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On the relation between Tc and structural properties of the A-15 compounds
Solid State Communications, Vol. 19, pp. 1107-1109, 1976.
Pergamon Press.
Printed in Great Britain
ON THE RELATION BETWEEN Te AND STRUCTURAL PROPERTIES OF THE A-I 5 COMPOUNDS L.P. Gor'kov and O.N. Dorokhov L.D. Landau Institute for Theoretical Physics, 142432, Chernogolovka, Moscow region, U.S.S.R. (Received 13 May 1976 by G.S. Zhdanov) It is shown that both the Te and Tm dependences on the deformations and composition for the A-15 compounds can be described at least qualitatively in the quasi-one-dimensional model developed by the authors previously. The mechanism for the superconductivity is supposed to be the same as in the BCS theory. The upper critical field near Te is obtained. IN THIS SHORT NOTE we report on the results of the Tc investigations of the A-15 compounds made on the base of the theory 1'2 developed previously for its structural properties. It is supposed in reference 1 and 2 that only d-electrons in the chains of the transition elements atoms are the carriers. The motion along the chains is quasi-one-dimensional. The 3d-corrections arise from the transfer integrals, B, between orthogonal chains. The scale energy for the three-dimensional spectrum of electrons is T* "" B2[AE (AE the bandwidth). The electronic density of states z,(e) has logarithmic singularities v(e) =
(0 ) ~2 In 32T* ~
(1)
near the two points et = 0 and e2 = -- 2T*. A strong dependence of the martensitic transformation, T~, on the way of preparing samples a allowed us to conclude that in the "best" samples the Fermi level position could be close to one of the two peaks (1). We investigate the Cooper effect on the same suggestion. The equation which defines the critical temperature is of the form: ~
3_
erl
(In reference 2 the interaction with the sublattice distortion Uz of the transition element atoms has been omitted). The interaction R in equation (2) is proportional to (.ta) and is supposed to be short-ranged. Using the single-type pairing of the BCS theory, the superconducting gaps, A ~ , should be chosen to be symmetric with respect to the indices of the zone: A~# = a(°~ + ~(O~x + i3,(0~z.
The evaluation of the right side of equation (2) shows that 7 o) = 0 and ~O are much smaller than a (0. After some lengthy enough calculations one gets a system of three equations which define To. Only the first row in this set of equations for the chain along (x) is written down:
+ k, X(a (y~ + a~)).
ft~
= vp,~.~' +ft,)~,,
(5)
We supposed that the tetragonal distortion A = dl [(a/c) -- 1] = d18 (or an external deformation) concerns ( x , y ) chains only. Therefore, in equation (5) written for cttz) the dependence on A in F-functions is absent. The following notations are used:
(2)
The Green functions of the electron are the sixorder mattrices. The upper indices refer to the type of chains (i.e., along directions (x, y, z)), the Greek indices arise due to the two-fold degeneration of the onedimensional electronic term at the point X. r x-matrices correspond to the vertices of the electron-phonon interaction. Their origin becomes clear if one writes the onedimensional part of the Hamiltonian 1'2 for the z-chain electron in the form:
(4)
, 270 X = m ~-
(0 is the BCS-I)ebye cut off)
32T* P(X) = X 2 + 2X In - - 7 - + 1.32
F(z) =
f
0
dUth(ulzl)ln U
(6)
1- ~ 1 I
The asymptotic expansion of function F(z) are: / 7~'(3__.__))z2. lr2 ,
(3)
where f ~ ) is linear on the displacement of the lattice. For the interactions with the small momentum phonons f i s reduced to the form x'z f~) = dl(exx -- %:~) + ")'uz. 1107
z'~ 1
F(z) =
(7) - - - ~ + 1.32;
z~'l.
1108
STRUCTURAL PROPERTIES OF THE A-15 COMPOUNDS
Equation (5) as a whole are of the form suggested in reference 4 for the set of the weakly coupled linear chains in the Labbe-Friedel model. However, the coupling kl = In
cons
r!
