- Email: [email protected]
Effect of intercalation on structural instability and superconductivity of layered 2H-type NbSe2 and NbS2
Pergamon
\
I
.I. Phys. Chum Solids Vol 57, Nos 6-8. pp. 1091-1096, 1996 Copyright 0 1996 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0022-3697196 0.00
si5.00+
EFFECT OF INTERCALATION ON STRUCTURAL INSTABILITY AND SUPERCONDUCTIVITY OF LAYERED 2H-TYPE NbSe2 AND NbS2 KAZUKO
MOTIZUKIt,
YOSHIMASA
NISHIO&
NAOSHI
MASAFUMI
SHIRAIS and
SUZUKIJ
TDepartment of Applied Physics, Faculty of Science, Okayama University of Science, Ridai-cho I-1, Okayama 700, Japan fDepartment of Material Physics, Faculty of Engineering Science, Osaka University, Machikaneyama-cho 1-3, Toyonaka 560, Japan (Received 28 May 1995; accepted in revisedform 31 May 1995)
Abstract-Effects ofintercalation on the lattice instability and the superconductivity ofNbSet and NbSz are studied in the framework of the rigid band approximation. In the case of doping acceptor-type intercalants into NbSe2, the point of lattice instability in q-space is moved to q = rM from q = ZI’M. Further, the frequency renormalization is enhanced and the CDW transition temperature is expecte d to increase. In the case of doping donor-type intercalants, on the other hand, the frequency renormalization is weakened and the lattice instability is suppressed. The same tendency is found also in NbS2. The effect of intercalation on the superconducting transition temperature T, of NbS2 is investigated by calculating the functional derivative of T, with respect to the spectral function. The donor-type intercalant lowers T, as observed in the case of alkali-metal doping. On the other hand, the acceptor-type intercalant raises T, as long as the lattice instability does not take place. Keyworcis: A. chalcogenides, D. superconductivity.
D. charge density wave, D. electronic structure, D. lattice dynamics,
1. INTRODUCTION
Niobium and tantalum dichalcogenides, 2H-MXz (M = Nb,Ta; X = S, Se), take a layered structure, which consists of a sequence of X-M-X sandwich layers (see Fig. 1). Many atoms and molecules can be intercalated into the so-called van der Waals gap layers between the adjacent sandwiches and hence they have been important materials as mother crystals of intercalation compounds. These four compounds have attracted much interest also from an academic point of view because all the compounds except NbS2 reveal charge density wave (CDW) transition accompanying structural transition (lattice instability) [l, 21. Furthermore, all of these four compounds become superconductors at low temperatures [3]. The electronic band structures of these compounds have been calculated by first-principles methods such as the APW method [4] and the layer method [5]. According to their results the electronic band structures of the four compounds are nearly the same. We have reproduced the first-principles band structures by a tight-binding (TB) fitting method with use of Slater-Koster transfer integrals t(ddo), t(dpa), etc. as
§Present address: MEC Lab., Daikin Industries Co. Ltd., Miyukigaoka 3, Tsukuba 305, Japan.
fitting parameters [6,7]. We have tried to reproduce as precisely as possible the two d-p hybridization bands, which cross the Fermi level. The density of states (DOS) of the d-p hybridization bands obtained by our TB calculation is shown in Fig. 2. There is certainly a non-negligible contribution from the p component of S or Se atoms, but the main component is the Nb 4d state. As seen from Fig. 2 a sharp peak exists in the DOS curve and the Fermi level EF is located at the higher energy side of the peak. The value of the DOS at EF, N(Er), is 2.56 states/(eV - spin. unit cell), which agrees fairly well with 2.6 f 0.4 states/ (eV . spin * unit cell) determined by NMR measurements [8] or 3.0 states/(eV - spin - unit cell) obtained by the layer method [5]. We have developed the microscopic theory of electron-phonon interaction and lattice dynamics for NbSez and NbS2 [6, 71on the basis of our realistic TB bands. We have found that the characteristic dependence of electron-phonon interaction on phonon mode and wave vector, in addition to the effect of the Fermi surface nesting, causes significant frequency renormalization (softening) for the C, phonon mode around q = iI’M. The CDW transition in NbSez is driven by complete softening of this q = $l’M Ci phonon. In NbS2, on the other hand, the frequency renormalization of the Cl phonon is not
1091
1092
K. MOTIZUKI et al. The results show that the electron-phonon interaction and the renormalized phonon frequency play significant roles in raising the superconducting transition temperature of NbS2 (T, = 6 K) and in explaining the temperature dependence of the electrical and thermal resistivities. In the present paper we investigate the effect of intercalation on the lattice instability of NbSez and NbS2 and the superconductivity of NbS2 in the framework of the rigid band approximation.
vl
2. LATTICE INSTABILITY
2.1.
