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On the pressure dependence of martensitic and superconducting transition temperature in intermetallic compounds
Solid State Communications, Vol. 28, pp. 17—19. ©Pergamon Press Ltd. 1978. Printed in Great Britain.
0038—1098/78/1001—0017 $02.00/0
ON THE PRESSURE DEPENDENCE OF MARTENS ITIC AND SUPERCONDUCTING TRANSITION TEMPERATURE IN INTERMETALLIC COMPOUNDS S.K. Ghatak I.I.T.,
Kharagpur—2, India and D.K. Ray
Laboratoire Louis N6e1, C.N.R.S., i66x, 380142 Grenoble—cgdex, France Received 27 July 1978 by E.F. Bertaut In order to explain the large pressure dependence of the cubic to tetragonal transition temperature TM in LaAg an expression has been derived from 3TM/3P for a two—fold degenerate electronic band interacting with the tetragonal strain mode. Analo~’ with the pressure dependence of the ferromagnetic transition temperature T~in an itinerant system is pointed out. The nature of variation of the superconducting transition temperature with pressure is also discussed.
Many interinetallic compounds undergo cubic to tetragonal structural transition followed by superconducting transition at a lower tempera— 1. Band Jahn—Teller mechanism has been ture suggested as the cause of this structural
such systems in order to find the origin of the large aTM/aP observed in LaAg and explain the change sign of ~Tc/~P with pressure observed in manyinintermetallic compounds including that of LaAg6,7. For a doubly degenerate electronic band in presence a tetragonal strain one gets the following expression for the structural transi— tion temperature I~ M = ~ (J_1)1/2 I(t~’(~F)2 Dtt(CF)I1/2 kBT D (CF)~ — D (1) 402 where J = J D(EF) with J = ~- ~ 0 is the
transition2. One of the systems which has been extensively studied is LaAg 3. Band calculations on LaAg indicate that the Fermi 1 —xlnx ener~r EF passing through a doubly degenerate e —band might be responsible for the observed c~bic to tetragonal transition in this alloy system. Ghatak et a11 recently derived an
~
analytical expression of the transition temperature TM for the case of a doubly degenerate electronic band and pointed out the analo~rof this case with that of ferromagnetic transition in an itinerant electronic systemS. In order to find out the relationship between the electronic structure and the structural and superconducting transitions, studies are being made on the variation of TM and Tc under pressure for different intermetallic compounds. Pure LaAg did not show any structural transition though it was found to be a superconductor with Tc = 1 K3. But under pressure LaAg shows 6 structural transitions with ~TM/aP ~ 20 K/kbar. This is one of the largest pressure dependence ever observed (~T) 6). 4/aP % 0.2 K/kbar and Tc, on K/kbar the other both increase and -0.15 for hand, Nb3Sn shows and V3Si respectively decrease as a function of pressure. This apparent lack of correlation between the pressure dependence of TM and Tc indicates that variation of the density of states at the 3T~T.~f~ 31J B ~
coupling constante for the tetragonal strain with eg electrons and C is the corresponding elastic constant. Without specifying the exact form of the density of states expression (1) can be rewritten in the form 1/2 kBTM = aB f(EF/B) (2)
(4i~)
where B is the band width, f(EF/B) is a func— tion which depends on the particular form of the density of states and a is a numerical constant. The volume derivative of Equ. (2) 3TM gives TM ~B =
~
TM =
-
TM (~!)_1 ~J ~ W
1 2
+
TM ~f f(EF/B) W
Using the expression for J we get finally TM 2(J—1)
Fern~’ener~rD(EF) with pressure is not the principle source of pressure dependence of T~and Tc. We here, therefore, derive an analytical expression for 3TM/3P and aTc/aP for
I
+
T W
i~l~iaTM~fI av av g av
(4)
TWI
where we have taken D(CF) = 1/B g(c~/B). Since the volume compressibility = — ~,vav/ap we get
v 1 j~— (aBf)2 Ii 3C B ~V 2JKBTM ~
17
1
—x
v
~B
IV
—
1 ~I g
~p
+
TM -~-
~
(5)
18
PRESSURE DEPENDENCE OF TRANSITION TEMPERATURE IN INTERNETALLIC COMPOUNDS
This shows that the pressure dependence of TM depends essentially on three terms e.g. variations of the band width, the elastic constant and the Fermi ener&y CF with pressure. 8 has shovu that V/B aB/~V= — 5/3 and Heine consequently we can write :
aTM
~——
—
(~ 5 x+~~)T 1 ~f M
—
(aBf)2 2JkBTM
cap 1 ~C + 3 X
—
Vol. 28, No.1
inside the bracket associated with the second term in Equ. (6). We should consider the case of LeAg in this context. Since La.Ag is cubic at normal pressure, the effective parameter J should be less than 1 but since structural transition occurs under small pressure 3 must be close to 1. If D(p’) increases rapidly under
(~]
J-g .~I aPI
This is the general analytical expression for the pressure dependence of TM that we get for
pressure then there is expected to be correlation between aTM/~Pand aT
