Pram,~-J. Phys., Vol. 25, No. 5, November 1985, pp. 617-633. © Printed in India.
On the critical temperature of a superconducting system
C M SRIVASTAVA
Department of Physics and Advanced Centre for Research in Electronics, Indian Institute of Technology, Bombay 400 076, India
MS received 28 February 1985; revised 12 August 1985
Abstract. Based on the assumption that in the ground acs state the net gain in energy is equivalent to the repulsive electron-ion and electron-electron Darwin interactions, an expression for T, has been obtained which depends on only a few atomic parameters. The theory provides a criterion for the occurrence of superconductivity and yields satisfactory values of T, for metals and alloys, and ternary chalcogenides and borides. It explains the difference in T, in the crystalline and amorphous states as well as the pressure dependence of T,.
The possibility of occurrence of high temperature superconductivity has been explored.
Keyword$. Superconductivity; amorphous alloys; crystalline alloys; high temperature super- conductors; heavy electron superconductors.
PACS No. 74-10
1. I n t r o d u c t i o n
Despite the phenomenal success of the acs theory (Bardeen et a11957), the criterion for superconductivity is still not understood properly. Consequently, the present Bcs theory has not been very helpful in the search for high temperature superconductors and it has not been possible to cross the upper critical temperature limit of 23.2 K (Gamota 1981) reached in 1973. Even the empirical rules like that of Matthias (1957) regarding the dependence of Tc on e/a in the periodic table showed significant departure when ternary chalcogenides (Fisher et al 1975) and borides (Matthias et al 1977) appeared as new class of superconducting materials. In fact, in cheveral phase ternary chalcogenides the maximum Tc occurs (Yvon 1979) at valence electron concentration per MO6 cluster of 22 which gave 3.7 for the value o f e/a for which Matthias rules do not favour high T¢.
The most widely used relation today for T~ is given by McMillan (1968) for strongly coupled superconductors and is based on the Bcs electron-phonon mechanism. This expression relates 1~ to the normal state properties of the superconductor like the Debye temperature, ®, the electron-phonon interaction dependent parameter, 2 and the coulomb pseudopotential,/~*. Consequently this relation has been used mostly to derive the normal state parameters of the metal from the known values of T¢ and has been of little use in the search for high temperature superconductors.
In a recent review Shrivastava and Sinha (1984) have reviewed the present status of our understanding of the magnetic superconductors.
In the past few years a number of compounds of U (Ottet a11983; Stewart et a11984;
Smith 1984) and Ce (Steglich et al 1979) have shown superconductivity in which 617
618 C M Srivastava
parallel-spin effect interactions are attractive and antiparallel interactions repulsive leading to triplet superconducting energy gaps. This arises due to the quenching of the magnetism of the f-shell ion by the Kondo effect, resulting in a Fermi liquid of extremely heavy electrons. In the conventional 'singlet' superconductors T¢ ,~ 0o ,~ TI while in these unconventional 'triplet' superconductors (Steglich et al 1979) T~ < T I
< 0 n. Here 0 n and T~- are the Debye temperature and the degeneracy temperature of the Fermi liquid respectively. Recently Anderson (1984) has applied group theory to discuss the structure of the 'triplet' superconductors but a microscopic understanding of these states is still not available.
We have recently proposed (Srivastava 1984) that Cooper pairing in the Bcs ground state enables the system to lower the energy arising from the electron-ion and electron- electron Darwin interactions. It has been shown by us that in the Bcs ground state the repulsive Darwin interaction energy vanishes and this gain in energy results in the binding of the Cooper pair. Since the Bcs binding energy is 2Ao where Ao is the energy gap parameter, 2Ao equals the sum of the electron-ion and electron-electron Darwin interaction energies of the two electrons of the pair. As the Darwin interaction energy can be expressed in terms of a few atomic parameters it is easier to calculate T~ using this approach.
It has also been shown that for the Bcs ground state to be established the electrons within the energy shell +hcoc around the Fermi surface should be continuously Bragg- scattered by the lattice planes. This imposes the condition that gmin/2kr should be less than 1 for superconductivity to occur. Here grain is the shortest reciprocal lattice vector and k Fis the Fermi wave-vector. This accounts for the absence of superconductivity in metals of some groups of the periodic table.
We have also shown that odd parity pairing leading to the triplet state is consistent with the microscopic theory of superconductivity based on Darwin interaction.
After a brief outline of our theory we present calculations of T~ for transition series alloys and ternary chalcogenides and show that the agreement with experiment is reasonably good. We also calculate the change in T~ due to pressure and show that this is also in agreement with experiment. Finally we discuss the problem of high temperature superconductivity in §10.