Phonon anomaly
The free energy change due to lattice distortion is expressed generally by [2]:
where u:(q) denotes Fourier component of the displacement of the pth atom in the unit cell along the (Y direction (o = x, y, z) and x$(q), which is called the generalized electronic susceptibility, is expressed in the following form:
a Fig. 1.The crystal structure of MX2. large enough to cause the lattice instability. These results are consistent with the observations. Furthermore we have studied the superconductivity [7] and the transport properties, such as electrical and thermal resistivities [9] of NbSz, by using the electron-phonon interaction and the renormalized phonon frequencies. I
I -total
_
-------d
state
P $.tate
-7
-a ENERGY
(eY)
Fig. 2. The electronic DOS of NbXz
x$(q)
= 2xxg’““(nk,n’k n,n’ k xf&‘k-q)
- q)gV4(nk,n’k - q)*
-f(‘%k) (2) En,
-
En’k-q
’
Here Enk represents the one electron band energy (n denotes the band index) of the undistorted structure, ,f(&k) is the Fermi distribution function, and gFa(nk, n’k - q) represents the strength of the coupling between two electronic states (nk) and (n’k -9) caused by displacement of the pth atom in the (Y direction. The quantity g““(nk, n’k - q) is called the electron-lattice coupling coefficient. In the TB model, the electron-lattice interaction arises from the modulation of transfer energies due to lattice distortion, and gw(nk, n’k - q) is expressed explicitly in terms of the derivatives of Slater-Koster transfer integrals and the coefficients of transformation which diagonalizes the undistorted TB Hamiltonian [2]. We have estimated the derivatives of the Slater-Koster transfer integrals by using the semiempirical rule that the transfer integrals are proportional to R-” where R is the inter-atomic distance. We have used n = 5 for transfer integrals between d orbitals and n = 2 for those between d and p orbitals. We have calculated gpa(nk,n’k - q) for the two conduction bands which cross the Fermi level. The calculated gpa(nk,n’k - q) depends remarkably on wave-vectors, k and q, and also on mode of atomic displacements or vibrations. For q = rM or l?K
Effect of intercalation on instability and superconductivity
1093
direction, the coupling coefficients caused by the Nb atom displacements in the c plane are larger than those due to the other modes. Particularly, the coupling coefficients due to the chalcogen atom displacements are quite small. Next we have calculated the phonon frequencies of NbSz and NbSez by diagonalizing the dynamical matrix 0$(q) which is given by D;!(q) = R;!(q) + x$(q). >
E2g
30 w AI, z
g
W
10 Bl, E2g n
r
M WAVE-VECTOR
b)
NbS2
50 AI,
2
40 EZC
0
M
r WAVE-VECTOR
Fig. 3. The phonon dispersion curves along the pM direction: (a) NbSe* and (b) NbS2. The neutron scattering data of NbSe, are taken from Ref. [lo], the Raman scattering data of NbSe, from Ref. 11I], and the Raman scattering data of NbS2 from Refs [12]and [13].
(3)
Here x;!(q) is the generalized electronic susceptibility defined by eqn (2) and it corresponds to the Fourier transform of effective long-range atomic forces caused by the electron-lattice interaction. We have calculated x$(q) by usingg““(nk, n’k - q) and the energy eigenvalues of the electronic bands. It is noted that x$(q) is the same for both NbSz and NbSez since we have assumed the same TB bands and the same electron-lattice coupling coefficients for these two compounds. The remaining part R;!(q) represents contributions arising other than from x$(q) and it is usually expressed as the Fourier transform of the short-range forces. We have assumed five kinds of short-range force constants of stretching-type and determined them so as to reproduce the observed phonon frequencies [IO-131, which are not affected by the electron-phonon interaction or x$(q). The phonon dispersion curves calculated along the IM direction are shown in Fig. 3(a) for NbSez and in Fig. 3(b) for Nb&. As seen in these figures, the lower Ci mode which consists mainly of Nb atom displacements in the c plane reveals considerable frequency renormalization around q = $IM. In NbSez the frequencies of the lower Ci mode around q = $l?M are imaginary, while in NbS2 they remain real. These results indicate that the 2H-type structure NbSe, is unstable against lattice distortion corresponding to the C, mode at q = !I’M, while the lattice instability does not occur in NbSz. The results are consistent with the experimental results. It should be noted that remarkable frequency renormalization of the C, phonon mode around q = 3rhf is caused due to characteristic dependences of the electron-phonon interaction on wave vectors and phonon modes, as well as the nesting effect of the Fermi surface. Further, it should be pointed out that the frequency of the Ci phonon mode is affected sensitively by the short-range force constant f,-, for neighboring X (= S or Se) ions on different X-layers in the same X-Nb-X sandwhich, namely, for the larger value off,_, the frequency of the Ci phonon mode becomes the higher. Therefore, the value of this force constant primarily determines whether the lattice instability occurs or does not occur. In fact, the
1094
K. MOTIZUKI
er al.
value offX-, for NbS2, 5.5 x IO4 dyne/cm, is larger than that for NbSez, 4.25 x lo4 dyne/cm. This difference in the magnitude off,+ is regarded to be the main reason why lattice instability occurs in NbSe, but not in NbSz.