(8)
is not small (~@-~AE ~ 0.1). This coupling removes the degeneracy of superconductivity between three sets of chains and provides a correct symmetry of the energy gap in the cubic or tetragonal phase. Let us enumerate some representative consequencies from equation (5). The dependence of T~ on the weak deformation
ATe _ TO
7~'(3) (~e) 2 ln-' 1-647T*1 12rr2 k~J
(9)
being compared with the corresponding change 2 of Tm:
ATmT ° _
7f(3) ~_~Amf47r 2
(10)
shows precisely the experimentally observed weaker sensitivity of Tc to the stresses applied. (For simplicity, in equations (9) and (10) ~t is supposed to be exactly at the peak (1)). Taking A = d15 with dl --~ 1 eV one can estimate T*, comparing (9) with the experimental data for VaSi [see equation (11)3]. In this way one gets T~,,si --~ 500-800 K. Unlike the behaviour of Tin, the dependence of Te on the composition of/a is rather weak. Even at # >> T° one still gets: In TO ~ In 2 t 2 " / # t / 2 I n647T* It is easy to obtain the G - L equation defining the upper critical field He2 near re. For the cubic phase one has: 481r2Te2 7~'(3)v2 ( ~ c +
-
647T* , , . T ) [ 2 ~ In lrTc] tp(r) ~(r)
Vol. 19, No. 11
= 0.
The factor in the square brackets obtained with the logarithmic accuracy (G = 0.916) corresponds to the peak (1) in the density of states. Altogether, larger T~, smaller v and the logarithmic factor are able to increase the critical fields in the pure NbaSn or V3Si at least by an order of magnitude, as compared with Nb and V. In the presence of the stress applied He2 becomes anisotropic. The last question concerns the change of Te due to the structural transformation. In Fig. 1 the curve for Tm(I.t) and the more fiat one for Te(la) are drawn
I
)~-
?,
Fig. 1. Schematic dependence on the composition of Tc and Tin. The dotted-line parts for each curve are supposed to be calculated in the neglection of another phase transition. The small jump shown for Tc at/l* is the result of the first order character of the martensitic transformation. schematically. The dotted parts of both curves are supposed to be calculated only as a function of the composition # neglecting the possibility of another transition. Actually, at the point of intersection and at ]#[ < #* the developing tetragonal distortion A reduces To. If the martensitic transformation is the first order phase transition (as follows from the symmetry considerations 5), the finite magnitude of A6u*) would result in the jump of Tc at this point. Exactly the same behaviour has been observed for VaSi. 6 This result 6 could therefore be considered as an experimental manifestation of the first order character of the martensitic transformation. The third order group invariant in deformations [ e x , , ( e ~ - ezz) 2 + e ~ ( e z z
- exx) 2 + e~(e~,,, - e ~ ) 2]
in the Landau expansion for the thermodynamic potential is proportional to the interaction d2 introduced in reference 2. According to reference 2 0 = d2/dl ~ 1. The Landau expansion in the tetragonal phase now has the form." = -- 3rr2
ln-~-- + s ~ 1
---~s'
-~
and the spontaneous deformation g.o at the transition is: #--~sp = 20 dl
(12)
The last result is correct at small/s ~ T° . Provided
#/T > 1.91, s"(g/T) becomes positive and the phase transition cannot be described by the Landau type expansion (I 1) (see reference 2 for more details.)
Vol. 19, No. 11
STRUCTURAL PROPERTIES OF THE A-15 COMPOUNDS
The change of the sign of tetragonality with composition has been observed for the alloys NbsSnl-xSbx.7 However, we do not know whether this result 0 2 ) could be affected by the omitted interaction 3,uz with the corresponding optical phonon mode. To estimate the effect of alloying in the systems A3BI_xCx we suppose that atom C just adds or removes one electron in comparison with B-atom. A crude estimation of the corresponding change in the chemical potential 8# can be obtained 2 using v(0) (in eV -1 atom-l): 2x = 8v(0)St~. In this way one gets the correct
1109
experimental value for the characteristic x ~ 0.12 at 8/a ~ 50 K for NbsSn. Thus, it is clear that most of the striking peculiarities of the superconductive properties of the A-15 materials can be explained, at least qualitatively, on the base of the phonon induced BCS-superconductivity. In our opinion there are no evidences in favour of the hypothetical new mechanism of the "enhancement of superconductivity by the lattice instability" repeatedly mentioned in literature (see, however, the concluding remarks in reference 2).
REFERENCES 1.
GOR'KOV L.P., ZhETF6$, 1658 (1973);ZhETFLett. 20,571 (1974).
2.
GOR'KOV LP. & DOROKHOV O.N., J. Low Temp. Phys. 22, 1 (1976).
3.
TESTARDI L.R., PhysicalAcoustics 10, 4 (1973).
4.
BARISIC S. & DEGENNES P., Solid State Commun. 6, 281 (1968).
5.
ANDERSON P.W. & BLOUNT E.I., Phys. Rev. Lett. 14, 217 (1965).
6.
BERTHEL K.H., ECKERT D. & HANDSTEIN A., IV-th Int. Syrup. Reinststoffe in Wissenschaft und Technik. Dresden (1975).
7.