2. Darwin interaction and the pairing hamiitonian
It has been shown recently (Srivastava 1984) that the electron momentum pairing of the type envisaged by the ground state of the Bcs-reduced Hamiltonian leads to the vanishing of the repulsive Darwin interaction and thereby provides the binding energy for the pair. The two electrons of the pair in the coherent state with vanishing total momentum are continuously Bragg-scattered by the lattice. In the presence of phonons the satellites to the reciprocal lattice vectors provide additional planes for scattering. In such a case the Hamiltonian can be written in the following form,
H = ~e~C~Ck+ ~
VkgsCk++g+s~C+-k-g-s~C-k~,Cky
(1)k g, s, k
[gl < 2kr'
where g is the reciprocal lattice vector, g + s are the satellites to g produced by phonons, ek is the Hartree-Fock energy of the electron in state k. The graphical representation of
Critical temperature of a superconducting system 619
-g-s +_9*s
) _ k -k-g-s -k
k+g+~
g._s -g-s
Figure 1. Graphical representation of the scattering process.
the scattering process is shown in figure 1. Vkos is given by
-- N (2mc) 2 ~b~'+a+s(Xl)~b*-k-o-s(X2) E(x1)'p(xl) i 2
-- E(XI - x2)" { p ( X l ) - ptx2)}) ~'-k(X2)~k(X,) dx I dx 2. (2) Here E(x,) is the electric field seen by the ith (i = 1, 2) electron due to the electron-ion interaction and p(xi) is its momentum. E ( x l - x2) is the field due to the Coulomb repulsion interaction, Nk is the normalization constant and is equal to the number of the degenerate pair states with momentum I k, - k ). For the electron wavefunction we take the Bloch form uk(x)exp (ik. x). Using this in (2) we obtain (Srivastava 1984)
Vko~- 1 ~ { Z e f f < t ~ 3 ( x , ) > _ < 6 3 ( x l _ x 2 ) > } " (3) s , 2
Here Zef r is the effective nuclear charge seen by the s-electron.
The energy gap equation is given by
A~ = ~ Vkg~Ck+o+~tC-k-o-,~C-ktCkT. + + (4)
@, $ Igl <2kr Using the acs wave function
~b = r~ (Uk + Vkb~ ) I0 ), (5)
k
we obtain
~0 ~O3c
A~
= NAO)~/kg~UkVk
d~f h6~ c
= VEo~ jo N,(O)ukvk d~.
Here N,(0) is the s-electron density at the Fermi surface. Let fo ~c N~(O)ukvk de = N'k,
then
n / e h \2 where 'l = N ~/N~.
(6)
(7)
(8)
620
C M Srivastava
To obtain the number of degenerate pair states,
Nk,
we sum over all the possible g and s vectors which are responsible for the Bcs ground stateNk=N-~ ~, IS(g)12 f~ F(cO)UkVkdCO.
(9)~g] < 2ke-
Here N is the number of atoms and S is the structure factor per atom
S(9) = n- 1 ~ , exp( - ig" Rj), (10)
J
the summation on j being over the n atoms in the unit cell and the prime in (10) denotes that the summation does not include g = 0. The summation over s gives the term under the integration over the phonon density of state F(~o)
F(co) = ~ ( ~ 6(co- mp~), (11)
,I. J t i
where 2 is the polarization of the phonon. The term
IlkD k
arises due to the coherence factor for the pair state [k T, - k ~ >.Since Ak is independent of k, using the Bcs relation between the gap parameter, A0, at 0 K and T~ in the weak coupling approximation, we obtain,
2A0 = 3-5 k B T~
\ m c /
(12)
~
o~,N~(O)ukvk de
where t / = (13)
N-I Z lS(g)12 f~ F(W)UkV, d~
0 < 2k~
The cut-off phonon frequency o9c in (13) would account for the isotope effect. For hydrogenic s-type wavefunction in the He atom (Bethe and Salpeter 1957),
<63(x~ -
x2)> = ~ < 63 (x~)>. (14) To estimate Tc we may therefore neglect < 6 3 ( x t - x2)> in the first approximation.Then,
lit l
(eh
~2Zerf<f3(x,)>" (15)T~ - 3"5 ka \m-~c /
Thus Tc is directly proportional to the s-electron density at the nucleus arising from the electrons at the Fermi Surface.
3. Criterion for superconductivity
The condition for superconductivity can be obtained from (13). Here r/denotes the ratio of the number of s-like electrons at the Fermi surface which can form coherent pairs to the total number of g + s vectors which provide the lattice planes that can Bragg scatter the pair I k T, - k ~ >. This is shown in figure 1. Note that the scattering of l k T, - k ~ >
to [ k + g + s T, - k - g - s ~ > state is a recoilness process.
Critical temperature of a superconducting system 621 Since each pair of electrons in the coherent state is scattered to a new state by one Bragg plane, the number of available degenerate states to a given pair state is given by the sum of all possible g and s in (4). The coherence factor UkVk in both the numerator and denominator in (13) indicates that of all the possible state only a fraction is occupied to permit continuous Bragg scattering of the pairs. From the condition ~/<~ 1 it follows from (13) that for superconductivity to occur
gmin/2k~-
~< 1. (16)We may assume that r / = 0 if
gmin/2kt:
> 1. It also follows that the smaller the value ofgmin/2kr
the larger is the value oft/and superconductivity is likely to occur more readily.Also the case gmin/2kv ",~ 1 is unlikely to produce superconductivity. It also follows that the higher the s-electron density at the Fermi surface the higher is the expected value of T~.
In order to generate a larger number of degenerate states to a given pair state it is necessary that
x IS(9)12 g < 2k r
should be large. In that case the Fermi surface is cut by BriUouin zone boundaries producing a number of distinct pieces of the Fermi surface. In this case the cross-section for the umklapp scattering process is enhanced which favours superconductivity. This is shown in figure 2 where the degenerate states k, k + g and k + g + s are shown which can be connected through umklapp process. It follows that in case of many zone planes cutting the Fermi surface the contribution from the umklapp process is likely to be large and superconductivity is expected to occur readily as long as there is sufficient density of s-electrons at the Fermi Surface.