( a) A, /MS% 40
30
2.2. Effect of intercalation We investigate the effect of intercalation on the lattice instability of NbSe* and NbSz in the framework of the rigid band approximation. Intercalation changes the shape of Fermi surfaces as well as the Fermi level. Therefore the nature of phonon renormalization is affected through the change of the generalized electronic susceptibility x$(q). We have calculated x$(q) and the renormalized phonon frequencies for two types of intercalant, acceptor-type and donor-type. As to the short-range forces we have used the same values as for pure NbSe2. The calculated phonon dispersion curves of intercalated NbSez are shown in Fig. 4(a) and (b). In this calculation the Fermi level is shifted by -0.15 eV or +O.l2eV which corresponds to doping monovalent acceptor of l/5 per formula or donor of l/8 per formula. In case of doping the acceptor-type intercalants into NbSez, the point of instability in q-space has moved to q = FM from q = +rM. Further, reflecting an increase of the electronic DOS at the Fermi level the frequency renormalization is enhanced and hence the CDW transition temperature is expected to become higher. In case of doping the donor-type intercalants, on the other hand, the frequency renormalization is weakened and hence the lattice instability is suppressed. The same tendency has been found for NbSz. Therefore, the lattice instability is expected also in NbSz as the concentration of the acceptortype dopant increases.
3. SUPERCONDUCTIVITY
% G
20
u w Z W
E
10
$ 2 0
-10
M
IWAVE-VECTOR
(b) 40
I
D,/dbSQ I
1
1
I
I
I
1
OF NbS2
3.1. Spectralfunction 2H-NbSz becomes a superconductor at 6 K directly from the undistorted normal state. We have studied the superconductivity of this compound in the framework of a phonon-mediated pairing mechanism. One of the most important quantities in the Eliashberg theory is the superconducting spectral function cr2Fs(w) which is defined by:
r JF,(w)
nk ’
%%k
-
n’k’
k’-k
?
EF)~(J%rkr
M WAVE-VECTOR
= &~~~‘~~~~;~‘~‘)I’
-
EF)b(w
Fig. 4. The phonon dkpersjoncurvesof intercalated NbSe, -
along the TM direction: (a) acceptor-type intercalation and (b) donor-type intercalation.
w;,_k)’
(4) where N(&) is the DOS at the Fermi level, w’pdenotes
Effect of intercalation on instability and superconductivity the renormalized phonon frequency and P(nk, n’k’) is the electron-phonon coupling coefficient given by:
V7(nk,n’k)
= c -c7,pu(k’ ’ lb0 fi
- k)gFa(nk,n’k’).
Here y specifies the phonon mode, c7,+(q) represents the phonon polarization vector, and MP is the mass of the @h atom in the unit ceil. By using the obtained electron-phonon coupling and the renormalized phonon frequencies, we have calculated the spectral function a2Fs(w) [7]. Calculation has been done with use of 502 k points on the Fermi surfaces in the l/24 reduced Brillouin zone. The calculated a*F,(w) is shown in Fig. 5(a) together with the phonon DOS. The spectral function a2Fs(w) has significant values only in the low frequency range (w < 20meV) where longitudinal vibrations of Nb atoms are dominant. For comparison we have calculated a2Fs(w) by using the unrenormalized bare phonon frequencies, i.e. frequencies obtained by neglecting x$(q). The results are shown in Fig. 5(b). As seen apparently from Fig. 5(a) and (b), the electron-phonon interaction drastically alters a2Fs(w). The dimensionless electron-phonon coupling strength X given by 2Jdwa’F,(w)/w is evaluated as X = 0.554.
NbS2 0.6 I I I _ (a) Renormalized
I
I 1 7 2
1095
3.2. Superconducting properties We have evaluated the superconducting transition temperature T, by solving the linearized isotropic Eliashberg equation [14, 151. We have obtained T, = 2.9 K for the effective screened Coulomb constant p* = 0.1. The estimated value of T, is smaller by a factor 2 than the experimental value. This discrepancy may be partly removed by solving the anisotropic Eliashberg equation. We also evaluated T, by using a2Fs(w) which is obtained by using the unrenormalized bare phonons. The value of T, is 0.02 K for ZL*= 0.1. These results clearly indicate that the renormalization of phonon frequencies plays a significant role in raising the superconducting transition temperature of NbS2. We have solved also the isotropic Eliashberg equation at finite temperatures and determined the gap-function A(w) as a function of temperature. The superconducting gap A, = ReA(A9) has been evaluated to be 0.37meV at T = 0.1 T,. The ratio 2A0/ksT, = 3.72 is slightly larger than the value 3.53 predicted in the BCS weak-coupling theory. By using the obtained A(w) we have calculated the tunneling spectra, i.e. the differential conductance defined by:
and its derivative d21/dV 2. The results obtained for T = 0.1 T, are shown in Fig. 6. The curves of dZ/d V and d2Z/d V 2 show prominent structures, which reflect the peak structure in a2Fs(w), and deviate from the results of BCS theory which replaces A(w) with A, in the right-hand side of eqn (6). As far as we know, there is no experimental study on dZ/dV and d2Z/d V 2. Measurements of the tunneling spectra are desired. 3.3. E#ect of intercalation
If the shape of the spectral function cr2Fs changes a
NbS2 0.6 no.4 3
I
I _
(b)
I
I
I
‘.0°8I2
7 3
Bare
i -2
E
-
0.996
ENERGY
0
’
(mev)
I
I
I
10
20
30
BIAS VOLTAGE
404
(mv)
Fig. 5. The spectral function a2FE(w)(full curves) and the phonon DOS (broken curves) of NbS2: (a) the results
Fig. 6. The tunneling spectra df/dV (full curves)
obtained
d*I/ dV* (broken
by using the renormalized phonons and (b) the results obtained by using the bare phonons.
