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Superconducting transition temperature and electronic structure in the pseudobinaries Nb3Al-Nb3Sn and Nb3Sn-Nb3Sb
J. Phys. Chem. Solide
Pergamon
Press 1967. Vol. 28, pp. 889-895.
SUPERCONDUCTING ELECTRONIC
Printed in Great Britain.
TRANSITION
STRUCTURE
TEMPERATURE
AND
IN THE PSEUDOBINARIES
Nb,Al-Nb3 Sn AND Nb, Sn-Nb, Sb and J. B. GOODENOUGH
F. J. BACI-INER Lincoln Laboratory,*
Massachusetts
Institute of Technology,
Lexington,
Massachusetts
02173
H. C. GATOS Department
of Metallurgy,
Massachusetts
Institute of Technology,
Cambridge,
Massachusetts
02139
(Received 25 August 1966; in revisedform 15 December 1966) Abstract-The system NbsA1,Snl _ y exhibits complete solid solubility, a positive Seebeck voltage that increases monatomically from 1.8 to over 12 pV/“K, and a superconducting transition temperature T, that has a broad minimum of T, SW15~3°K at about y = 0.5. The pseudobinary NbsSnl - .Sb,, on the other hand, contains two phases, each with the Al 5 structure, in the interval 0.3 < x < O-8. Whereas the Seebeck voltage of the low-Sb phase is positive and has a Tc > 14”K, the high Sb phase has a negative Seebeck voltage and is not superconducting above 4~2°K. These properties have been interpreted with the aid of a qualitative band-structure scheme. Thus, the rapid drop in T, to below 4.2”K in the system NbsSnl _ ,Sb = is believed to be associated with the filling of the bonding, interchain d, bands of the Nb subarray.
INTRODUCTION
ATTEMPTSto relate variations in superconducting transition temperatures to changes in the occupancy of electron energy bands is greatly hindered in intermetallic compounds by the lack of firm band energies. This problem is especially acute if one or more constituents is a transition metal. However, the possibility of correlating superconducting behavior with band occupancy is of fundamental importance, and the present study represents an attempt in that direction for the systems Nb&&_ II and NbsSn,_ $b,. The compound NbsSn has the beta-tungsten structure (Al 5) and a superconducting transition temperature T, = 18*2”K, which is the highest yet measured. Except for almost pure NbsSb, all compositions have the A15 structure. The number of outer electrons per molecule increases monaton&ally from 18 for NbsAl to 19 for NbsSn to 20 for NbsSb. * Operated with support from the U.S. Air Force. 889
In order to correlate the data with band occupancies, it would suffice to have only a qualitative energy-band scheme. Such a scheme is presented. It has relatively little ambiguity in the region of the Fermi energy, so that meaningful correlations are possible. EXPERIMENTAL All the compounds used in this investigation were made by powder metallurgical techniques from powders or filings of the elements. Niobium and tin powders (99.8 and 99.5 per cent pure respectively) and aluminum and antimony filings (ingot purity-99.999 for both elements) were used. In no case were impurities in the starting materials found to have a significant effect either on the formation of the desired compound or on its properties. The elements were intimately mixed in the desired proportions, cold pressed at 50,000 p.s.i. into ingots, and sintered in purified argon at the desired temperatures. The Nb-Sn-Al compounds were prepared at 1400” and the Nb-Sn-Sb
890
F.
J.
BACHNER,
and H.
J. B. GOODENOUGH
C.
GATOS
compounds at 1200”. The same temperature was used for all the compounds in a particular pscudobinary in order to minimize the effect of transition temperature degradation due to the nontransitionmetal atom vaporizing with subsequent niobiumatom disorder.(l) This problem is especially acute in the compounds of composition Nb,A1,Snl_,. The Seebeck voltage was measured on ingots prepared as described above. The measurements were made at room temperature by establishing a temperature. gradient of a few degrees across the sample (32” vs. room temperature). The temperature and potential differences from one side of the sample to the other were measured by means of two iron-constantan thermocouples placed in thermal and electrical contact with the sample. Even though these measurements were made on porous ingots, the values of Seebeck voltage reported are believed to be accurate to _t 2 per cent. The superconducting transition temperature was measured by the self-inductance method described by REED et al. c2) For these measurements the oscillating coil, the sample, and the thermocouple were placed in intimate contact with a copper block (split to prevent the establishment of circular currents induced by the coil). This established a heat sink so that heat would not be conducted down the thermocouple wires from the bead-thus resulting in a thermocouple-bead temperature lower than the sample temperature. Both copper-constantan and copper-gold: 2.1 per cent cobalt thermocouples were employed. Calibration against samples of known transition temperature permitted an accuracy of of:O.l”K. The lattice parameters were determined from - 325 mesh powder in a 114.6 mm Debye-Sherer camera. Lattice parameters were calculated directly from measured line positions by Cohen’s method programmed for the 7094 computer.
mately 18~4°K at y = 0.1. This behavior has been previously reported by RBED et al.(2) and by ROTHWARP. Because samples of high ahnninum content were difficult to prepare free of the phase NbaAl when reacted at 1400”, the accuracy of the
RESULTS AND DISCUSSION A. Experhnental results All the lattice parameters, Seebeck voltages and superconducting transition temperatures for the two pseudobinaries are presented in Figs. l-6. In the system NbsAl,Snl_ ,,, the lattice parameter (Fig. 1) of the A15 phase shows a positive deviation from Vegard’s law. The variation of the transition temperature in this system (Fig. 2) has a broad minimum at y ZZ 0.5 and a maximum of approxi-
FIG. 2. Superconducting transition temperature vs. composition in the pseudobinary Nbs(A1,SnI - y).
