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Fermi surface and charge density waves in niobium diselenide
Solid State Communications, Vol. 22, pp. 33 1—333, 1977.
Pergamon Press.
Printed in Great Britain
FERMI SURFACE AND CHARGE DENSITY WAVES IN NIOBIUM DISELENIDE B. Riccó* Cavendish Laboratory, Madingley Road, CB3 OHE Cambridge, England (Received 20 December 1976 by R. Fieschi) The Fermi energy of niobium diselenide is calculated and the result is used to compute the wave-vector dependent susceptibility x(q). This is found to have a fairly narrow peak at a wave-vector in very good agreement with that of the periodic lattice distortion developed by the solid at low temperature and such a peak can be seen to arise from good nesting of the Fermi surface.
TFIE TRANSITION METAL layer compound 2H-NbSe2 at T 26K undergoes a phase transition developing charge density waves (CDW) together with a periodic lattice distortion (PLD). This latter consists of the formarion ofa slightly inconunensurate 3 x 3 superlattice which is then described in reciprocal space by a wavevector q~,very nearly equal 1/3 of the smallest reciprocal vector g0 parallel to the layers [1J. Already at room temperature the solid shows a clear tendency towards that particular PU) [2] so that it seems meaningful, from a theoretical point of view, to look for properties of the undistorted crystal reflectlag such a tendency. It is generally accepted that for the solid to develop a PU) with wave-vector q~,the
susceptibility ~
f(E(k q)} —f{E(k)} E(k)+—E(k +~ (wheref represents the Fermi function and the presence of matrix elements has been neglected) should be large at q = q~[3]. It should in fact be large enough to satisfy Chan and Heine’s criterion [4] and this aspect of the problem, as far as 2H-NbSe2 is concerned as well for the other group V metallic compounds, is discussed in details in reference [5] where the condition for the formation of CDW is found to be correctly fulfilled, The reason why x(~c)becomes large enough instead is still somewhat debated. Some mechanisms have been proposed: one suggests it could be due to the presence on the Fermi Surfaces (FS) of saddle points at distance apart [6], another, which is probably the most widely accepted, indicates the cause in the presence of parallel bits of FS separated by q~,[7, 81 (usually ~ ferred to as nesting condition). All these theories look only at the FS as if the rest of the band structure was not essential in dealing with
qc
*
CDWso that it should be easy, in principle, to determine q~once the FS is known. In the case of the group V metallic compounds, on the other hand, a recent FS calculation did not seem to confirm any of the theories mentioned above [9] in that the first seemed to suggest the wrong kind of PU) whereas only poor nesting was found. The question however depends very strongly upon the value E~of the Fermi energy and here we present, in the case of 2H-NbSe2, a new calculation of EF and FS which seems to indicate very clearly good nesting as the cause of CDW. Furthermore x(q) as given in equation (1) is also calculated with the new value ofEF (entering the equation through the Fermi functions) and it is found to have fairly almost temperature independent peak ajust at qnarrow, = ~g0in very good agreement with wavevector of the PU). l’his partially modifies the results of reference [5], In where Fermi the energy given in reference [9] was used. fact,the although value of x(~c) is not
essentially changed so that all the quantitative considerations are not affected, the shape of x(q) does no longer have a broad maximum about q~,but a narrow peak as it should be according with the theories based on the properties of reference the sole FS. While in [9] EF was obtained using a Fourier fit of the first principle band structure computed with the layer method, in the present work it is calculated by means of a tight-binding fit of the same E(k) followed by a calculation of density of states (d.o.s.). The method is particularly convenient because in order to compute a fast algorithm to evaluate E(k) over thex(q) BZone and needs this can be satisfactorily achieved using the tight-binding fitting procedure of Slater and Koster Once thehopping fit is carried out,from however, also has[10J. the so-called integrals whichone the d.o.s. can be directly calculated using the recursion method of Haydock, Heine and Kelly [11]. Then EF
Now at UniversitI di Bologna, Istituto di Elettronica, Viale Risorgimento 2, Bologna, Italy. 331
332
CHARGE DENSITY WAVES IN NIOBIUM DISELENIDE
Vol. 22, No.5
Table 1. Parametersfor the recursion method E
0.715 0.603 = ~ —0.612 —0.758 E~ —0.686 intrasandwicb E3z2_r3,y 0.065 E3z2_r2,z 0.048 Ex2_y2.z 0.049 Ex2_y2,v —0.048 ~ —0.002 0.031 ~ ~ 0.098 z 0.086 dda 0.002 ddir 0.010 ddô 0.003 PPO 0.059 PP1~ 0.031 3j3_~2
= Ex2_y2
‘
-
Fig. 2. The Fermi line of the two-dimensional model of 2H-NbSe2 (EF = 0.29 Rj~)and the same shifted by ~go(dotted line go = 2AL).