curves) of NbS2 calculated of the Eliashberg theory.
and on the basis
1096
K. MOTIZUKI el al.
here that the present argument cannot be applied to intercalation compounds of NbSe* because the superconductivity of NbSez occurs in the CDW phase. Experimental study of the effects of alkali-metal doping on superconductivity has been done for NbS2 by Kanzaki er al. [16] and by McEwen and Sienko [17]. According to their results the superconducting transition temperature of NbS2 has certainly decreased by the alkali-metal doping. Experimental study of acceptor-type doping into NbS2 is desired.
0.6 A,
0
0
I
I
I
I
I
10
20
30
40
50
ENERGY (mev) Fig. 7. The functional derivative bT,/s(a’F,(w))
of NbS2.
little as a2Fs + a2Fs + Soi2Fs, the resulting change in T, can be expressed as 6T, =
x dwKb’F,(w)l 6(craFs(w))
s0
6(~2Fs(w))’
(7)
To study the effect of intercalation on T, of NbS2, we have calculated functional derivative of T, with respect to the spectral function, STc/S((r2Fs(w)) [15]. Figure 7 shows the calculated result. According to the results in Section 2, the donor-type intercalant weakens the renormalization of the phonon frequencies and the accepter-type intercalant, on the other hand, enhances the renormalization. Then we can expect that in case of donor-type intercalation, the shape of cx’F,(w) shown in Fig. 5(a) approaches that of (Y*F~(w) shown in Fig. 5(b) which was obtained by using the bare phonons. In the case of acceptor-type intercalation an enhancement of a2Fs(w) is expected in the lower energy range below 15meV. Considering such a change of ct2Fs(w) and the shape of the functional derivative shown in Fig. 7, it can be expected that the acceptor-type intercalant increases the superconducting transition temperature as long as lattice instability does not occur. On the other hand, the donor-type intercalant lowers the transition temperature of NbS2. It should be noted
REFERENCES 1. Wilson J. A. and Yoffe A. D., Adv. Phys. 18,193 (1969). 2. Motizuki K. and Suzuki N., Structural Phase Transition in Layered Transition Meial Compounds (Edited by K. Motizuki), p. 1. Reidel, Dordrecht (1986). 3. Wilson J. A., DiSalvo F. J. and Mahajan S., Adv. Phys. 24, 117 (1975). 4. Mattheiss L. F., Phys. Rev. B 8.3719 (1973). 5. Wexler G. and Wooley A. M., J. Phys. C: Solid State Phys. 9, 1185 (1976). 6. Motizuki K. and Ando E., .I. Phys. Sot. Jpn 52, 2849 (1983). 7. Nishio Y., Shirai M., Suzuki N. and Motizuki K., J. Phys. Sot. Jpn 63, 156 (1994); Inr. J. Mod. Phys. B 7, 188 (1993). 8. Wada S., Nakamura S., Aoki R. and Molinie P., J. Phys. Sot. Jpn 48,786 (1980). 9. Nishio Y., J. Phys. Sot. Jpn 63,223 (1994). 10. Moncton D. E., Axe J. D. and DiSalvo F. J., Phys. Rev. B 16, 801 (1977). 11. Wang C. S. and Chen J. M., Solid State Commun. 14,
1145 (1974). 12. Nakashima S., Tokuda Y., Mitsuishi A., Aoki R. and Hamaue Y., Solid State Commun. 42,601 (1982). 13. McMullan W. G. and Irwin J. C., Can. J. Phys. 62,789 (1984). 14. Allen P B. and MitroviC B., Solid Stare Physics (Edited by H. Ehrenreich, F. Seitz and D. Turnbull), Vol. 37, p. 1. Academic Press, New York (1982). 15. Bergmann G. and Rainer D., Z. Phys. 263,59 (1973). 16. Kanzaki Y., Konuma M. and Matsumoto O., Physica lOSB, 205 (1981). 17. McEwen C. S. and Sienko M. J., Revue de Chimie minkafe 19,309 (1982).