5.300‘
-I
0
0.2
Nb3Al
FIG.
1.
04 (1-y)-c
0.6
0.6
,O
I,
Nb3Sn
Lattice parameters vs. composition pseudobinary Nbs(Al,Snl - y).
in
the
T------
ISI Nb,AI
_I 0
(I-y)*
Nb3Srb
Seebeck voltage was questionable, since the samples were two phase. Therefore, samples were prepared in the system Nb,Al-Nb,Sn, which also has the Al5 structure across the entire pseudobinary, but is single phase when heat treated at 1400”. The Seebeck voltage variation for both pseudobinaries is ~~shown ~~~~ ~~in~~Fia.D 3. The ~~~ similaritv ~~ , of the two curves
PSEUDOBINARIES
NbaAl-NbaSn
is taken as a justilication for the assumption that the NbsAl phase does not significantly perturb the measurement of Seebeck voltage on samples in which that phase is present. 14 1 . Nb3 (AlySn,_y) 12--
0
Nb,
,
(AlySn,_+
w
NOb,AI
AND
891
NbsSn-NbsSb
Although there is no comparable miscibility gap at 1400” in the pseudobinary system Nb3Al,,Snl_Y, there is evidence, from a broadening of the X-ray lines and a widening of the temperature range over which the superconducting transition occurs, of considerable clustering of the tin and aluminum atoms. Apparently there is a pronounced tendency for the nontransition-metal atoms to stabilize like atoms as nearest neighbors within their bodycentered-cubic subarray. However, the fact that there is a miscibility gap in the system NbsSnNbsSb, where the lattice constants are similar, and not in the system NbsAl-NbsSn, where they vary more widely, is striking. It suggests that the miscibility gap is of electronic origin.
NbSSn
(1-y)--
FIG. 3. Seebeck voltage vs. composition in the pseudobinariee Nbs(A1,Snl.,) and Nb4(A1,Sn1 _ y).
a”
_____ [Ie> ,,6 ,
OQC5.300~
2
Ll ZE
a
a ;z
ij
a”
l
0 Nb3Sn FIG.
FIG. 5. Superconducting transition temperature vs. composition in the pseudobinary Nbr(Snl_,Sbz). For x 2 0.8, any transition temperature is < 4*2”K.
5.280
a’
4.
0.2
-5. Nbf$b
:‘=; 0.6
0.4
0.8
X+
Lattice parameter vs. composition pseudobinary Nbs(Snl _ ,Sb,).
Ia Nb3Sb in
the
The pseudobinary system Nb,Sn,_ ,Sb,, on the other hand, exhibits a miscibility gap at 1200” within the interval 0.3 < x < 0.7, as can be seen from the lattice parameters plotted in Fig. 4. The two phases (designated a’ and a” for the tin-rich and antimony-rich, respectively) both have the A15 structure, but differ in lattice parameter.
ii
i\
I -4
1 0
Nb,Sn
1 0.2
I I
I
1
0.4
1
06
1
I ’
o.8
I-
FIG. 6. Seebeck voltage vs. composition binary NbB(Snl- .Sb,).
‘\ 1
I( Nb$ib
in the pseudo-
892
F.
J.
BACHNER,
J.
B.
GOODENOUGH
From Fig. 5, the measurable superconducting transition temperatures (T, > 4.2”K) are clearly associated with the a’ phase. The fact that the transition temperature is not constant across the two-phase region is probably due to a sensitivity of T, on the macroscopic dimensional@ of the superconducting phase. As seen in Fig. 6, the a’ phase has positive Seebeck voltages and the a” phase mostly negative Seebeck voltages. Within the miscibility gap, the measured value reflects weighted contributions
5p--5s
and
H.
C.
GATOS
face of the cube, i.e. a total of six Nb atoms per unit cell. The Nb-Nb distance along the chains is (a,/2) < 2.65 A, which is smaller than the nearest neighbor Nb-Nb distance 2/3(a,/2) = 2.86 A in elemental, b.c.c. niobium. The B atom is always more electronegative than the A atom, and GELLER(*) has noted that the individual atomic radii determined from nearest-neighbor A-B separations are always the same for each atom regardless of their partners in the Al5 structure. Construction of a qualitative energy-band
-
Nb[5]
Nb,Sb
Sb[fl
FIG. 7. Schematic band structure for NbaSb, with 20 outer electrons per molecule. The Nb-Nb sp band contains [3] states per molecule below the Fermi energy E,. Each Nb orbital of left-hand column is weighted by [3] and each Sb orbital of right-hand column is weighted by [l], corresponding to the number of atoms per molecule. Each atomic orbital must also be weighted by [2] for spin degeneracy.