a two fold degenerate band. It is evident from this expression that depending on the relative importance of the different terms ~TM/aP can have both signs i.e., TM can either increase or decrease with pressure. There are two terms in Equ. (6) — one linearly proportional to TM and another inversely proportional to TM 5. to If TM large the first term analogous theiscase of then itinerant ferro— in Equ. (6.) dominates and the sign of ~ magnets depends on the sign of af/~P. It can be shown that for small pressure ag/aP or af/aP an/W % 0, where n is the number of electrons in the band and consequently the expression given by Equ. (6) further simplifies to aTM 5 aBf 1 ac = x TM - 23 kBTM ~ + (7)
other hand, if the variation of the elastic constant C with pressure is large, then aTM/aP and aTc/aP are expected to be uncorrelated as has been found experimentally to be the case for Lakg. In structural phase transition C is expected to decrease with pressure so that we should have aC/aP < 0 which might give aTM/aP for LaAg might be dueincrease to the very value positive. The large of TM small with pressure of TM and possibly large value of aC/aP for this compound. In order to verify the correct— ness of our analysis we suggest the following two experiments 1 — To measure the dependence of TM for LaAg~.~In~ alloy system for different values of x so that one can get aTM/3P for different initial values of TM and 2 — To measure the elastic constant C under pressure for LaAg which will indicate the importance of 1/C 3C/~P in 3TM/aP. We now consider the pressure dependence of Tc. In BCS approximation9 T 0 is given by 1 T0 = 1.114 huD exp (— D(CF)B~ (10)
0/3P. On the
So for compounds having large value of TM~TM should always increase with pressure at least for small applied pressures which has been observed experimentally for lanthanum inter— metallic compounds. For large applied pressure the shift of EF with pressure becomes increasingly important and might be more important than the increase of bandwidth with pressure represented by the tern 5/3 TM ~ _______
aT
It
where WD is the Debye frequency and VB is the effective electron—phonon coupling parameter. In the formalism used earlier we get
1
=
Tc
alnV~~
ln ‘~D + D(EF)VB
I~
Equ. (6). In order to see how f varies with pressure it is necessary to assume a particular form of the density of states. For semielliptical form2we have : 1 — (CF/B)
1
+
yV -
(c~/B)~ ~
+
1
~
D(CF)VB ~ g(C~/B)~
(ii)
with the change of Ep under pressure small. The first term associated with changeisof frequency with pressure can be related to Debye the
This gives : For compounds having large value of
1
X 3 D(EF)VB
The first two terms in this expression are always positive whereas the third term is negative For smalland pressure the fourth the fourth can have term both associated signs.
(CF/B)2lhl~2
laf
B
Gri~neisen constant10. This term is in general larger than the third term and consequently
TM~ ___
(CF/B)
2(3—1) 1
J > 1 and CF/B 1. Consequently the tern inside the bracket is expected to be negative, 2 is a small quantity in such a case. Also, We 1 — might, (CF/B)therefore, have a fairly high and negative value for 1/f af/aP. So TM should at first increase and then decrease as a T function of applied pressure for Such systems. where Such variation from symmetry has been considerations observed fo: a two La3Se4 fold degenerate band is expected to lie near the Fermi ener~’. For compounds having small values of’ TM~ the sign of ~TM/3P depends on that of’ the term
______
—
~1
+ +
(CF/B)21 (CF/B)2~
(9) 1
aTc/3P is in general positive for small pressures (e.g. La be cases where the 3S4 third andterm LaAgis). more But impo~tamt there may and T 0 decreases with pressure (e.g. La3Te ). But with larger applied pressures the fourth term can become more important. This can be states. Taking a semi—elliptical seen if we specify the form of thedensity densityof of states such that 2 2 1/2 D(CF) = ~ (B — CF) (12) TtB we get
Vol. 28, No.1
—
Tc
PRESSURE DEPENDENCE OF TRANSITION TEMPERATURE IN INTERNETALLIC COMPOUNDS
WD + D(CF)VB
VB
3
D(cF~VB— D(CF)VB 1
If the position of’ EF with respect to the density of’ states is such that E~.increases under pressure then the last term makes negative contribution and can become more important than other terms in Equ. (12). Decrease of Tc with pressure for high values of applied pressure observed in many compounds? might be related to this term. The oscillatory behaviour of ~T 0/aP
C/B 2 x ~ (CF/B)
—
(cF/B)~
19 (13)
for LaAg might be related to the complex nature of this term. It will, therefore, be useful to do band calculations for LaAg to determine EF for different values of lattice parameters and thus. get an idea of the relative importance of the last term. Also, it is to be noted that aTc/3P is always linearly proportional to Tc as has been found experimentally?.