We may now distinguish three cases having different possibilities for the relationship between the Fermi surface and the Brillouin zone plane. This is shown in figure 3 in the single band gap model. In (a) the Fermi Surface is far away from the Brillouin zone boundary and in such a case the system will not superconduct as ~ I S (g) I 2 vanishes.
g ~< 2k F
In (b) the Fermi Surface forms a neck. The number of degenerate states connected through umklapp processes remain confined to a very small number of Bragg planes
Figure 2. Off diagonal matrix elements are produced by the Bragg scattering of k to degenerate states k + g and k + g + s using umklapp processes.
622 C M Srivastava
(al (b) (e)
Figure 3. Relationship between the Fermi surface and the Brillouin zone plane in the single band gap model. (a) Fermi surface far away from the Brillouin zone (BZ) (b) BZ cuts the Fermi surface forming a neck (c) BZ cuts the Fermi surface and forms distinct pieces of the Fermi surface.
and superconductivity does not occur. Moreover, it is shown by Ziman (1966) that in such cases the scattering of neck electrons is predominantly by the normal processes. In case (c) the Fermi surface comprises of distinct pieces. It is only in this case that superconductivity can occur as a large number of degenerate states can be formed which are connected by umklapp processes.
4. Triplet pairing
In case of triplet pairing the interaction Hamiltonian may be expressed in the form
Hint = ~
I/kos[C~+o+srC+-k-g-sTC-kTCkT
g , $, k
+ +
+ ½(C[+g+~TC+-k-g-~- Ck++ g +,~C-+~- ~-~ r)
× (C_kF, T-C_krC, i)], (17)
where we have assumed that in all the three states Ms = 0, _ 1 electron pairs have the same coupling strength. Since the Darwin interaction does not include spin-dependent interaction the integral in (2) remains basically unchanged except of course for the pair electron wavefunctions which will have a different symmetry now compared to the singlet state. In this case the expectation values of (63 (x t ) ) and (63 (xt - x 2)
) in
(3) will be modified.It has been experimentally demonstrated that in heavy fermion compounds like CeSu2Si2 the superconductivity neither arises from phonon mechanism nor from d- electron states (Steglich et al 1979). In the presence of Kondo condensates, a triplet pairing is possible and if a pair state like (k T, - k T) is scattered to a state (k + g + s T, - k - g - s T), with s ,,, 0 the effective mass is enhanced accounting for the behaviour of the 'heavy fermion' compounds.
5. Estimate o f Zen
To use (15) for the calculation of T~ it is necessary to have an estimate of Z a v For the atomic case Goudsmit (1933) and Fermi and Segre (1933) have assumed
( 3 3 ( r t ) ) = Z, Zff2(1 - dtr/dnaf)/na3nn~r, (18)
Critical temperature o f a superconducting system 623 where Z~ is the nuclear charge, Z~ is the degree of ionization (unity for neutral atom, 2 for singly ionized atom etc.), a and n are the quantum defect and the principal quantum number respectively, a n is the B6hr radius and nef r is the effective quantum number.
In the metallic case an estimate of (6S(rl)) can be obtained from the Knight shift, K,. However, K, in general, has contributions from four different sources,
K , = K s + Kd + Kor b + t~Kdi a. (19)
Here the contribution K, is from the Fermi contact interaction due to unpaired s-electrons at E r, K a is from the induced hyperfine field due to unpaired d electrons, Kor b is the orbital contribution to Knight shift and 3Kdi a is due to chemical shift. Of these only K~ gives an estimate of ( f i 3 ( r 0 ) . Only in simple cases, therefore, where K, dominates over the other contributions that Knight shift could give an estimate of
( 6 3 ( r 0 ) . In such a case,
n [ e h ~ 2 ~. Ks(O)
T~ = 3"5 k~ ~mcc ] ~lz¢fr [" 8 ~ . "~ (20)
where X~ is the Pauli spin susceptibility, ~ =/z2N (0), and Ks(0) is the Knight shift at normal pressure. Here/z 8 is the B6hr magneton and N (0) is the density of single spin state at the Fermi surface.
The analysis of T~ of elements using (15) shows that in most cases the delta function is given by
3 3 3
Z dr/naunef f. (21)
For hydrogenic ls-type wavefunction in He atom (Bethe and Salpeter 1957) ( 6 3 ( r , - r 2 ) ) = 1
We may assume then that
z, ]
= y ,, [ 1 - 1
T¢ 3.5 \ m c ] k n nagne½ L
~
"Since I/8Zef r ,~ 1, to a first approximation, we may neglect it in the estimate for T~. Thus
~ mc2~t4 Ze'lr = 4-802r/ (22)
- - 3 '
T~ 0"28D/ k8 neff neir where at is the fine structure constant.
We may take the Slater orbital values of nee r as these give reasonably good values for free atoms. For Zef f it is found that the experimental data fits the relation (Srivastava 1984)
Zeff = a(rs - b), (23)
where r~ is the mean electronic separation parameter and a and b are constants which are given as follows.
For systems with valency Zo ~ 4, a = 2-07 and b = 1-28 (class 1) while for Zo >/5, a = 5"29 and b = 1.50 (class 2). It has been shown that there is good agreement with experiment if T~ is calculated using (22) and (23) for most elements and alloys.
624 C M Srivastava
6. Superconductivity in the periodic table
In this section we discuss the occurrence of superconductivity in the periodic table and its explanation on the basis of our model.