from each phase. It appears that the change from positive to negative Seebeck voltages occurs within the a” phase, but close to the phase boundary. B. Qualitative energy band scheme The cubic A15 structure is formed by compounds having the nominal composition A,B, where A is a transition-metal atom of Group IVb, Vb, or VIb. Although the B atom may be a transition-metal atom, in this investigation it is Al, Sn, or Sb. The B atoms form a body-centeredcubic (b.c.c.) subarray, and the A atoms, which are Nb atoms in this study, form three orthogonal chains within the three orthogonal faces of the b.c.c. unit cell: there are two Nb atoms on each
scheme can proceed as shown in Fig. 7. The leftand right-hand columns, respectively, correspond to the atomic energy levels at the Nb and the B atoms in the presence of crystalline fields. The electronegativity difference between the ions produces the relative stabilization ErE, for the B atoms, where ENI and Er are the Madelung and ionization energies corresponding to the “effective” atomic charges. The constancy of the atomic radii in the A-B direction indicates that this electronegativity difference is large enough that the overlapping s and p orbitals of neighboring A and B atoms always form a filled valence band and an empty conduction band, which are labeled VB and CB in the figure. Each of these bands contains
PSEUDOBINARIES
NbaAl-NbsSn
eight states per molecule, including the twofold spin degeneracy per spatial orbital. The energy gap between these bands is completely overlapped by a Nb-Nb sp band containing the remaining 16 sp states per molecule. Since this band is always partially filled, compounds with the A15 structure are always metallic. The small Nb-Nb distance within the Nbsubarray chains indicates especially strong intrachain bonding and strong interchain bonding. If the z-axis is always taken along a chain, the do = dzr orbitals are directed along the chain to form bonding and antibonding, intrachain do and d,* bands that are split from one another, since there are two Nb atoms per unit length along a chain. Similarly, the d,& orbitals overlap both within and between chains to form separated bonding and antibonding d, and d,* bands. Finally there are nonbonding dzydz9_-ys orbitals that form a very narrow, nonbonding band d,+. There are a total of 9 bonding and 9 antibonding d states per molecule and of 12 nonbonding d,+ states, including the spin degeneracy. The Al5 structure is stabilized by a complete occupancy of the bonding d, band and a complete, or nearly complete, occupancy of the bonding d, band. This means that the Fermi level EF falls between the top of the valence band VB and the bottom of the nonbonding da+ band, If the d, bands are filled in NbsSb, which has 20 outer electrons per molecule, it means that there must be approximately three Nb-Nb sp states below the top of the bonding d, band. An augmented-plane-wave calculation for V,X compounds substantiates the essential features of this qualitative model, particularly in the relevant energy interval about the Fermi level.(5) C. Discussim From the qualitative band model of Fig. 7, the density-of-states vs. energy curves (N(E) vs. E) for the various energy bands in the interval about EF is schematically as shown in Fig. 8. It is possible to locate EF approximately within this schematic diagram by means of the variation of the Seebeck voltage as a function of the electron : atom ratio in the system NbsAl,Sn,_, and Nb,Sb,_,Sb,. From Figs. 3 and 6, the Seebeck voltage is seen to be positive for all values of y and for x < 04; to increase monotonically for all values ofy, reaching
AND
893
NbaSn-NbsSb
a broad maximum at x R 0.2; and to fall off rapidly at the a” phase boundary x % 0.8, becoming negative for x > 0.85. It is reasonable to assume that this change from positive to negative Seebeck voltage is associated with the passage of the Fermi
E,
(Nb3S
EF (Nb3S
DENSITY
OF
STATES
-
Fro. 8. Schematic density of states vs. energy curves for Nb&n. The Fermi energy for Nb&b, corresponding to Fig. 7, is also indicated. The energies Ep and E,, are defined.
energy through the top of the bonding d, band. In the Appendix it is shown that for a rigid-band model, this interpretation requires that in NbsSn,
where the subscripts n and p refer to the broad Nb-Nb sp band and to the narrow d,, band, respectively. The top of the d, band is at EF+E,, the bottom of the broad band at EF- E,,. The constraint on the mobility ratio ~J,J~L~ may be rationalized if the principal charge-carrier scattering mechanism is due to a jumping from one overlapping band to the other. The Bardeen-Copper-Schrieffer (BCS) theory of the superconducting transition temperature To
a94
F.
J.
BACHNER,
J.
B.
GOODENOUGH
predicts a strong dependence of T, on the product N(E,)V, where V is a measure of the strength of the electron-electron interactions responsible for electron-pair formation in the superconducting state. Even if the interaction energy V is not due to a phonon-mediated mechanism in transition-metal compounds,(6*7) nevertheless the transition temperature should reflect the magnitude of the product N(E,)V. Although N(E,)V appears to be relatively constant, it tend8 to be larger for larger N(Ep).(*) It follows that the transition temperature has its maximum value where EF is at a maximum in the N(E) vs. E curve. Figure 8 has therefore been constructed so as to provide such a maximum at x x O-2. This maximum presumably occurs near the top of the d, band, since T, falls below 4.2”K and the Seebeck voltage becomes negative for x > 0% In conclusion, it appears from our correlation of Seebeck voltage, superconducting transition temperature, and band model that the large transition temperature8 in the alloys having A15 structure are to be associated with a partially tilled, narrow, bonding d, band. This band, although restricted to the Nb subarray, is both an intrachain and interchain band, so that it belongs to a three-dimensional array. Further, the fact that the bonding orbitals are nearly filled greatly reduces the possibility that band ferromagnetiam can be a competitive alternative.(O) Finally, the rather surprising fact that the system NbsSn-NbsSb exhibit8 a miscibility gap whereas the system NbsAl-NbsSn doe8 not is compatible with a sharp maximum in N(E,) vs. x at one side of the gap and a deep minimum in N(E,) vs. x at the other, as indicated in Fig. 8. Since the binding energy EB changes with the Fermi level a8 dE,ldEF = N(EF)EF, the free energy of the crystal is optimum where N(E,) is a maximum. The fact that the second phase also has the Al5 structure is presumable due to the gap between bonding and nonbonding d states. As in the case of conventional semiconductors, a structure having filled bonding orbitals and empty antibonding and nonbonding orbitals is relatively stable.
and
H.
C.