\
I
.I. Phys. Chum Solids Vol 57, Nos 6-8. pp. 1091-1096, 1996 Copyright 0 1996 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0022-3697196 0.00
si5.00+
EFFECT OF INTERCALATION ON STRUCTURAL INSTABILITY AND SUPERCONDUCTIVITY OF LAYERED 2H-TYPE NbSe2 AND NbS2 KAZUKO
MOTIZUKIt,
YOSHIMASA
NISHIO&
NAOSHI
MASAFUMI
SHIRAIS and
SUZUKIJ
TDepartment of Applied Physics, Faculty of Science, Okayama University of Science, Ridai-cho I-1, Okayama 700, Japan fDepartment of Material Physics, Faculty of Engineering Science, Osaka University, Machikaneyama-cho 1-3, Toyonaka 560, Japan (Received 28 May 1995; accepted in revisedform 31 May 1995)
Abstract-Effects ofintercalation on the lattice instability and the superconductivity ofNbSet and NbSz are studied in the framework of the rigid band approximation. In the case of doping acceptor-type intercalants into NbSe2, the point of lattice instability in q-space is moved to q = rM from q = ZI’M. Further, the frequency renormalization is enhanced and the CDW transition temperature is expecte d to increase. In the case of doping donor-type intercalants, on the other hand, the frequency renormalization is weakened and the lattice instability is suppressed. The same tendency is found also in NbS2. The effect of intercalation on the superconducting transition temperature T, of NbS2 is investigated by calculating the functional derivative of T, with respect to the spectral function. The donor-type intercalant lowers T, as observed in the case of alkali-metal doping. On the other hand, the acceptor-type intercalant raises T, as long as the lattice instability does not take place. Keyworcis: A. chalcogenides, D. superconductivity.
D. charge density wave, D. electronic structure, D. lattice dynamics,
1. INTRODUCTION
Niobium and tantalum dichalcogenides, 2H-MXz (M = Nb,Ta; X = S, Se), take a layered structure, which consists of a sequence of X-M-X sandwich layers (see Fig. 1). Many atoms and molecules can be intercalated into the so-called van der Waals gap layers between the adjacent sandwiches and hence they have been important materials as mother crystals of intercalation compounds. These four compounds have attracted much interest also from an academic point of view because all the compounds except NbS2 reveal charge density wave (CDW) transition accompanying structural transition (lattice instability) [l, 21. Furthermore, all of these four compounds become superconductors at low temperatures [3]. The electronic band structures of these compounds have been calculated by first-principles methods such as the APW method [4] and the layer method [5]. According to their results the electronic band structures of the four compounds are nearly the same. We have reproduced the first-principles band structures by a tight-binding (TB) fitting method with use of Slater-Koster transfer integrals t(ddo), t(dpa), etc. as
§Present address: MEC Lab., Daikin Industries Co. Ltd., Miyukigaoka 3, Tsukuba 305, Japan.
fitting parameters [6,7]. We have tried to reproduce as precisely as possible the two d-p hybridization bands, which cross the Fermi level. The density of states (DOS) of the d-p hybridization bands obtained by our TB calculation is shown in Fig. 2. There is certainly a non-negligible contribution from the p component of S or Se atoms, but the main component is the Nb 4d state. As seen from Fig. 2 a sharp peak exists in the DOS curve and the Fermi level EF is located at the higher energy side of the peak. The value of the DOS at EF, N(Er), is 2.56 states/(eV - spin. unit cell), which agrees fairly well with 2.6 f 0.4 states/ (eV . spin * unit cell) determined by NMR measurements [8] or 3.0 states/(eV - spin - unit cell) obtained by the layer method [5]. We have developed the microscopic theory of electron-phonon interaction and lattice dynamics for NbSez and NbS2 [6, 71on the basis of our realistic TB bands. We have found that the characteristic dependence of electron-phonon interaction on phonon mode and wave vector, in addition to the effect of the Fermi surface nesting, causes significant frequency renormalization (softening) for the C, phonon mode around q = iI’M. The CDW transition in NbSez is driven by complete softening of this q = $l’M Ci phonon. In NbS2, on the other hand, the frequency renormalization of the Cl phonon is not
1091
1092
K. MOTIZUKI et al. The results show that the electron-phonon interaction and the renormalized phonon frequency play significant roles in raising the superconducting transition temperature of NbS2 (T, = 6 K) and in explaining the temperature dependence of the electrical and thermal resistivities. In the present paper we investigate the effect of intercalation on the lattice instability of NbSez and NbS2 and the superconductivity of NbS2 in the framework of the rigid band approximation.
vl
2. LATTICE INSTABILITY
2.1.