6.1 Alkali and noble metals
The alkali metals and noble metals do not show superconductivity. For alkali metal, gmin/2kF calculated on the free electron model and ideal Bcc structure is 1.14 while for noble metals with vcc structure it is 1.11. In both the cases according to condition (16) q = 0.
The Fermi surface of noble metals does show that these touch the Brillouin zones at the centre of the hexagonal face. This, therefore, forms case (b) of figure 3 and hence these metals do not superconduct.
Amongst the alkali metals only Cs has been found to be superconductiving under pressure. The pressure dependence of the Knight shift in alkali metals has been studied by Benedek and Kushida (1958). The Knight shift at atmospheric pressure and room temperature, K(0) and the value of [K(V)/K(O)]v/Vo, the ratio of Knight shift K(V) at the reduced volume V to K (0), that at volume, Vo, corresponding to the atmospheric pressure are given in table 1.
Using (20), it is seen that Cs has largest value of Ks(0) as well as dK/dp, the s-electron density at Fermi surface in this metal is relatively high and this as well as drl/dp increases with pressure. Li and Na can never become superconducting because in these metals K (0) decreases with pressure. Rb may become superconducting at very high pressures since the change in Knight shift for about the same volume change for Rb is only 6.8 ~ , while for Cs it is 41.5 9/0.
6.2 Divalent metals (Z = 2)
Amongst group IIA metals only Be is superconducting. It has a T~ of 0-03 K. Using (22) and (23) we obtain for Be, T~ = 1"43q, so q = 0-021. The low value oft/is supported by the experimental result on Knight shift (Wick et al 1981). Be is probably the only element which has a negative Knight shift indicating that the s-electron contribution from Fermi surface is very small.
The reason that q vanishes in most of group IIA metals is that the Fermi surface just touches the BZ boundary in the first zone and small pockets of holes and electrons extend over several zones (Ziman 1960). This does not assist in the scattering of electrons by umklapp processes.
In metals of group IIB, the q values for Zn, Cd and Hg are 0.47, 0-13 and 0"95 respectively. In Zn and Cd which have hcp structure but depart significantly from the
Table 1. The Knight shift K (0) of alkali metals at room temperature and ambient pressure (Benedek and Kushida 1958). Also given are the pressure dependence of K(0). K(V) is the Knight shift at the reduced volume V/Vo.
Li Na Rb Cs
K(0) x 103 0-249 1-13 6-53 14-9
[K(V)/K(O)]v/~ [0-9884]0.o 2 [0"98692]o.a~ [1'067971o.77 [1"41501o.76
Critical temperature of a superconducting system 625 ideal value of c/a = 1"632 the small values of ~/are not surprising. When c/a departs from the ideal value the Fermi surface elongates in some directions and overlap on the different faces of the Brillouin zones is increased or decreased according to the direction of elongation. In table 2 is included the deviation ofc/a from the ideal value along with the variation of T~ with r/. It follows from this table that r/initially increases with c/a and then begins to decrease.
Hg has a special rhombohedral lattice which generates (Cohen and Heiner, 1970) three sets of reciprocal lattice vectors (100), (110), and (111) for which g/2k r is less than or nearly equal to 1. Hence r/is large and close to the maximum value possible.
It is possible to explain the observed values of T~ in Mg, Ca, Sr and Ba observed in thin films or under pressure. In both the cases the structural changes induce superconductivity. The main difficulty is regarding the estimate of Zef f. Using (23), Zef r is 1"21, 2"84, 4"06, 4"73 and 5"00 for Be, Mg, Ca, Sr and Ba respectively. Except for Be, all other values are large compared to Slater atomic values. It is obvious that (23) cannot be used for these elements. In these cases we have estimated the values of Zef r from the observed values of To. These are given in table 3. It is interesting to note that these values are close to those obtained using Slater rules for the atomic system.
6.3 Trivalent metals (Z = 3)
With three valence electrons per atom we expect one and half zones to be filled thus expecting a larger area of Fermi surface and a larger number of reciprocal lattice vectors
Table 2. Dependence of r/on the deviation of c/a from the ideal value I'632 for hexagonal structure of the metals of group liB.
Be Zn Cd
c/a 1.575 1'861 1.890
A(c/a) --0"057 +0"229 +0"258
T,(K) 0-026 0.88 0-53
r/ 0.021 0-47 0"13
n/latc/a)l 0"30 2.05 0"505
Table 3. Transition temperature in thin film (or under pressure) for metals of group IIA.
Mg Ca Sr Ba
Constraint Thin film Thin film Thin film Pressure
T~(K) 5.5 4.3 3.6 5.4
Observed
Zef f 2.36 2-60 2.58 3'02
(,1 = 1)
Slater atomic 2.85 2'85 2.85 2"85
Zeff
The experimental values of Zef r obtained from ~ using ~/= 1 are in good agreement with the values obtained from Slater rules for atomic orbitals.
626 C M Srivastava
for which g/2k r < 1. For example, for FCC lattice with e/a = 3, G ( l l l ) / 2 k F is 0"78 for (111) and 0.88 for (200). In this case we expect r/to be close to 1. This is borne out by table 4 for all the elements which are superconducting. We also include in table 4 those which are superconducting only at high pressures.
Scandium and yttrium both having hcp structure are superconducting only at high pressure. The reason seems to be the low density of s-electrons state at the Fermi surface at ambient pressure which makes q tend to zero (see eq. 13).