GATOS
2. Bmrn T. B., GATOS H. C., LAFLEIJRW. J. and RODDY J. T., Metuliwygv of Advanced Electronic editor G. E. BROCK, Interscience, New York, 71 (1963). ROTHWARF F., personal communication, Frankford Arsenal, Philadelphia, Pennsylvania. GBUER S., Actu Cty&lZogr. 9,885 (1956). MA~HBISS L. F., Phyr. Reu. 138,All2 (1965). MTTHIAS B. T., G~~ALLBT. H. and COMPTON V. B., Rev. Mod. Phys. 35, 1 (1963). GEBALLE T. H.,Reu. Mod. Phys. 36,134 (1964). GEBALLHT. H., MATTHUB ti. T., RBMBIKAJ. P.,
Muter&&,
3. 4. 5. 6. 7.
8.
cLO(IsTON
A.
M..
COMPTON
v.
B..
~~AITAJ.P.
and WILLIAMS H.. J., Physics 2,293.(1966).
9.
GOODENOUGH
_ J. B., J. appl. Phys. (Suppl.) (to be
published 1967).
APPENDIX In order that the proposed band scheme be correct, it is necessary to account for the Seebeck voltages as a function of x, where s is detlned as either
NbsAI1_zlSn,l
or
NbsSnl_,lSb,,
(1)
Since the niobium matrix is fixed for all x, except for changes in lattice parameter, and since the Fermi energy Ea lies, according to the model, only within niobiumsublattice sp and d, bands, a rigid two-band model is assumed for all x. With a two-band model,
pfa+ = 7#+)(kTIE~)
and
yl N
(2/3)M
(3)
The El are measured from the bottom of the broad electron band or the top of the narrow hole band, and for arbitrarily shaped bands in the Sommerfeld model for a metal
E,., = g,,n”
Ep = gpp”
and
(4)
For parabolic bands, Y and n each equal 3; for square bands Y and w are each equal to unity. Therefore $
and
#
(5)
From the rigid-band model of Fig. 8, we have
P =%
-x+An
n = p+a+x
(6)
(7)
where p. is the number of holes in NbaAl if x = xi, in NbsSn if x = xs, and
REFERENCES 1. BACHNER F. J. and GATOS H. C., Truns. Am Inst. Min. Engrs. 226, 1261 (1966).
a=
1
wherex=xr,
a=2
where x = xg (8)
PSEUDOBINARIES
NbsAl-NbsSn
I
([N(&)I,I([N(WI,
0
8 = (!z+?Ml+r),
P = Pa(I
-Y
+ !IJ?
n = (Pa + a)(1 + kr
+V)
a =
1 >Q. > 0 and q1 > 0,
all
the
quantities
N/D fi: aa+aIy-qy2
(24)
(IO)
al/A =
S(l-R)+&(l-v)-(1-r)
(25)
(II)
aZjA =
8(1-R)+(3v-l)p+(3x-l)q-6(a,/A)
(12) (13)
R<
ql, (14)
l&?lya
E,,(l-lryf[3rrq+~~-l)yal}
E ,,a = gnh+4Y,
(15) (16)
Epcr= g&z”
(17)
This means that
1,
+pppap
2 AP&pdo(
-
Y =
(18)
where
(28) (29) (30)
b{T,m,*/T,m,*},
b, ZSQ
b2 ~3
and
(31)
where bI and ba refer to (p,+a)/po at xr and ~a, respectively. Although we expect m,*/ms* N 10, nevertheless N qo,since electrons are primarily we also expect (7./rg). scattered by jumping from one band to the other. It follows that Y -
6 > 0.4
- l)lY%
(27)
+tj,“)
R = r&J&z
-[(3v-l)qf+(v-l)q,alya)+(1-(1-n)y - K37r - l)!I +b(r
> 9u-gcJ
From equations (19) and (22),
Thus the requirements provided
R(1+ (1 -~)!LY
(26)
> SaJA+Kl
q,-0.05, - np,a,
-Tr)
a(1 -R)
-R)+q+q
is very small (~0.01) where a = 1, and is probably still < O-1 where a = 2. Substitution of equations (10) and (11) into equation (4) gives to order ys, +v@+[~v~+++-
-V)fj,a+r(l
In order to conform with the a vs. x curves of Figs. 3 and 6, it is necesssry to have all ai > 0. For the case of parabolic bands (V = n = #), this imposes the following requirements :
S(1 + [N(W&I
(23)
1-R
i&, and + are positive. In fact, 4a = ([N(~~)I,I([N(&)I,
(2), (18) and (20) it follows
a,/A =
Q = qzJ&J,+4
N =
hall~,)(Pa+~)/Pa
where
-&(l
q = qIP,/u -q2 Y = 41 -qcr)h, ga s [40/U -ax.AI/(h+~)l~
(21)
(22) Therefore from equations that to order ys
where we define
Ep =
t =
+t)
(9)
where [JV(&)], is the density of states of the ith band at the Fermi energy, 1x1 Q 1, and we restrict ourselves to powers of 5’. Note that a miscibility gap occurs well before the Seebech voltage changes sign, so that [iV(E,)], > [N(E,)]. may be assumed throughout. Therefore equations (6) and (7) become
E A s E,,,{l
8 E (1 -r&)/(1
=(l+t),
de
+CN(&)lJW
w pax-I-q#
Since
895
NbsSn-NbaSb
where we define
t
An =
AND
1
and
Rz
0.2
(32)
of equation (27) are satisfied or
Y
(33)
which is self-consistent. Note that the constraint on r is not quite so stringent if tr > #, so that the assumption of parabolic bands is not critical. Further, given r w 1, we have 6 =
8 =
W&
Q %
i!ll,
!?x !71
and equation (28) gives the requirement
and
q1 > (19)
Further,
D = -np,,+Pp,
4,
= P&do(l
-
SY+ W)
GO)
0.06
(34)
which is of the same order of magnitude as qo. This does not appear to be a difficult constraint to satisfy, at least near the top of the d band. We conclude that the observed dependence of Seebeck voltage on composition is compatible with our proposed band model.
