critical temperature is too low to be observable. This is mainly a consequence of the low density of TLS. However, it would be sufficient to increase the number of TLS by two orders of magnitude in order to obtain an observable effect: A0 102 gives T~ 10 mK in the absence of any pairing induced by phonons. Finally we remark that near a structural transition the density of two-level systems is larger and the superconductivity due to electron pairing induced by TLS could be observable.
~‘
—
4’,J2 [T~(T~ T)] 1/2 —
,
T~
,
A0
‘~
1
.
(15)
The pre-factor is therefore distinct from the BCS case: 5.6 instead of 3.1. In conclusion we find that TLS-induced superconductivity (characterized by a flat TLS-distribution function) eq. (1)leads to expressions for T~,L~(T 0) and ~ near T~,which differ from the corresponding expressions of phonon-induced superconductivity. In particular for weak coupling (2A0 ‘~ I) the critical
336
The authors would like to thank Dr. 0. Béthoux for pertinent suggestions and discussions. References [1] J.L.
Black, Low energy excitations in metallic glasses, in:
Metallic glasses, ed. H.J. Güntherodt (Springer, New York), to be published. [2] K.L. Ngai and T.L. Reinecke, Phys. Rev, B16 (1977) 1077. 131 G.M. Vuji~ié,V.L. Aksenov, N.M. Plakida and S. Stamenkovic, Phys. Lett. 73A (1979) 439. [4] U. Brandt, J. Low Temp. Phys. 2 (1970) 573. [5] P. Fulde, L.L. Hirst and A. Luther, Z. Phys. 230 (1970) 155. [6] D.J. Scalapino, in: Superconductivity, ed. R.D. Parks (Dekker, New York, 1969) Ch. 10. [7] A.A. Abrikosov, Physics 2(1965)5. [8] W.L. McMilan, Phys. Rev. 167 (1968) 331. [91 J.L. Black, B.L. Gyorffy, J. J~ickle,Phil. Mag. B40 (1979) 331.