—
—
—
(i) The fit has been designed such as to be particularly accurate for the central so called d32 band (cut across by the Fermi level). This can be achieved very satisfactorily by heavily weighting the errors involving the ~ band in the total error function to be minimised for the fit [13] and is justified by the fact that all the other bands have been found to give a contribution to x which is almost independent of q and estimated ~ 10%. (ii) It has been assumed that the weak interactions between the sandwiches of the layer structure can be
—
~Ryd]
30 000
~
0
I I
10 I
I
neglected [14]. With this assumption E(k) becomes twodimensional (independent of k2) and it has been necessary to choose an horizontal plane in the BZ to be fitted. It has been taken the top face (k~= ir/c) since there the bands stick together [9] and the intersandwich interactions have only second order effects [15]. The resulting tight-binding hopping integrals are reported in Table 1 where a standard notation has been used and from them, in the way briefly outlined above, we have obtained EF = 0.620 Ry instead of the published value 0.629. Using a Gaussian quadrature procedure for both the k~and lc~,directions we have then computed x(q) including in the integrand only the d~2 band [see point (i)]. The result is shown in Fig. 1: as can be seen, although the new estimate of EF differs —
—
~ 0
0.1
0.2
0.3 0.333
0.4
0.5
q/g0
Fig. I. The calculated x(q) for 2H-NbSe2 for T = 1 K(o) and T = 300 K(+). q in a g0 direction.
from that used in reference [5] by only 1.5%, it sharpens up considerably the peak in x(q) about a wave-vector in very good agreement with that experimentally found to
can be simply obtained filling up such a d.o.s. with the number of available electrons. A calculation of d.os. of 2H-NbSe2 performed along these lines is presented in reference [12] where a more detailed description of the whole procedure can be found. For the present work we have used the same method (same orbitals, number of points to fit, etc.) with the following two modifications:
be associated with the PU). In Fig. 2 instead the Fermi line E(k) = EF is plotted (full line) and shifted by qc = ~g0 to show that this is just the shift producing the best possible nesting. The agreement between experimental evidence and the nest. ing theory of CDW as we have used it here seems better than that of reference [5]. This can either be due to a sheer question of consistency, in that the use of two
Vol. 22, No.5
CHARGE DENSITY WAVES IN NIOBIUM DISELENIDE
different fits (one to approximate E(k) and the other to determine E,) is not good enough for detailed calculations, or simply to the fact that the value of EF used in the present work is more realistic. In this case, finally, it would perhaps be worth pointing out how, although the recursion methods gives the d.o.s. of a finite cluster of atoms (which is then only an approximation of that
333
of the whole solid) together with tight-binding it works better than the purely mathematical Fourier fit, quite probably because of the physics embedded in it. Acknowledgements The author is very grateful to DJ. Titterington for his invaluable assistance and help with the computing of this work. —
REFERENCES 1. 2.
WILUAMS P.M., SCRUBY C.B. & TATLOCK G., Solid State Commun. 17, 1197(1975). MONCTON D.E., AXE J.D. & DI SALVO FJ.,Phys. Rev. Lett. 34,734, (1975).
3.
FRIEDEL J., The Physics ofMetals (Edited by ZIMAN J.M.), p. 340. Cambridge (1969).
4.
CHAN S.K. & HEINE V.,J. Phys. F3, 795 (1972).
5.
DORAN NJ., WEXLER G., HEINE V. & RIccO B. (in press), to appear in II Nuovo Clmento, 37B, 2(1977).
6. 7. 8.
RICE J.M. & SCOTT G.K.,Phys. Rev. Lett. 35, 120, (1975). TOSATTI E., Festkorperprobleme (Adv. Solid State Phys.) (Edited by QUEISSER HJ.), Vol. XV, p. 113. Pergamon/Wieweg Brauschweig (1975). OVERHAUSER A.W.,Phys. Rev. 167, 691, (1968).
9.
WEXLERG.&WOLLEYA.M.,J.Phys. C. 9,1185(1976).
10.
SLATER G.C. & KOSTER GJ., Phys. Rev. 94, 1498, (1954).
11.
HAYDOCK R., HEINE V. & KELLY M.J.,J. Phys. C. 5,2845 (1972).
12. 13.
RICCO B., Phys. Status Solidi (b) 77,287 (1976). RICCO B., Ph.D. Thesis, University of Cambridge (U.K.), (1975).
14.
YOFFE A.D., Festkorperprobleme XIII (Adv. SolidState Phys.) (Edited by QUEISSER HJ.) p. 1. Pergamon/ Vieurg ‘(1973).
15.
WEXLER G. (Private communication).
Pergamon Press.