Pergamon
Press 1967. Vol. 28, pp. 889-895.
SUPERCONDUCTING ELECTRONIC
Printed in Great Britain.
TRANSITION
STRUCTURE
TEMPERATURE
AND
IN THE PSEUDOBINARIES
Nb,Al-Nb3 Sn AND Nb, Sn-Nb, Sb and J. B. GOODENOUGH
F. J. BACI-INER Lincoln Laboratory,*
Massachusetts
Institute of Technology,
Lexington,
Massachusetts
02173
H. C. GATOS Department
of Metallurgy,
Massachusetts
Institute of Technology,
Cambridge,
Massachusetts
02139
(Received 25 August 1966; in revisedform 15 December 1966) Abstract-The system NbsA1,Snl _ y exhibits complete solid solubility, a positive Seebeck voltage that increases monatomically from 1.8 to over 12 pV/“K, and a superconducting transition temperature T, that has a broad minimum of T, SW15~3°K at about y = 0.5. The pseudobinary NbsSnl - .Sb,, on the other hand, contains two phases, each with the Al 5 structure, in the interval 0.3 < x < O-8. Whereas the Seebeck voltage of the low-Sb phase is positive and has a Tc > 14”K, the high Sb phase has a negative Seebeck voltage and is not superconducting above 4~2°K. These properties have been interpreted with the aid of a qualitative band-structure scheme. Thus, the rapid drop in T, to below 4.2”K in the system NbsSnl _ ,Sb = is believed to be associated with the filling of the bonding, interchain d, bands of the Nb subarray.
INTRODUCTION
ATTEMPTSto relate variations in superconducting transition temperatures to changes in the occupancy of electron energy bands is greatly hindered in intermetallic compounds by the lack of firm band energies. This problem is especially acute if one or more constituents is a transition metal. However, the possibility of correlating superconducting behavior with band occupancy is of fundamental importance, and the present study represents an attempt in that direction for the systems Nb&&_ II and NbsSn,_ $b,. The compound NbsSn has the beta-tungsten structure (Al 5) and a superconducting transition temperature T, = 18*2”K, which is the highest yet measured. Except for almost pure NbsSb, all compositions have the A15 structure. The number of outer electrons per molecule increases monaton&ally from 18 for NbsAl to 19 for NbsSn to 20 for NbsSb. * Operated with support from the U.S. Air Force. 889
In order to correlate the data with band occupancies, it would suffice to have only a qualitative energy-band scheme. Such a scheme is presented. It has relatively little ambiguity in the region of the Fermi energy, so that meaningful correlations are possible. EXPERIMENTAL All the compounds used in this investigation were made by powder metallurgical techniques from powders or filings of the elements. Niobium and tin powders (99.8 and 99.5 per cent pure respectively) and aluminum and antimony filings (ingot purity-99.999 for both elements) were used. In no case were impurities in the starting materials found to have a significant effect either on the formation of the desired compound or on its properties. The elements were intimately mixed in the desired proportions, cold pressed at 50,000 p.s.i. into ingots, and sintered in purified argon at the desired temperatures. The Nb-Sn-Al compounds were prepared at 1400” and the Nb-Sn-Sb
890
F.
J.
BACHNER,
and H.
J. B. GOODENOUGH
C.
GATOS
compounds at 1200”. The same temperature was used for all the compounds in a particular pscudobinary in order to minimize the effect of transition temperature degradation due to the nontransitionmetal atom vaporizing with subsequent niobiumatom disorder.(l) This problem is especially acute in the compounds of composition Nb,A1,Snl_,. The Seebeck voltage was measured on ingots prepared as described above. The measurements were made at room temperature by establishing a temperature. gradient of a few degrees across the sample (32” vs. room temperature). The temperature and potential differences from one side of the sample to the other were measured by means of two iron-constantan thermocouples placed in thermal and electrical contact with the sample. Even though these measurements were made on porous ingots, the values of Seebeck voltage reported are believed to be accurate to _t 2 per cent. The superconducting transition temperature was measured by the self-inductance method described by REED et al. c2) For these measurements the oscillating coil, the sample, and the thermocouple were placed in intimate contact with a copper block (split to prevent the establishment of circular currents induced by the coil). This established a heat sink so that heat would not be conducted down the thermocouple wires from the bead-thus resulting in a thermocouple-bead temperature lower than the sample temperature. Both copper-constantan and copper-gold: 2.1 per cent cobalt thermocouples were employed. Calibration against samples of known transition temperature permitted an accuracy of of:O.l”K. The lattice parameters were determined from - 325 mesh powder in a 114.6 mm Debye-Sherer camera. Lattice parameters were calculated directly from measured line positions by Cohen’s method programmed for the 7094 computer.
mately 18~4°K at y = 0.1. This behavior has been previously reported by RBED et al.(2) and by ROTHWARP. Because samples of high ahnninum content were difficult to prepare free of the phase NbaAl when reacted at 1400”, the accuracy of the
RESULTS AND DISCUSSION A. Experhnental results All the lattice parameters, Seebeck voltages and superconducting transition temperatures for the two pseudobinaries are presented in Figs. l-6. In the system NbsAl,Snl_ ,,, the lattice parameter (Fig. 1) of the A15 phase shows a positive deviation from Vegard’s law. The variation of the transition temperature in this system (Fig. 2) has a broad minimum at y ZZ 0.5 and a maximum of approxi-
FIG. 2. Superconducting transition temperature vs. composition in the pseudobinary Nbs(A1,SnI - y).
5.300‘
-I
0
0.2
Nb3Al
FIG.