VIELAND L.J., J. Phys. Chem. Solids 31, 1449 (1970).
Pergamon Press.
Printed in Great Britain
ON THE RELATION BETWEEN Te AND STRUCTURAL PROPERTIES OF THE A-I 5 COMPOUNDS L.P. Gor'kov and O.N. Dorokhov L.D. Landau Institute for Theoretical Physics, 142432, Chernogolovka, Moscow region, U.S.S.R. (Received 13 May 1976 by G.S. Zhdanov) It is shown that both the Te and Tm dependences on the deformations and composition for the A-15 compounds can be described at least qualitatively in the quasi-one-dimensional model developed by the authors previously. The mechanism for the superconductivity is supposed to be the same as in the BCS theory. The upper critical field near Te is obtained. IN THIS SHORT NOTE we report on the results of the Tc investigations of the A-15 compounds made on the base of the theory 1'2 developed previously for its structural properties. It is supposed in reference 1 and 2 that only d-electrons in the chains of the transition elements atoms are the carriers. The motion along the chains is quasi-one-dimensional. The 3d-corrections arise from the transfer integrals, B, between orthogonal chains. The scale energy for the three-dimensional spectrum of electrons is T* "" B2[AE (AE the bandwidth). The electronic density of states z,(e) has logarithmic singularities v(e) =
(0 ) ~2 In 32T* ~
(1)
near the two points et = 0 and e2 = -- 2T*. A strong dependence of the martensitic transformation, T~, on the way of preparing samples a allowed us to conclude that in the "best" samples the Fermi level position could be close to one of the two peaks (1). We investigate the Cooper effect on the same suggestion. The equation which defines the critical temperature is of the form: ~
3_
erl
(In reference 2 the interaction with the sublattice distortion Uz of the transition element atoms has been omitted). The interaction R in equation (2) is proportional to (.ta) and is supposed to be short-ranged. Using the single-type pairing of the BCS theory, the superconducting gaps, A ~ , should be chosen to be symmetric with respect to the indices of the zone: A~# = a(°~ + ~(O~x + i3,(0~z.
The evaluation of the right side of equation (2) shows that 7 o) = 0 and ~O are much smaller than a (0. After some lengthy enough calculations one gets a system of three equations which define To. Only the first row in this set of equations for the chain along (x) is written down:
+ k, X(a (y~ + a~)).
ft~
= vp,~.~' +ft,)~,,
(5)
We supposed that the tetragonal distortion A = dl [(a/c) -- 1] = d18 (or an external deformation) concerns ( x , y ) chains only. Therefore, in equation (5) written for cttz) the dependence on A in F-functions is absent. The following notations are used:
(2)
The Green functions of the electron are the sixorder mattrices. The upper indices refer to the type of chains (i.e., along directions (x, y, z)), the Greek indices arise due to the two-fold degeneration of the onedimensional electronic term at the point X. r x-matrices correspond to the vertices of the electron-phonon interaction. Their origin becomes clear if one writes the onedimensional part of the Hamiltonian 1'2 for the z-chain electron in the form:
(4)
, 270 X = m ~-
(0 is the BCS-I)ebye cut off)
32T* P(X) = X 2 + 2X In - - 7 - + 1.32
F(z) =
f
0
dUth(ulzl)ln U
(6)
1- ~ 1 I
The asymptotic expansion of function F(z) are: / 7~'(3__.__))z2. lr2 ,
(3)
where f ~ ) is linear on the displacement of the lattice. For the interactions with the small momentum phonons f i s reduced to the form x'z f~) = dl(exx -- %:~) + ")'uz. 1107
z'~ 1
F(z) =
(7) - - - ~ + 1.32;
z~'l.
1108
STRUCTURAL PROPERTIES OF THE A-15 COMPOUNDS
Equation (5) as a whole are of the form suggested in reference 4 for the set of the weakly coupled linear chains in the Labbe-Friedel model. However, the coupling kl = In
cons
r!
(8)
is not small (~@-~AE ~ 0.1). This coupling removes the degeneracy of superconductivity between three sets of chains and provides a correct symmetry of the energy gap in the cubic or tetragonal phase. Let us enumerate some representative consequencies from equation (5). The dependence of T~ on the weak deformation
ATe _ TO
7~'(3) (~e) 2 ln-' 1-647T*1 12rr2 k~J
(9)
being compared with the corresponding change 2 of Tm:
ATmT ° _
7f(3) ~_~Amf47r 2
(10)
shows precisely the experimentally observed weaker sensitivity of Tc to the stresses applied. (For simplicity, in equations (9) and (10) ~t is supposed to be exactly at the peak (1)). Taking A = d15 with dl --~ 1 eV one can estimate T*, comparing (9) with the experimental data for VaSi [see equation (11)3]. In this way one gets T~,,si --~ 500-800 K. Unlike the behaviour of Tin, the dependence of Te on the composition of/a is rather weak. Even at # >> T° one still gets: In TO ~ In 2 t 2 " / # t / 2 I n647T* It is easy to obtain the G - L equation defining the upper critical field He2 near re. For the cubic phase one has: 481r2Te2 7~'(3)v2 ( ~ c +
-
647T* , , . T ) [ 2 ~ In lrTc] tp(r) ~(r)
Vol. 19, No. 11
= 0.