Phonon anomaly
The free energy change due to lattice distortion is expressed generally by [2]:
where u:(q) denotes Fourier component of the displacement of the pth atom in the unit cell along the (Y direction (o = x, y, z) and x$(q), which is called the generalized electronic susceptibility, is expressed in the following form:
a Fig. 1.The crystal structure of MX2. large enough to cause the lattice instability. These results are consistent with the observations. Furthermore we have studied the superconductivity [7] and the transport properties, such as electrical and thermal resistivities [9] of NbSz, by using the electron-phonon interaction and the renormalized phonon frequencies. I
I -total
_
-------d
state
P $.tate
-7
-a ENERGY
(eY)
Fig. 2. The electronic DOS of NbXz
x$(q)
= 2xxg’““(nk,n’k n,n’ k xf&‘k-q)
- q)gV4(nk,n’k - q)*
-f(‘%k) (2) En,
-
En’k-q
’
Here Enk represents the one electron band energy (n denotes the band index) of the undistorted structure, ,f(&k) is the Fermi distribution function, and gFa(nk, n’k - q) represents the strength of the coupling between two electronic states (nk) and (n’k -9) caused by displacement of the pth atom in the (Y direction. The quantity g““(nk, n’k - q) is called the electron-lattice coupling coefficient. In the TB model, the electron-lattice interaction arises from the modulation of transfer energies due to lattice distortion, and gw(nk, n’k - q) is expressed explicitly in terms of the derivatives of Slater-Koster transfer integrals and the coefficients of transformation which diagonalizes the undistorted TB Hamiltonian [2]. We have estimated the derivatives of the Slater-Koster transfer integrals by using the semiempirical rule that the transfer integrals are proportional to R-” where R is the inter-atomic distance. We have used n = 5 for transfer integrals between d orbitals and n = 2 for those between d and p orbitals. We have calculated gpa(nk,n’k - q) for the two conduction bands which cross the Fermi level. The calculated gpa(nk,n’k - q) depends remarkably on wave-vectors, k and q, and also on mode of atomic displacements or vibrations. For q = rM or l?K
Effect of intercalation on instability and superconductivity
1093
direction, the coupling coefficients caused by the Nb atom displacements in the c plane are larger than those due to the other modes. Particularly, the coupling coefficients due to the chalcogen atom displacements are quite small. Next we have calculated the phonon frequencies of NbSz and NbSez by diagonalizing the dynamical matrix 0$(q) which is given by D;!(q) = R;!(q) + x$(q). >
E2g
30 w AI, z
g
W
10 Bl, E2g n
r
M WAVE-VECTOR
b)
NbS2
50 AI,
2
40 EZC
0
M
r WAVE-VECTOR
Fig. 3. The phonon dispersion curves along the pM direction: (a) NbSe* and (b) NbS2. The neutron scattering data of NbSe, are taken from Ref. [lo], the Raman scattering data of NbSe, from Ref. 11I], and the Raman scattering data of NbS2 from Refs [12]and [13].
(3)
Here x;!(q) is the generalized electronic susceptibility defined by eqn (2) and it corresponds to the Fourier transform of effective long-range atomic forces caused by the electron-lattice interaction. We have calculated x$(q) by usingg““(nk, n’k - q) and the energy eigenvalues of the electronic bands. It is noted that x$(q) is the same for both NbSz and NbSez since we have assumed the same TB bands and the same electron-lattice coupling coefficients for these two compounds. The remaining part R;!(q) represents contributions arising other than from x$(q) and it is usually expressed as the Fourier transform of the short-range forces. We have assumed five kinds of short-range force constants of stretching-type and determined them so as to reproduce the observed phonon frequencies [IO-131, which are not affected by the electron-phonon interaction or x$(q). The phonon dispersion curves calculated along the IM direction are shown in Fig. 3(a) for NbSez and in Fig. 3(b) for Nb&. As seen in these figures, the lower Ci mode which consists mainly of Nb atom displacements in the c plane reveals considerable frequency renormalization around q = $IM. In NbSez the frequencies of the lower Ci mode around q = $l?M are imaginary, while in NbS2 they remain real. These results indicate that the 2H-type structure NbSe, is unstable against lattice distortion corresponding to the C, mode at q = !I’M, while the lattice instability does not occur in NbSz. The results are consistent with the experimental results. It should be noted that remarkable frequency renormalization of the C, phonon mode around q = 3rhf is caused due to characteristic dependences of the electron-phonon interaction on wave vectors and phonon modes, as well as the nesting effect of the Fermi surface. Further, it should be pointed out that the frequency of the Ci phonon mode is affected sensitively by the short-range force constant f,-, for neighboring X (= S or Se) ions on different X-layers in the same X-Nb-X sandwhich, namely, for the larger value off,_, the frequency of the Ci phonon mode becomes the higher. Therefore, the value of this force constant primarily determines whether the lattice instability occurs or does not occur. In fact, the
1094
K. MOTIZUKI
er al.
value offX-, for NbS2, 5.5 x IO4 dyne/cm, is larger than that for NbSez, 4.25 x lo4 dyne/cm. This difference in the magnitude off,+ is regarded to be the main reason why lattice instability occurs in NbSe, but not in NbSz.