REFERENCES
1. 2. 3.
4. 5. 6. 7. 8. 9.
GEATAK, S.K., RAY, D.K., and ¶~ANNOUSC., Phys. Rev. (in press) and the references therein. IABBE, J., and FRIEDEL, 3., J. de Pbysique~~,153 (1966) ; Thid., ~ 303 (1966) Thid., ~, 708 (1966). IRRIG, H., VIGREN, D.T., KUBLER, 3., and METHFESSEL, S., Phys. Rev. B 8, 4525 (1973) BALSTER, H., IHRIG, H., KOCKEL, A., and METHFESSEL,S., Z.physikB2l, 21~1 (1975). HASEGAWA, A., BRE~4ICKER,B. and KUBLER, 3., Z. Physik B22, 231 (1975) ; TANNOUS, C., RAY, D.K., and BELAKHOVSKY, M., J. Phys. F Metal Physics 6, 2091 (1976). EDWARDS, D.M., and WOHLFARTH, E.P., Proc. Roy. Soc. (LondonT A 303, 127 (1968) EDWARDS, L.R., and BARTEL, L.C., Phys. Rev. B ~, 1064 (1972). SCHILLING, J.S., NETHYESSEL, S., and SHELTON, R.N., Sol. Stat. Commun. (1977). SHELTON, R.N., L&WSON, A.C., and JOENSTON, D.C., Mat. Res. Bull. 10, 297 (1975) SHELTON • R .N., MOODENBAUCH, A.R., DERNIER, P .D., and MATTHIAS, B.T., Mat. Res. Bull. 10, 1111 (1975). HEINE, V., Phys. Rev. .j,~, 673 (1967). BARDEEN, J., COOPER, L.N., and SCHRIEFFER, J.R., Phys. Rev. 108, 1175 (1957).
0038—1098/78/1001—0017 $02.00/0
ON THE PRESSURE DEPENDENCE OF MARTENS ITIC AND SUPERCONDUCTING TRANSITION TEMPERATURE IN INTERMETALLIC COMPOUNDS S.K. Ghatak I.I.T.,
Kharagpur—2, India and D.K. Ray
Laboratoire Louis N6e1, C.N.R.S., i66x, 380142 Grenoble—cgdex, France Received 27 July 1978 by E.F. Bertaut In order to explain the large pressure dependence of the cubic to tetragonal transition temperature TM in LaAg an expression has been derived from 3TM/3P for a two—fold degenerate electronic band interacting with the tetragonal strain mode. Analo~’ with the pressure dependence of the ferromagnetic transition temperature T~in an itinerant system is pointed out. The nature of variation of the superconducting transition temperature with pressure is also discussed.