The factor in the square brackets obtained with the logarithmic accuracy (G = 0.916) corresponds to the peak (1) in the density of states. Altogether, larger T~, smaller v and the logarithmic factor are able to increase the critical fields in the pure NbaSn or V3Si at least by an order of magnitude, as compared with Nb and V. In the presence of the stress applied He2 becomes anisotropic. The last question concerns the change of Te due to the structural transformation. In Fig. 1 the curve for Tm(I.t) and the more fiat one for Te(la) are drawn
I
)~-
?,
Fig. 1. Schematic dependence on the composition of Tc and Tin. The dotted-line parts for each curve are supposed to be calculated in the neglection of another phase transition. The small jump shown for Tc at/l* is the result of the first order character of the martensitic transformation. schematically. The dotted parts of both curves are supposed to be calculated only as a function of the composition # neglecting the possibility of another transition. Actually, at the point of intersection and at ]#[ < #* the developing tetragonal distortion A reduces To. If the martensitic transformation is the first order phase transition (as follows from the symmetry considerations 5), the finite magnitude of A6u*) would result in the jump of Tc at this point. Exactly the same behaviour has been observed for VaSi. 6 This result 6 could therefore be considered as an experimental manifestation of the first order character of the martensitic transformation. The third order group invariant in deformations [ e x , , ( e ~ - ezz) 2 + e ~ ( e z z
- exx) 2 + e~(e~,,, - e ~ ) 2]
in the Landau expansion for the thermodynamic potential is proportional to the interaction d2 introduced in reference 2. According to reference 2 0 = d2/dl ~ 1. The Landau expansion in the tetragonal phase now has the form." = -- 3rr2
ln-~-- + s ~ 1
---~s'
-~
and the spontaneous deformation g.o at the transition is: #--~sp = 20 dl
(12)
The last result is correct at small/s ~ T° . Provided
#/T > 1.91, s"(g/T) becomes positive and the phase transition cannot be described by the Landau type expansion (I 1) (see reference 2 for more details.)
Vol. 19, No. 11
STRUCTURAL PROPERTIES OF THE A-15 COMPOUNDS
The change of the sign of tetragonality with composition has been observed for the alloys NbsSnl-xSbx.7 However, we do not know whether this result 0 2 ) could be affected by the omitted interaction 3,uz with the corresponding optical phonon mode. To estimate the effect of alloying in the systems A3BI_xCx we suppose that atom C just adds or removes one electron in comparison with B-atom. A crude estimation of the corresponding change in the chemical potential 8# can be obtained 2 using v(0) (in eV -1 atom-l): 2x = 8v(0)St~. In this way one gets the correct
1109
experimental value for the characteristic x ~ 0.12 at 8/a ~ 50 K for NbsSn. Thus, it is clear that most of the striking peculiarities of the superconductive properties of the A-15 materials can be explained, at least qualitatively, on the base of the phonon induced BCS-superconductivity. In our opinion there are no evidences in favour of the hypothetical new mechanism of the "enhancement of superconductivity by the lattice instability" repeatedly mentioned in literature (see, however, the concluding remarks in reference 2).
REFERENCES 1.
GOR'KOV L.P., ZhETF6$, 1658 (1973);ZhETFLett. 20,571 (1974).
2.
GOR'KOV LP. & DOROKHOV O.N., J. Low Temp. Phys. 22, 1 (1976).
3.
TESTARDI L.R., PhysicalAcoustics 10, 4 (1973).
4.
BARISIC S. & DEGENNES P., Solid State Commun. 6, 281 (1968).
5.
ANDERSON P.W. & BLOUNT E.I., Phys. Rev. Lett. 14, 217 (1965).
6.
BERTHEL K.H., ECKERT D. & HANDSTEIN A., IV-th Int. Syrup. Reinststoffe in Wissenschaft und Technik. Dresden (1975).
7.
VIELAND L.J., J. Phys. Chem. Solids 31, 1449 (1970).