( a) A, /MS% 40
30
2.2. Effect of intercalation We investigate the effect of intercalation on the lattice instability of NbSe* and NbSz in the framework of the rigid band approximation. Intercalation changes the shape of Fermi surfaces as well as the Fermi level. Therefore the nature of phonon renormalization is affected through the change of the generalized electronic susceptibility x$(q). We have calculated x$(q) and the renormalized phonon frequencies for two types of intercalant, acceptor-type and donor-type. As to the short-range forces we have used the same values as for pure NbSe2. The calculated phonon dispersion curves of intercalated NbSez are shown in Fig. 4(a) and (b). In this calculation the Fermi level is shifted by -0.15 eV or +O.l2eV which corresponds to doping monovalent acceptor of l/5 per formula or donor of l/8 per formula. In case of doping the acceptor-type intercalants into NbSez, the point of instability in q-space has moved to q = FM from q = +rM. Further, reflecting an increase of the electronic DOS at the Fermi level the frequency renormalization is enhanced and hence the CDW transition temperature is expected to become higher. In case of doping the donor-type intercalants, on the other hand, the frequency renormalization is weakened and hence the lattice instability is suppressed. The same tendency has been found for NbSz. Therefore, the lattice instability is expected also in NbSz as the concentration of the acceptortype dopant increases.
3. SUPERCONDUCTIVITY
% G
20
u w Z W
E
10
$ 2 0
-10
M
IWAVE-VECTOR
(b) 40
I
D,/dbSQ I
1
1
I
I
I
1
OF NbS2
3.1. Spectralfunction 2H-NbSz becomes a superconductor at 6 K directly from the undistorted normal state. We have studied the superconductivity of this compound in the framework of a phonon-mediated pairing mechanism. One of the most important quantities in the Eliashberg theory is the superconducting spectral function cr2Fs(w) which is defined by:
r JF,(w)
nk ’
%%k
-
n’k’
k’-k
?
EF)~(J%rkr
M WAVE-VECTOR
= &~~~‘~~~~;~‘~‘)I’
-
EF)b(w
Fig. 4. The phonon dkpersjoncurvesof intercalated NbSe, -
along the TM direction: (a) acceptor-type intercalation and (b) donor-type intercalation.
w;,_k)’
(4) where N(&) is the DOS at the Fermi level, w’pdenotes
Effect of intercalation on instability and superconductivity the renormalized phonon frequency and P(nk, n’k’) is the electron-phonon coupling coefficient given by:
V7(nk,n’k)
= c -c7,pu(k’ ’ lb0 fi
- k)gFa(nk,n’k’).
Here y specifies the phonon mode, c7,+(q) represents the phonon polarization vector, and MP is the mass of the @h atom in the unit ceil. By using the obtained electron-phonon coupling and the renormalized phonon frequencies, we have calculated the spectral function a2Fs(w) [7]. Calculation has been done with use of 502 k points on the Fermi surfaces in the l/24 reduced Brillouin zone. The calculated a*F,(w) is shown in Fig. 5(a) together with the phonon DOS. The spectral function a2Fs(w) has significant values only in the low frequency range (w < 20meV) where longitudinal vibrations of Nb atoms are dominant. For comparison we have calculated a2Fs(w) by using the unrenormalized bare phonon frequencies, i.e. frequencies obtained by neglecting x$(q). The results are shown in Fig. 5(b). As seen apparently from Fig. 5(a) and (b), the electron-phonon interaction drastically alters a2Fs(w). The dimensionless electron-phonon coupling strength X given by 2Jdwa’F,(w)/w is evaluated as X = 0.554.
NbS2 0.6 I I I _ (a) Renormalized
I
I 1 7 2
1095
3.2. Superconducting properties We have evaluated the superconducting transition temperature T, by solving the linearized isotropic Eliashberg equation [14, 151. We have obtained T, = 2.9 K for the effective screened Coulomb constant p* = 0.1. The estimated value of T, is smaller by a factor 2 than the experimental value. This discrepancy may be partly removed by solving the anisotropic Eliashberg equation. We also evaluated T, by using a2Fs(w) which is obtained by using the unrenormalized bare phonons. The value of T, is 0.02 K for ZL*= 0.1. These results clearly indicate that the renormalization of phonon frequencies plays a significant role in raising the superconducting transition temperature of NbS2. We have solved also the isotropic Eliashberg equation at finite temperatures and determined the gap-function A(w) as a function of temperature. The superconducting gap A, = ReA(A9) has been evaluated to be 0.37meV at T = 0.1 T,. The ratio 2A0/ksT, = 3.72 is slightly larger than the value 3.53 predicted in the BCS weak-coupling theory. By using the obtained A(w) we have calculated the tunneling spectra, i.e. the differential conductance defined by:
and its derivative d21/dV 2. The results obtained for T = 0.1 T, are shown in Fig. 6. The curves of dZ/d V and d2Z/d V 2 show prominent structures, which reflect the peak structure in a2Fs(w), and deviate from the results of BCS theory which replaces A(w) with A, in the right-hand side of eqn (6). As far as we know, there is no experimental study on dZ/dV and d2Z/d V 2. Measurements of the tunneling spectra are desired. 3.3. E#ect of intercalation
If the shape of the spectral function cr2Fs changes a
NbS2 0.6 no.4 3
I
I _
(b)
I
I
I
‘.0°8I2
7 3
Bare
i -2
E
-
0.996
ENERGY
0
’
(mev)
I
I
I
10
20
30
BIAS VOLTAGE
404
(mv)
Fig. 5. The spectral function a2FE(w)(full curves) and the phonon DOS (broken curves) of NbS2: (a) the results
Fig. 6. The tunneling spectra df/dV (full curves)
obtained
d*I/ dV* (broken
by using the renormalized phonons and (b) the results obtained by using the bare phonons.