In some cases the experimental values of q exceed 1. This indicates the inadequacy of our calculation specially of Ze~ r.
6.4 Tetravalent metals (Z = 4)
In this group are Ti, Zr and Hf amongst the transition series elements and Si, Ge, Sn and Pb amongst the simple metals. Si and Ge are superconducting only at high pressure. The parameters are given in table 5. Except for Hf the values of r/are close to 1.
The values of T~ are small for the transition series elements but are large for the simple metals. This is due to the lower degree of ionization, Z*, for the sp metals compared to the transition series elements. Si and Ge when made metallic under pressure follow the same trend for Z as other elements of group IVB.
6.5 Multivalent metals (4 < Z ~< 8)
We consider here the elements in group VA to VIIIA, VB and V1B. The elements with magnetic order (Cr, Mn, Fe, Co, Ni) are not superconductors. Besides non-metals like N and O, Pt and Po are non-superconductors, P, As, Sb, Bi, S, Se and Te are superconducting under pressure, Pd is superconducting under thin film form while the rest of the elements are superconducting under normal conditions.
According to the empirical rules of Matthias, amongst the transition series elements T~ has a maximum for e/a equal to 5 or 7. In the alloys the maxima occur at 4.75 and 6.5.
In the periodic table this leads to high T¢ values for groups for elements of V and VIIB.
Table 4. ~ values for trivalent metals.
At Ga In TI La(hcp) Sc* Y*
T,(K) 1.20 1.10 3-40 2,39 4.90 0"3 2-5 Observed
q 0"99 0-97. 1-04 0"97 1-01 0"12 1'36
* Under pressure.
Table 5. Z* and q values for tetravalent metals.
Ti Zr Hf Sn Pb Si t Ge t
T, 0"39 0"52 0"09 3"72 7"19 7"0 5"3
Z* 4 4 4 2'5 2 2 2
q 1"26 0"79 0"18 1.04 0"97 0"91 0-92
* Under pressure.
Critical temperature of a superconducting system 627 The degree o f ionization in the transition series elements depends on the filling of the half-shells o f the crystals field split d-states. In VB and VIIB the electronic ground state is generally nd3(n + l)s z and ndS(n + 1)s 2 respectively. In VB therefore tzg band is half full while in VIIB both t2g and eg bands are half filled. The half filling ensures that the electrons in these bands are more localised and less mobile than the s-electrons, thus leading to smaller degrees of ionization and larger Zdr which accounts for thc high T~.
This is shown in table 6. Also given in this table arc the values of Z* and ~/. All the elements in these groups arc of class 2 type. T~ is found to vary from 325/~K to 9.5 K.
The degree o f ionization is close to the dominant valency in most cases.
Superconductivity in Pt is not observed. For Z* equal to 4, T~ is expected to be 0"58 K. Thus, in this case ~/seems to be vanishingly small. This may be caused by the ground state configuration o f the atom viz 5d ~°6sl which makes the probability of 6s electrons at the Fermi surface small.
The elements in the group VA and VIA though non-superconducting in the normal state show relatively high T~ in the thin film form or at higher pressures. In these cases the values o f ~/listed in table 7 deviate significantly from 1. The low observed values of ~/
indicate that both in Sb and Bi it may be possible to enhance T~ by suitable synthesis programmes to at least twice their present values.
7. Crystalline vs amorphous systems
In the transition from crystalline to amorphous state T~ in simple metals and alloys is often found to be enhanced although the opposite is also known to occur. In Be for
Table 6. Z* and ~/values for the group VB, VIB, VIIB and VIIIB metals.
VB VIB VIIB VIIIB
V Nb Ta Mo W Tc Re Ru Os Rh Ir Pdt
To(K) 5"3 9"5 4"48 0"92 0"01 8'82 1'70 0-49 0-66 325 (~ 11 3"2
Obs x 10 -6
Z* 2"5 3"0 3'5 4 6 3 3'5 4 3"0 6 4"0 3"0 t/ 0.89 1"02 1"10 1"05 1"0 1-00 1"40 1"53 0.56 1"13 0"30 0.93
t in thin film form
Table 7, All the elements in group VA and VIA are non- superconducting but in the thin film form or under pressure these become superconductors. The values of r/ have been deduced using Z*=5.
V A VIA
P As Sb Bi S Sc Te
~(K) 5.8 0.4 3.5 8.5 7-0 6.9 4-5
Obs
Z* 5 5 5 5 5 5 5
r/ 1.37 1-46 0-29 0-45 0.76 1.00 0.6
628 C M Srivastava
example T~ is found to increase from 0"03 K (crystalline) to 9 K (amorphous).
In amorphous metals the evaluation of Zef r for electrons at the Fermi surface is not easy. On account of higher localisation of conduction electron states here we may examine if the observed trend of the variation of T~ in simple metals and transition metals in the periodic table could be accounted for by the Slater rules for the screening constant s for the atomic orbitals. These rules would not be applicable for systems specially with more than one unfilled band but at least the trends may be explained.
In the transition series elements we assume that when the nd subshell is more than half filled the contribution to screening for (n + 1)s electrons from the half filled shell electrons is 1 and not 0.85 as required by Slater rules. Using (22) we then obtain the values of T~ for some amorphous simple metals and transition metals which are given in table 8.