Many interinetallic compounds undergo cubic to tetragonal structural transition followed by superconducting transition at a lower tempera— 1. Band Jahn—Teller mechanism has been ture suggested as the cause of this structural
such systems in order to find the origin of the large aTM/aP observed in LaAg and explain the change sign of ~Tc/~P with pressure observed in manyinintermetallic compounds including that of LaAg6,7. For a doubly degenerate electronic band in presence a tetragonal strain one gets the following expression for the structural transi— tion temperature I~ M = ~ (J_1)1/2 I(t~’(~F)2 Dtt(CF)I1/2 kBT D (CF)~ — D (1) 402 where J = J D(EF) with J = ~- ~ 0 is the
transition2. One of the systems which has been extensively studied is LaAg 3. Band calculations on LaAg indicate that the Fermi 1 —xlnx ener~r EF passing through a doubly degenerate e —band might be responsible for the observed c~bic to tetragonal transition in this alloy system. Ghatak et a11 recently derived an
~
analytical expression of the transition temperature TM for the case of a doubly degenerate electronic band and pointed out the analo~rof this case with that of ferromagnetic transition in an itinerant electronic systemS. In order to find out the relationship between the electronic structure and the structural and superconducting transitions, studies are being made on the variation of TM and Tc under pressure for different intermetallic compounds. Pure LaAg did not show any structural transition though it was found to be a superconductor with Tc = 1 K3. But under pressure LaAg shows 6 structural transitions with ~TM/aP ~ 20 K/kbar. This is one of the largest pressure dependence ever observed (~T) 6). 4/aP % 0.2 K/kbar and Tc, on K/kbar the other both increase and -0.15 for hand, Nb3Sn shows and V3Si respectively decrease as a function of pressure. This apparent lack of correlation between the pressure dependence of TM and Tc indicates that variation of the density of states at the 3T~T.~f~ 31J B ~
coupling constante for the tetragonal strain with eg electrons and C is the corresponding elastic constant. Without specifying the exact form of the density of states expression (1) can be rewritten in the form 1/2 kBTM = aB f(EF/B) (2)
(4i~)
where B is the band width, f(EF/B) is a func— tion which depends on the particular form of the density of states and a is a numerical constant. The volume derivative of Equ. (2) 3TM gives TM ~B =
~
TM =
-
TM (~!)_1 ~J ~ W
1 2
+
TM ~f f(EF/B) W
Using the expression for J we get finally TM 2(J—1)
Fern~’ener~rD(EF) with pressure is not the principle source of pressure dependence of T~and Tc. We here, therefore, derive an analytical expression for 3TM/3P and aTc/aP for
I
+
T W
i~l~iaTM~fI av av g av
(4)
TWI
where we have taken D(CF) = 1/B g(c~/B). Since the volume compressibility = — ~,vav/ap we get
v 1 j~— (aBf)2 Ii 3C B ~V 2JKBTM ~
17
1
—x
v
~B
IV
—
1 ~I g
~p
+
TM -~-
~
(5)
18
PRESSURE DEPENDENCE OF TRANSITION TEMPERATURE IN INTERNETALLIC COMPOUNDS
This shows that the pressure dependence of TM depends essentially on three terms e.g. variations of the band width, the elastic constant and the Fermi ener&y CF with pressure. 8 has shovu that V/B aB/~V= — 5/3 and Heine consequently we can write :
aTM
~——
—
(~ 5 x+~~)T 1 ~f M
—
(aBf)2 2JkBTM
cap 1 ~C + 3 X
—
Vol. 28, No.1
inside the bracket associated with the second term in Equ. (6). We should consider the case of LeAg in this context. Since La.Ag is cubic at normal pressure, the effective parameter J should be less than 1 but since structural transition occurs under small pressure 3 must be close to 1. If D(p’) increases rapidly under
(~]
J-g .~I aPI
This is the general analytical expression for the pressure dependence of TM that we get for
pressure then there is expected to be correlation between aTM/~Pand aT
a two fold degenerate band. It is evident from this expression that depending on the relative importance of the different terms ~TM/aP can have both signs i.e., TM can either increase or decrease with pressure. There are two terms in Equ. (6) — one linearly proportional to TM and another inversely proportional to TM 5. to If TM large the first term analogous theiscase of then itinerant ferro— in Equ. (6.) dominates and the sign of ~ magnets depends on the sign of af/~P. It can be shown that for small pressure ag/aP or af/aP an/W % 0, where n is the number of electrons in the band and consequently the expression given by Equ. (6) further simplifies to aTM 5 aBf 1 ac = x TM - 23 kBTM ~ + (7)
other hand, if the variation of the elastic constant C with pressure is large, then aTM/aP and aTc/aP are expected to be uncorrelated as has been found experimentally to be the case for Lakg. In structural phase transition C is expected to decrease with pressure so that we should have aC/aP < 0 which might give aTM/aP for LaAg might be dueincrease to the very value positive. The large of TM small with pressure of TM and possibly large value of aC/aP for this compound. In order to verify the correct— ness of our analysis we suggest the following two experiments 1 — To measure the dependence of TM for LaAg~.~In~ alloy system for different values of x so that one can get aTM/3P for different initial values of TM and 2 — To measure the elastic constant C under pressure for LaAg which will indicate the importance of 1/C 3C/~P in 3TM/aP. We now consider the pressure dependence of Tc. In BCS approximation9 T 0 is given by 1 T0 = 1.114 huD exp (— D(CF)B~ (10)
0/3P. On the
So for compounds having large value of TM~TM should always increase with pressure at least for small applied pressures which has been observed experimentally for lanthanum inter— metallic compounds. For large applied pressure the shift of EF with pressure becomes increasingly important and might be more important than the increase of bandwidth with pressure represented by the tern 5/3 TM ~ _______
aT
It
where WD is the Debye frequency and VB is the effective electron—phonon coupling parameter. In the formalism used earlier we get
1
=
Tc
alnV~~
ln ‘~D + D(EF)VB
I~
Equ. (6). In order to see how f varies with pressure it is necessary to assume a particular form of the density of states. For semielliptical form2we have : 1 — (CF/B)
1
+
yV -
(c~/B)~ ~
+
1
~
D(CF)VB ~ g(C~/B)~
(ii)
with the change of Ep under pressure small. The first term associated with changeisof frequency with pressure can be related to Debye the
This gives : For compounds having large value of
1
X 3 D(EF)VB
The first two terms in this expression are always positive whereas the third term is negative For smalland pressure the fourth the fourth can have term both associated signs.