1.
04 (1-y)-c
0.6
0.6
,O
I,
Nb3Sn
Lattice parameters vs. composition pseudobinary Nbs(Al,Snl - y).
in
the
T------
ISI Nb,AI
_I 0
(I-y)*
Nb3Srb
Seebeck voltage was questionable, since the samples were two phase. Therefore, samples were prepared in the system Nb,Al-Nb,Sn, which also has the Al5 structure across the entire pseudobinary, but is single phase when heat treated at 1400”. The Seebeck voltage variation for both pseudobinaries is ~~shown ~~~~ ~~in~~Fia.D 3. The ~~~ similaritv ~~ , of the two curves
PSEUDOBINARIES
NbaAl-NbaSn
is taken as a justilication for the assumption that the NbsAl phase does not significantly perturb the measurement of Seebeck voltage on samples in which that phase is present. 14 1 . Nb3 (AlySn,_y) 12--
0
Nb,
,
(AlySn,_+
w
NOb,AI
AND
891
NbsSn-NbsSb
Although there is no comparable miscibility gap at 1400” in the pseudobinary system Nb3Al,,Snl_Y, there is evidence, from a broadening of the X-ray lines and a widening of the temperature range over which the superconducting transition occurs, of considerable clustering of the tin and aluminum atoms. Apparently there is a pronounced tendency for the nontransition-metal atoms to stabilize like atoms as nearest neighbors within their bodycentered-cubic subarray. However, the fact that there is a miscibility gap in the system NbsSnNbsSb, where the lattice constants are similar, and not in the system NbsAl-NbsSn, where they vary more widely, is striking. It suggests that the miscibility gap is of electronic origin.
NbSSn
(1-y)--
FIG. 3. Seebeck voltage vs. composition in the pseudobinariee Nbs(A1,Snl.,) and Nb4(A1,Sn1 _ y).
a”
_____ [Ie> ,,6 ,
OQC5.300~
2
Ll ZE
a
a ;z
ij
a”
l
0 Nb3Sn FIG.
FIG. 5. Superconducting transition temperature vs. composition in the pseudobinary Nbr(Snl_,Sbz). For x 2 0.8, any transition temperature is < 4*2”K.
5.280
a’
4.
0.2
-5. Nbf$b
:‘=; 0.6
0.4
0.8
X+
Lattice parameter vs. composition pseudobinary Nbs(Snl _ ,Sb,).
Ia Nb3Sb in
the
The pseudobinary system Nb,Sn,_ ,Sb,, on the other hand, exhibits a miscibility gap at 1200” within the interval 0.3 < x < 0.7, as can be seen from the lattice parameters plotted in Fig. 4. The two phases (designated a’ and a” for the tin-rich and antimony-rich, respectively) both have the A15 structure, but differ in lattice parameter.
ii
i\
I -4
1 0
Nb,Sn
1 0.2
I I
I
1
0.4
1
06
1
I ’
o.8
I-
FIG. 6. Seebeck voltage vs. composition binary NbB(Snl- .Sb,).
‘\ 1
I( Nb$ib
in the pseudo-
892
F.
J.
BACHNER,
J.
B.
GOODENOUGH
From Fig. 5, the measurable superconducting transition temperatures (T, > 4.2”K) are clearly associated with the a’ phase. The fact that the transition temperature is not constant across the two-phase region is probably due to a sensitivity of T, on the macroscopic dimensional@ of the superconducting phase. As seen in Fig. 6, the a’ phase has positive Seebeck voltages and the a” phase mostly negative Seebeck voltages. Within the miscibility gap, the measured value reflects weighted contributions
5p--5s
and
H.
C.
GATOS
face of the cube, i.e. a total of six Nb atoms per unit cell. The Nb-Nb distance along the chains is (a,/2) < 2.65 A, which is smaller than the nearest neighbor Nb-Nb distance 2/3(a,/2) = 2.86 A in elemental, b.c.c. niobium. The B atom is always more electronegative than the A atom, and GELLER(*) has noted that the individual atomic radii determined from nearest-neighbor A-B separations are always the same for each atom regardless of their partners in the Al5 structure. Construction of a qualitative energy-band
-
Nb[5]
Nb,Sb
Sb[fl
FIG. 7. Schematic band structure for NbaSb, with 20 outer electrons per molecule. The Nb-Nb sp band contains [3] states per molecule below the Fermi energy E,. Each Nb orbital of left-hand column is weighted by [3] and each Sb orbital of right-hand column is weighted by [l], corresponding to the number of atoms per molecule. Each atomic orbital must also be weighted by [2] for spin degeneracy.