The reason for the disappearance of the peaks in T~ at e/a = 4"75 and 6"5 accruing in crystalline transition metals and alloys on going to the amorphous state is now explained. In the amorphous state Zet r varies continuously with e/a until the subshell is half-filled. The crystal field effects which stabilise the t2g and eg sub-bands in crystalline state disappear in the amorphous state and only the effect of half filling of the d band is observed in this state.
8. Superconductivity in ternary compounds
Several ternary chalcogenides, borides and stannides have been found to be super- conducting with relatively high To. These have become important as these also exhibit long range magnetic order in temperature intervals which overlap the region of the superconducting state. Another distinguishing feature of these compounds is their relatively high critical fields. Ternary compounds contain three elements, each of which occupies a distinct set or sets of crystallographic sites. The unique magnetic and superconducting properties of these compounds is derived from their crystal structure and stoichiometry since T~ depends both on the weight factors of the Bragg planes for which g/2kr ~< 1 as well as on rs. If these weight factors are high q tends to 1.
Table 8. T~ of metals in the amorph- ous state.
T~(K) T~(K)
Zeff q cal Obs
AI 3"5 0'28 7"47 4.8
Ga 3"8 0-28 5'54 8.4
Pb 4.45 0-28 7.12 7.16
Bi 5-10 0'28 12-27 6-I
Y 1.8 3-29 2"82 0'5
Zr 1"95 3.29 3'88 3-0
Nb 2"05 3.29 4-74 5-0
Mo 2"20 3.29 6"30 7.8
Tc 2"35 3-29 8-20 82
Ru 1.8 3.29 2.82 2-0
Be 1 "98 1.00 8'86 9.0
Critical temperature o f a superconducting system 629 We shall examine the dependence of T~ in ternary compounds in the following two classes: (i) Chevral phase compounds MMo6X8 where X is a chalcogen and M is a metal atom including the rare earths. (ii) Tetragonal borides MRh4B4 where M can be Y, Th or one of nine RE atoms.
Many of the REMo6Xa and RERh4B4 compounds exhibit the coexistence of superconductivity and antiferromagnetic order (Johnston and Braun 1982). For the heavy RE members, Tc in RE Rh4B4 compounds varies with the de Gennes factor (g - 1) 2 J( J + 1) where g is Lande' g factor and J is the total angular momentum of the ground state of the rare earth ion in the trivalent state according to Hund's rule. This result is the dilute limit of the Abrikosov-Gorkov (AG) theory that exchange interaction between the spin of the conduction electron and the local magnetic moment on RE reduces Tc below the value Tco which would be obtained in the absence of the exchange interaction. The universal curve predicted by the AC theory fits the data on RERh, B4 compounds with Tco = 11.4 K (Johnston and Braun 1982). Since the decrease in T~ due to the presence of the rare earth spin is accounted for we consider only the examples of nonmagnetic M atoms.
The valencies of X in MMo6Xa have been determined (Fradin et al 1982) and are - 2 - 1.75 and - 1 for S, Se and Te respectively. For boron we assume the valency as - 1.
The values of Tc calculated for the valencies of the different elements in the compounds are given in table 9. The experimental data is taken from Fisher (1978).
It can be seen that the valency of Mo in the molybdenum chalcogenides (MNo6Xs) varies from 4"5 to 6"0. For the sulphides with the metal atom M in groups IA, IIA or liB, the valency of Mo is 5-25 except for Li for which it is 6. With M from Group IB (Cu and Ag) it is 5, l i b (Zn and Cd) it is 5.25 and Group IVA (Sn and Pb) it is 4.5. The difference Ae in electronegatively (Pauling scale) of Mo and M atoms and the valency Z* of the Mo atom in sulphides and selenides are given in table 10.
It can be seen from table 10 that for sulphides there is a clear indication that with increase in Ae, Z* increases. In selenides such a correlation does not occur and for smaller Ae, the values of Z* are generally higher. The effect of electronegativity difference between implanted ions and Mo in implanted Mo films has been studied by Meyer (1976). He concludes that ions with larger values of electronegativity than Mo enhance the value of T~ which is in agreement with our conclusion in selenides.
The calculated values of Tc for some of the non-magnetic members of the rhodium borides are given in table 11. The agreement between the calculated and observed values of Tc in each of the cases is satisfactory.
9. Pressure dependence of T~
From (22) and (23) it is readily shown that dT, F l dr l 4 rs -I
= Tc l_ t/dp ~ K r-~--bJ" (24)
- d p ~
Here K is the compressibility. At present it is difficult to obtain the values of dt//dp since this depends on the effect of pressure on (i) the reciprocal lattice vectors i.e. the change of structure, (ii) the density of occupied s-states at the Fermi surface and (iii) and shielding of the nuclear charge by the valence and conduction electrons for the s-electrons at the Fermi surface. Of these (i) may be the dominating factor in some
630 C M Srivastava
Table 9. T~ in chevral phase molybdenum chalcogenides.