(CF/B)2lhl~2
laf
B
Gri~neisen constant10. This term is in general larger than the third term and consequently
TM~ ___
(CF/B)
2(3—1) 1
J > 1 and CF/B 1. Consequently the tern inside the bracket is expected to be negative, 2 is a small quantity in such a case. Also, We 1 — might, (CF/B)therefore, have a fairly high and negative value for 1/f af/aP. So TM should at first increase and then decrease as a T function of applied pressure for Such systems. where Such variation from symmetry has been considerations observed fo: a two La3Se4 fold degenerate band is expected to lie near the Fermi ener~’. For compounds having small values of’ TM~ the sign of ~TM/3P depends on that of’ the term
______
—
~1
+ +
(CF/B)21 (CF/B)2~
(9) 1
aTc/3P is in general positive for small pressures (e.g. La be cases where the 3S4 third andterm LaAgis). more But impo~tamt there may and T 0 decreases with pressure (e.g. La3Te ). But with larger applied pressures the fourth term can become more important. This can be states. Taking a semi—elliptical seen if we specify the form of thedensity densityof of states such that 2 2 1/2 D(CF) = ~ (B — CF) (12) TtB we get
Vol. 28, No.1
—
Tc
PRESSURE DEPENDENCE OF TRANSITION TEMPERATURE IN INTERNETALLIC COMPOUNDS
WD + D(CF)VB
VB
3
D(cF~VB— D(CF)VB 1
If the position of’ EF with respect to the density of’ states is such that E~.increases under pressure then the last term makes negative contribution and can become more important than other terms in Equ. (12). Decrease of Tc with pressure for high values of applied pressure observed in many compounds? might be related to this term. The oscillatory behaviour of ~T 0/aP
C/B 2 x ~ (CF/B)
—
(cF/B)~
19 (13)
for LaAg might be related to the complex nature of this term. It will, therefore, be useful to do band calculations for LaAg to determine EF for different values of lattice parameters and thus. get an idea of the relative importance of the last term. Also, it is to be noted that aTc/3P is always linearly proportional to Tc as has been found experimentally?.
REFERENCES
1. 2. 3.
4. 5. 6. 7. 8. 9.
GEATAK, S.K., RAY, D.K., and ¶~ANNOUSC., Phys. Rev. (in press) and the references therein. IABBE, J., and FRIEDEL, 3., J. de Pbysique~~,153 (1966) ; Thid., ~ 303 (1966) Thid., ~, 708 (1966). IRRIG, H., VIGREN, D.T., KUBLER, 3., and METHFESSEL, S., Phys. Rev. B 8, 4525 (1973) BALSTER, H., IHRIG, H., KOCKEL, A., and METHFESSEL,S., Z.physikB2l, 21~1 (1975). HASEGAWA, A., BRE~4ICKER,B. and KUBLER, 3., Z. Physik B22, 231 (1975) ; TANNOUS, C., RAY, D.K., and BELAKHOVSKY, M., J. Phys. F Metal Physics 6, 2091 (1976). EDWARDS, D.M., and WOHLFARTH, E.P., Proc. Roy. Soc. (LondonT A 303, 127 (1968) EDWARDS, L.R., and BARTEL, L.C., Phys. Rev. B ~, 1064 (1972). SCHILLING, J.S., NETHYESSEL, S., and SHELTON, R.N., Sol. Stat. Commun. (1977). SHELTON, R.N., L&WSON, A.C., and JOENSTON, D.C., Mat. Res. Bull. 10, 297 (1975) SHELTON • R .N., MOODENBAUCH, A.R., DERNIER, P .D., and MATTHIAS, B.T., Mat. Res. Bull. 10, 1111 (1975). HEINE, V., Phys. Rev. .j,~, 673 (1967). BARDEEN, J., COOPER, L.N., and SCHRIEFFER, J.R., Phys. Rev. 108, 1175 (1957).