from each phase. It appears that the change from positive to negative Seebeck voltages occurs within the a” phase, but close to the phase boundary. B. Qualitative energy band scheme The cubic A15 structure is formed by compounds having the nominal composition A,B, where A is a transition-metal atom of Group IVb, Vb, or VIb. Although the B atom may be a transition-metal atom, in this investigation it is Al, Sn, or Sb. The B atoms form a body-centeredcubic (b.c.c.) subarray, and the A atoms, which are Nb atoms in this study, form three orthogonal chains within the three orthogonal faces of the b.c.c. unit cell: there are two Nb atoms on each
scheme can proceed as shown in Fig. 7. The leftand right-hand columns, respectively, correspond to the atomic energy levels at the Nb and the B atoms in the presence of crystalline fields. The electronegativity difference between the ions produces the relative stabilization ErE, for the B atoms, where ENI and Er are the Madelung and ionization energies corresponding to the “effective” atomic charges. The constancy of the atomic radii in the A-B direction indicates that this electronegativity difference is large enough that the overlapping s and p orbitals of neighboring A and B atoms always form a filled valence band and an empty conduction band, which are labeled VB and CB in the figure. Each of these bands contains
PSEUDOBINARIES
NbaAl-NbsSn
eight states per molecule, including the twofold spin degeneracy per spatial orbital. The energy gap between these bands is completely overlapped by a Nb-Nb sp band containing the remaining 16 sp states per molecule. Since this band is always partially filled, compounds with the A15 structure are always metallic. The small Nb-Nb distance within the Nbsubarray chains indicates especially strong intrachain bonding and strong interchain bonding. If the z-axis is always taken along a chain, the do = dzr orbitals are directed along the chain to form bonding and antibonding, intrachain do and d,* bands that are split from one another, since there are two Nb atoms per unit length along a chain. Similarly, the d,& orbitals overlap both within and between chains to form separated bonding and antibonding d, and d,* bands. Finally there are nonbonding dzydz9_-ys orbitals that form a very narrow, nonbonding band d,+. There are a total of 9 bonding and 9 antibonding d states per molecule and of 12 nonbonding d,+ states, including the spin degeneracy. The Al5 structure is stabilized by a complete occupancy of the bonding d, band and a complete, or nearly complete, occupancy of the bonding d, band. This means that the Fermi level EF falls between the top of the valence band VB and the bottom of the nonbonding da+ band, If the d, bands are filled in NbsSb, which has 20 outer electrons per molecule, it means that there must be approximately three Nb-Nb sp states below the top of the bonding d, band. An augmented-plane-wave calculation for V,X compounds substantiates the essential features of this qualitative model, particularly in the relevant energy interval about the Fermi level.(5) C. Discussim From the qualitative band model of Fig. 7, the density-of-states vs. energy curves (N(E) vs. E) for the various energy bands in the interval about EF is schematically as shown in Fig. 8. It is possible to locate EF approximately within this schematic diagram by means of the variation of the Seebeck voltage as a function of the electron : atom ratio in the system NbsAl,Sn,_, and Nb,Sb,_,Sb,. From Figs. 3 and 6, the Seebeck voltage is seen to be positive for all values of y and for x < 04; to increase monotonically for all values ofy, reaching
AND
893
NbaSn-NbsSb
a broad maximum at x R 0.2; and to fall off rapidly at the a” phase boundary x % 0.8, becoming negative for x > 0.85. It is reasonable to assume that this change from positive to negative Seebeck voltage is associated with the passage of the Fermi
E,
(Nb3S
EF (Nb3S
DENSITY
OF
STATES
-
Fro. 8. Schematic density of states vs. energy curves for Nb&n. The Fermi energy for Nb&b, corresponding to Fig. 7, is also indicated. The energies Ep and E,, are defined.
energy through the top of the bonding d, band. In the Appendix it is shown that for a rigid-band model, this interpretation requires that in NbsSn,
where the subscripts n and p refer to the broad Nb-Nb sp band and to the narrow d,, band, respectively. The top of the d, band is at EF+E,, the bottom of the broad band at EF- E,,. The constraint on the mobility ratio ~J,J~L~ may be rationalized if the principal charge-carrier scattering mechanism is due to a jumping from one overlapping band to the other. The Bardeen-Copper-Schrieffer (BCS) theory of the superconducting transition temperature To
a94
F.
J.
BACHNER,
J.
B.
GOODENOUGH
predicts a strong dependence of T, on the product N(E,)V, where V is a measure of the strength of the electron-electron interactions responsible for electron-pair formation in the superconducting state. Even if the interaction energy V is not due to a phonon-mediated mechanism in transition-metal compounds,(6*7) nevertheless the transition temperature should reflect the magnitude of the product N(E,)V. Although N(E,)V appears to be relatively constant, it tend8 to be larger for larger N(Ep).(*) It follows that the transition temperature has its maximum value where EF is at a maximum in the N(E) vs. E curve. Figure 8 has therefore been constructed so as to provide such a maximum at x x O-2. This maximum presumably occurs near the top of the d, band, since T, falls below 4.2”K and the Seebeck voltage becomes negative for x > 0% In conclusion, it appears from our correlation of Seebeck voltage, superconducting transition temperature, and band model that the large transition temperature8 in the alloys having A15 structure are to be associated with a partially tilled, narrow, bonding d, band. This band, although restricted to the Nb subarray, is both an intrachain and interchain band, so that it belongs to a three-dimensional array. Further, the fact that the bonding orbitals are nearly filled greatly reduces the possibility that band ferromagnetiam can be a competitive alternative.(O) Finally, the rather surprising fact that the system NbsSn-NbsSb exhibit8 a miscibility gap whereas the system NbsAl-NbsSn doe8 not is compatible with a sharp maximum in N(E,) vs. x at one side of the gap and a deep minimum in N(E,) vs. x at the other, as indicated in Fig. 8. Since the binding energy EB changes with the Fermi level a8 dE,ldEF = N(EF)EF, the free energy of the crystal is optimum where N(E,) is a maximum. The fact that the second phase also has the Al5 structure is presumable due to the gap between bonding and nonbonding d states. As in the case of conventional semiconductors, a structure having filled bonding orbitals and empty antibonding and nonbonding orbitals is relatively stable.
and
H.
C.
GATOS
2. Bmrn T. B., GATOS H. C., LAFLEIJRW. J. and RODDY J. T., Metuliwygv of Advanced Electronic editor G. E. BROCK, Interscience, New York, 71 (1963). ROTHWARF F., personal communication, Frankford Arsenal, Philadelphia, Pennsylvania. GBUER S., Actu Cty&lZogr. 9,885 (1956). MA~HBISS L. F., Phyr. Reu. 138,All2 (1965). MTTHIAS B. T., G~~ALLBT. H. and COMPTON V. B., Rev. Mod. Phys. 35, 1 (1963). GEBALLE T. H.,Reu. Mod. Phys. 36,134 (1964). GEBALLHT. H., MATTHUB ti. T., RBMBIKAJ. P.,
Muter&&,
3. 4. 5. 6. 7.