Compound with V(A) 3 T¢ T~
assumed valency (unit cell) Ca[ Obs
Sulphides
N ~ I + L ~ ^ s . 2 s + e - 2 ,,2 . . . . 6 os 834'8 8"98 8"60
M g1.14,-,o6 2 + k i 5 . 2 5 + ~ - 2 "s 822"8 3"39 3'50 La 3 + M o s.2 s + S~ 2 826"9 6-65 7" 1 L 3+LA~5.75+e-2 U 1 . 2 t v a u 6 t-v8 796"1 2"01 2"0
CU~.sMo65 + S~ 2 815"4 12"56 10"8
S ~ 2 . 5 1 u [ ~ 4 . 5 + ~ - 2 " 1 . 2 . . . . 6 ° 8 8 2 9 " 4 1 3 ' 3 2 1 4 ' 2
2 + 4 5 + - 2
Pb Mo6~4 Sa 837'8 18-59 15"2
1 + 5 . 0 + - 2
Agl.6Mo6., t S s 814"7 9"26 9"1
• 1 + 6 + - 2
LI 2 M o 6 S s 863'3 3"39 4"2
Zn 2 + M o T M + S~ z 811"9 3"37 3"6
Cd 2 + MoS.Z 5 + S~ z 827'3 3"92 3"5
Selenides
La3 + Mo65.oo + Se~ 1~-1s 941.8 10-87 11'4
3 + 5.25"~ - 1 7 5
Yt.2Mo 6 Se a ' 907.2 5"61 6"2
3 + 5 0 0 + - 1 . 7 5
LUl.2Mo6' Sea 896"7 7"56 6'2
I + 5 7 5 + - 1 . 7 5
CUI.2M°6" Sea 926'6 5"43 5"9
I + 5 . 5 + - 1 . 7 5
Agl.2Mo6 Sea 911'3 7"08 5"9
S nl.5 , - 0 6 2 . 5 + k A 5 . 2 5 ~ , - I . 7 5 ~,a 936'3 6'35 6"8
2 + 5 5 - 1 . 7 5
Pbl.2Mo6' Sea 945"0 6"39 6'7
Mo~.Ts+ Se~ 1.7~ 884'2 5"70 6"3
Telluride
Mo~ "s + Re2 # +Te~ 1 1041 3"90 3"5
Mixed chalcooenides
$ 0 0 + 2 - I -
MO6' $6 Br2 817.2 12-37 13.8
5 0 0 2 - - 1
Mo6" $6 I2 840"8 13"96 14"0
Mo66 + Te6 Ii~ t 1062 2'91 2'6
metals. It is because of the contribution from (i) and (ii) that metals like Cs become superconducting at high pressures.
The data on pressure dependence of Tc in some metals is given in table 12. The ~ and r~ dependent parts of dT~/dp have been obtained by first calculating the latter from known values of K, T¢ and r~ and then substracting this from the observed dTJdp to obtain the former. The experimental values have been taken from (Narlikar and Ekbote 1983).
It may be assumed that the ~-dependent term will be small if Tc of the metal does not change significantly in going from crystalline to amorphous state. This is indeed observed in the case of Pb for which T~ does not change significantly.
Critical temperature of a superconducting system 631 Table 10. Dependence of the charge
state, Z*, of the Mo atom on the difference of the electronegativity of Mo and M in ternary molybdenum sulphides and selenides.
M.Mo6Ss M~,Mo6Sea
M Ae Z* for Mo Z* for Mo
Pb -0"17 4.5 5'5
Sn 0-20 4"5 5"25
Ag 0-25 5-0 5"5
Cu 0-26 5"0 5-75
Cd 0.47 5.25 - -
Zn 0"55 5-25 - -
Mg 0"83 5"25 - - Lu 0"89 - - 5"25
Y 0"94 - - 5-25
La 1"06 5"25 5"00
Na 1"23 5-25 - -
The electronegativity has been taken in the Pauling scale. In sulphides Z* increases with Ae.
Table 11. T, in non-magnetic rhodium borides.
V(A °.3)
Compound with Per formula T,(K) T,(K)
assumed valency unit Calc. Obs.
ya+Rh~'S+B]- 104"29 10"40 11'3
Lu3+Rh~'2S+B~ - 103"12 12"89 11"7
Th4+ Rh~'2s + BI- 108"12 4'52 4"3
Amongst all the simple metals only TI shows a positive coefficient o f pressure dependence o f T~. It is interesting that T1 assumes all three o f the simple structures bcc, fcc and hcp under different conditions o f temperature and pressure. The analysis o f Tl using pseudopotential theory (Cohen and Heine 1970) indicates that with pressure ~/is expected to change significantly c o m p a r e d to any other metal in the periodic table. This is reflected in o u r estimate o f d~I/dp given in table 8 for this metal.
10. High temperature superconductors
F r o m (22) it follows that T~ for lighter metals would be higher since for these metals ncf r is small. In the crystalline f o r m Tc for both Be and AI is small compared to their a m o r p h o u s state. Like the pressure dependence o f Tc in TI this can also be understood on the pseudopotential theory o f Be and AI. In both these elements the important first set o f 0's for fcc, hop and bec structures falls to the right o f qo and in Be (Cohen and
632 C M Srivastava
Table 12. The pressure dependence o f T, of simple metals and Nb,
Metal
T, dt/
r ~ KT,(rJr, -b)caj
T¢ (cmZ/kG) (dTc/dP)ex p t / d p
K ( x 10- 7) (0 K/atm) (0 K/atm) (0 K/atm)
La 4.9 35 4.34 - 4 - 3 + 0-04
Pb 7.2 23-7 4.1 - 3-8 + 0-30
AI 1.196 14 0-59 - 2 . 7 -2.11
Zn 0-88 16.9 0.44 - 1.6 - I. 16
Sn 3.72 18-8 1.87 - 5 . 0 - 3 . 1 3
Cd 0-56 22-5 0-33 - 1.0 - 0-63
Hg 4-15 34 3-6 - 7 . 2 - 3 . 6
TI 2.38 25 1.64 + 2.7 + 4.34
In 3"39 25 2'29 - 3'6 - 1-21
Nb 9"5 5.7 2"44 -0-28 + 2.16
K is the compressibility. The r, dependent term o f dTddp is calculated using the known values o f r,, b, T, and K and are given in column (4). The ~/dependent term in the last column is obtained by subtracting the r, dependent part from the observed values o f (dTc/dP)ex p given in column (5).