curves) of NbS2 calculated of the Eliashberg theory.
and on the basis
1096
K. MOTIZUKI el al.
here that the present argument cannot be applied to intercalation compounds of NbSe* because the superconductivity of NbSez occurs in the CDW phase. Experimental study of the effects of alkali-metal doping on superconductivity has been done for NbS2 by Kanzaki er al. [16] and by McEwen and Sienko [17]. According to their results the superconducting transition temperature of NbS2 has certainly decreased by the alkali-metal doping. Experimental study of acceptor-type doping into NbS2 is desired.
0.6 A,
0
0
I
I
I
I
I
10
20
30
40
50
ENERGY (mev) Fig. 7. The functional derivative bT,/s(a’F,(w))
of NbS2.
little as a2Fs + a2Fs + Soi2Fs, the resulting change in T, can be expressed as 6T, =
x dwKb’F,(w)l 6(craFs(w))
s0
6(~2Fs(w))’
(7)
To study the effect of intercalation on T, of NbS2, we have calculated functional derivative of T, with respect to the spectral function, STc/S((r2Fs(w)) [15]. Figure 7 shows the calculated result. According to the results in Section 2, the donor-type intercalant weakens the renormalization of the phonon frequencies and the accepter-type intercalant, on the other hand, enhances the renormalization. Then we can expect that in case of donor-type intercalation, the shape of cx’F,(w) shown in Fig. 5(a) approaches that of (Y*F~(w) shown in Fig. 5(b) which was obtained by using the bare phonons. In the case of acceptor-type intercalation an enhancement of a2Fs(w) is expected in the lower energy range below 15meV. Considering such a change of ct2Fs(w) and the shape of the functional derivative shown in Fig. 7, it can be expected that the acceptor-type intercalant increases the superconducting transition temperature as long as lattice instability does not occur. On the other hand, the donor-type intercalant lowers the transition temperature of NbS2. It should be noted
REFERENCES 1. Wilson J. A. and Yoffe A. D., Adv. Phys. 18,193 (1969). 2. Motizuki K. and Suzuki N., Structural Phase Transition in Layered Transition Meial Compounds (Edited by K. Motizuki), p. 1. Reidel, Dordrecht (1986). 3. Wilson J. A., DiSalvo F. J. and Mahajan S., Adv. Phys. 24, 117 (1975). 4. Mattheiss L. F., Phys. Rev. B 8.3719 (1973). 5. Wexler G. and Wooley A. M., J. Phys. C: Solid State Phys. 9, 1185 (1976). 6. Motizuki K. and Ando E., .I. Phys. Sot. Jpn 52, 2849 (1983). 7. Nishio Y., Shirai M., Suzuki N. and Motizuki K., J. Phys. Sot. Jpn 63, 156 (1994); Inr. J. Mod. Phys. B 7, 188 (1993). 8. Wada S., Nakamura S., Aoki R. and Molinie P., J. Phys. Sot. Jpn 48,786 (1980). 9. Nishio Y., J. Phys. Sot. Jpn 63,223 (1994). 10. Moncton D. E., Axe J. D. and DiSalvo F. J., Phys. Rev. B 16, 801 (1977). 11. Wang C. S. and Chen J. M., Solid State Commun. 14,
1145 (1974). 12. Nakashima S., Tokuda Y., Mitsuishi A., Aoki R. and Hamaue Y., Solid State Commun. 42,601 (1982). 13. McMullan W. G. and Irwin J. C., Can. J. Phys. 62,789 (1984). 14. Allen P B. and MitroviC B., Solid Stare Physics (Edited by H. Ehrenreich, F. Seitz and D. Turnbull), Vol. 37, p. 1. Academic Press, New York (1982). 15. Bergmann G. and Rainer D., Z. Phys. 263,59 (1973). 16. Kanzaki Y., Konuma M. and Matsumoto O., Physica lOSB, 205 (1981). 17. McEwen C. S. and Sienko M. J., Revue de Chimie minkafe 19,309 (1982).