8.
cLO(IsTON
A.
M..
COMPTON
v.
B..
~~AITAJ.P.
and WILLIAMS H.. J., Physics 2,293.(1966).
9.
GOODENOUGH
_ J. B., J. appl. Phys. (Suppl.) (to be
published 1967).
APPENDIX In order that the proposed band scheme be correct, it is necessary to account for the Seebeck voltages as a function of x, where s is detlned as either
NbsAI1_zlSn,l
or
NbsSnl_,lSb,,
(1)
Since the niobium matrix is fixed for all x, except for changes in lattice parameter, and since the Fermi energy Ea lies, according to the model, only within niobiumsublattice sp and d, bands, a rigid two-band model is assumed for all x. With a two-band model,
pfa+ = 7#+)(kTIE~)
and
yl N
(2/3)M
(3)
The El are measured from the bottom of the broad electron band or the top of the narrow hole band, and for arbitrarily shaped bands in the Sommerfeld model for a metal
E,., = g,,n”
Ep = gpp”
and
(4)
For parabolic bands, Y and n each equal 3; for square bands Y and w are each equal to unity. Therefore $
and
#
(5)
From the rigid-band model of Fig. 8, we have
P =%
-x+An
n = p+a+x
(6)
(7)
where p. is the number of holes in NbaAl if x = xi, in NbsSn if x = xs, and
REFERENCES 1. BACHNER F. J. and GATOS H. C., Truns. Am Inst. Min. Engrs. 226, 1261 (1966).
a=
1
wherex=xr,
a=2
where x = xg (8)
PSEUDOBINARIES
NbsAl-NbsSn
I
([N(&)I,I([N(WI,
0
8 = (!z+?Ml+r),
P = Pa(I
-Y
+ !IJ?
n = (Pa + a)(1 + kr
+V)
a =
1 >Q. > 0 and q1 > 0,
all
the
quantities
N/D fi: aa+aIy-qy2
(24)
(IO)
al/A =
S(l-R)+&(l-v)-(1-r)
(25)
(II)
aZjA =
8(1-R)+(3v-l)p+(3x-l)q-6(a,/A)
(12) (13)
R<
ql, (14)
l&?lya
E,,(l-lryf[3rrq+~~-l)yal}
E ,,a = gnh+4Y,
(15) (16)
Epcr= g&z”
(17)
This means that
1,
+pppap
2 AP&pdo(
-
Y =
(18)
where
(28) (29) (30)
b{T,m,*/T,m,*},
b, ZSQ
b2 ~3
and
(31)
where bI and ba refer to (p,+a)/po at xr and ~a, respectively. Although we expect m,*/ms* N 10, nevertheless N qo,since electrons are primarily we also expect (7./rg). scattered by jumping from one band to the other. It follows that Y -
6 > 0.4
- l)lY%
(27)
+tj,“)
R = r&J&z
-[(3v-l)qf+(v-l)q,alya)+(1-(1-n)y - K37r - l)!I +b(r
> 9u-gcJ
From equations (19) and (22),
Thus the requirements provided
R(1+ (1 -~)!LY
(26)
> SaJA+Kl
q,-0.05, - np,a,
-Tr)
a(1 -R)
-R)+q+q
is very small (~0.01) where a = 1, and is probably still < O-1 where a = 2. Substitution of equations (10) and (11) into equation (4) gives to order ys, +v@+[~v~+++-
-V)fj,a+r(l
In order to conform with the a vs. x curves of Figs. 3 and 6, it is necesssry to have all ai > 0. For the case of parabolic bands (V = n = #), this imposes the following requirements :
S(1 + [N(W&I
(23)
1-R
i&, and + are positive. In fact, 4a = ([N(~~)I,I([N(&)I,
(2), (18) and (20) it follows
a,/A =
Q = qzJ&J,+4
N =
hall~,)(Pa+~)/Pa
where
-&(l
q = qIP,/u -q2 Y = 41 -qcr)h, ga s [40/U -ax.AI/(h+~)l~
(21)
(22) Therefore from equations that to order ys
where we define
Ep =
t =
+t)
(9)
where [JV(&)], is the density of states of the ith band at the Fermi energy, 1x1 Q 1, and we restrict ourselves to powers of 5’. Note that a miscibility gap occurs well before the Seebech voltage changes sign, so that [iV(E,)], > [N(E,)]. may be assumed throughout. Therefore equations (6) and (7) become
E A s E,,,{l
8 E (1 -r&)/(1
=(l+t),
de
+CN(&)lJW
w pax-I-q#
Since
895
NbsSn-NbaSb
where we define
t
An =
AND
1
and
Rz
0.2
(32)
of equation (27) are satisfied or
Y
(33)
which is self-consistent. Note that the constraint on r is not quite so stringent if tr > #, so that the assumption of parabolic bands is not critical. Further, given r w 1, we have 6 =
8 =
W&
Q %
i!ll,
!?x !71
and equation (28) gives the requirement
and
q1 > (19)
Further,
D = -np,,+Pp,
4,
= P&do(l
-
SY+ W)
GO)
0.06
(34)
which is of the same order of magnitude as qo. This does not appear to be a difficult constraint to satisfy, at least near the top of the d band. We conclude that the observed dependence of Seebeck voltage on composition is compatible with our proposed band model.