Heine, 1970) these fall far away from qo- Here qo is the value of q where v(q) = ( k [ v [ k + q ) passes through zero. Consequently in the crystalline state ~/is small for Be. In the amorphous state T~. in Be is given by the Zefr value obtained by using Slater rules because the constraint imposed by the fixed set ofg's available in the crystalline form is relaxed and the system can reduce the total Darwin interaction energy by continuous Bragg scattering by several sets of local g's which are available in this state. If such a situation can be brought about in amorphous AI, T~ would be enhanced to about 27 K. AI does show an enhanced T~ of 3.3 K, when quench condensed (Buckel and Hilsch 1954). There is further enhancement in T~ to 8-35 K and 7"3 K when Ge and Si respectively are implanted (Neunier et al 1977) in AI. It should be possible to enhance T~ further in this metal by suitable synthesis procedures.
The A15 compounds show high To. We have outlined the rules by which T~ in alloys can be obtained (Srivastava 1984). In NbaSn, ~, = 2-235 which with nef f = 4 gives a value of 18'68 K which is close to the experimental value of 18.3 K. For Nb3Ge, is
= 2"255 using the value of nef r for Ge viz 3.7, T~ is 24.12 K which is again near the experimental value of 23"0 K. If r~ can be enhanced to a value of 2"40, T~ would be increased to 48"70 K for nef r = 3"7. Amongst the elements in the fourth period leaving aside the alkali metals the only element which has higher r~ than Ge is Ca (r~ = 3"272). It would appear that on adding Ca in NbaGe, if the lattice is not distorted, T~ should be enhanced.
11. Conclusion
A method has been outlined to calculate T~ of metals and alloys in crystalline and amorphous states using the scs theory based on the assumption that Cooper pairing occurs to reduce the Darwin interaction energy. The effect of pressure on T~ has also been explained on the basis of this approach.
Critical temperature of a superconductin 9 system 633 References
Anderson P W 1984 Phys. Rev. B30 1553, 4000
Bardeen J, Cooper L N and Schrieffer J R 1957 Phys. Rev. 108 1175 Benedek G B and Kushida T 1958 J. Phys. Chem. Solids 5 241
Bethe H A and Salpeter E E 1957 in Handbuch der Phys. (Berlin: Springer Verlag) 35/1 181 Buckel W and Hilsch R 1954 Z. Phys. 138 109
Cohen M L and Heiner V 1970 Solid State Phys. 24 184, 264, 411 Fermi E and Segre E 1933 Z. Phys. 82 729
Fisher O 1978 J. Appl. Phys. 16 1
Fisher O, Treyvand A, Cheverel R and Sergent M 1975 Solid State Commun. 17 721
Fradin F Y, Dunlap B D, Shenoy G K and Kinball C W 1982 in Superconductivity in ternary compounds I1 (eds) M B Maple and O Fischer, published under the series "Topics in current physics" (Berlin: Springer- Verlag) 34 201
Gamota G 1981 IEEE Trans. Mao. MAG-17 19 Goudsmit S 1933 Phys. Rev. 43 636
Johnston D C and Braum H F 1982 in Superconductivity in ternary compounds I1 (eds) M B Maple and O Fischer, published under the series "Topics in current physics" (Berlin: Springer-Verlag) 34 11 Matthias B T 1957 Pro0. Low Temp. Phys. (ed.) C J Gorter 2 138
Matthias B T, Corenznit E, Vandenberg J M and Barz H 1977 Proc. Nat. Acad. Sci. USA 74 1334 Matthias B T 1976 in Superconductivity in d- and f- hand metals (ed.) D H Douglass (New York: Plenum Press)
pp. 635
McMillan W L 1968 Phys. Rev. 167 331
Narlikar A V and Ekbote S N 1983 Superconductivity and superconductino materials (New Delhi: South Asian Publishers)
Neunier F, Pfenty P, Lamoise A M, Chaumont J, Bernas H and Cohen C 1977 J. Phys. Lett. 38 L435 Ott H R, Rudigier H, Fisk Z and Smith J L 1983 Phys. Rev. Lett. 50 1595
Shrivastava K N and Sinha K P 1984 Phys. Rep. 115 93 Smith J L 1984 J. Am. Phys. Soc. 29 245
Srivastava C M 1984 Bull. Mater. Sci. 6 273
Steglich F, Aarts J, Brede C D, Liebe W, Meschede D, Franz W and Schafer H 1979 Phys. Rev. Lett. 43 1892 Stewart G R, Fisk Z, Willis J O and Smith J L 1984 Phys. Rev. Lett. 52 679
Wiek W, Nusair M and Vosko S H 1981 Can. J. Phys. 59 585
Yvon K 1979 in Current topics in materials science (ed.) E Kaidis (New York: North Holland) 3 53 Ziman J M 1966 in Fermi surface (eds) W A Harrison and M B Webb (New York: John Wiley) pp. 301 Ziman J M 1960 Electrons and phonons (Oxford: Clarendon Press) pp. 114