Printed in Great Britain
FERMI SURFACE AND CHARGE DENSITY WAVES IN NIOBIUM DISELENIDE B. Riccó* Cavendish Laboratory, Madingley Road, CB3 OHE Cambridge, England (Received 20 December 1976 by R. Fieschi) The Fermi energy of niobium diselenide is calculated and the result is used to compute the wave-vector dependent susceptibility x(q). This is found to have a fairly narrow peak at a wave-vector in very good agreement with that of the periodic lattice distortion developed by the solid at low temperature and such a peak can be seen to arise from good nesting of the Fermi surface.
TFIE TRANSITION METAL layer compound 2H-NbSe2 at T 26K undergoes a phase transition developing charge density waves (CDW) together with a periodic lattice distortion (PLD). This latter consists of the formarion ofa slightly inconunensurate 3 x 3 superlattice which is then described in reciprocal space by a wavevector q~,very nearly equal 1/3 of the smallest reciprocal vector g0 parallel to the layers [1J. Already at room temperature the solid shows a clear tendency towards that particular PU) [2] so that it seems meaningful, from a theoretical point of view, to look for properties of the undistorted crystal reflectlag such a tendency. It is generally accepted that for the solid to develop a PU) with wave-vector q~,the
susceptibility ~
f(E(k q)} —f{E(k)} E(k)+—E(k +~ (wheref represents the Fermi function and the presence of matrix elements has been neglected) should be large at q = q~[3]. It should in fact be large enough to satisfy Chan and Heine’s criterion [4] and this aspect of the problem, as far as 2H-NbSe2 is concerned as well for the other group V metallic compounds, is discussed in details in reference [5] where the condition for the formation of CDW is found to be correctly fulfilled, The reason why x(~c)becomes large enough instead is still somewhat debated. Some mechanisms have been proposed: one suggests it could be due to the presence on the Fermi Surfaces (FS) of saddle points at distance apart [6], another, which is probably the most widely accepted, indicates the cause in the presence of parallel bits of FS separated by q~,[7, 81 (usually ~ ferred to as nesting condition). All these theories look only at the FS as if the rest of the band structure was not essential in dealing with
qc
*
CDWso that it should be easy, in principle, to determine q~once the FS is known. In the case of the group V metallic compounds, on the other hand, a recent FS calculation did not seem to confirm any of the theories mentioned above [9] in that the first seemed to suggest the wrong kind of PU) whereas only poor nesting was found. The question however depends very strongly upon the value E~of the Fermi energy and here we present, in the case of 2H-NbSe2, a new calculation of EF and FS which seems to indicate very clearly good nesting as the cause of CDW. Furthermore x(q) as given in equation (1) is also calculated with the new value ofEF (entering the equation through the Fermi functions) and it is found to have fairly almost temperature independent peak ajust at qnarrow, = ~g0in very good agreement with wavevector of the PU). l’his partially modifies the results of reference [5], In where Fermi the energy given in reference [9] was used. fact,the although value of x(~c) is not
essentially changed so that all the quantitative considerations are not affected, the shape of x(q) does no longer have a broad maximum about q~,but a narrow peak as it should be according with the theories based on the properties of reference the sole FS. While in [9] EF was obtained using a Fourier fit of the first principle band structure computed with the layer method, in the present work it is calculated by means of a tight-binding fit of the same E(k) followed by a calculation of density of states (d.o.s.). The method is particularly convenient because in order to compute a fast algorithm to evaluate E(k) over thex(q) BZone and needs this can be satisfactorily achieved using the tight-binding fitting procedure of Slater and Koster Once thehopping fit is carried out,from however, also has[10J. the so-called integrals whichone the d.o.s. can be directly calculated using the recursion method of Haydock, Heine and Kelly [11]. Then EF
Now at UniversitI di Bologna, Istituto di Elettronica, Viale Risorgimento 2, Bologna, Italy. 331
332
CHARGE DENSITY WAVES IN NIOBIUM DISELENIDE
Vol. 22, No.5
Table 1. Parametersfor the recursion method E
0.715 0.603 = ~ —0.612 —0.758 E~ —0.686 intrasandwicb E3z2_r3,y 0.065 E3z2_r2,z 0.048 Ex2_y2.z 0.049 Ex2_y2,v —0.048 ~ —0.002 0.031 ~ ~ 0.098 z 0.086 dda 0.002 ddir 0.010 ddô 0.003 PPO 0.059 PP1~ 0.031 3j3_~2
= Ex2_y2
‘
-
Fig. 2. The Fermi line of the two-dimensional model of 2H-NbSe2 (EF = 0.29 Rj~)and the same shifted by ~go(dotted line go = 2AL).
—
—
—
(i) The fit has been designed such as to be particularly accurate for the central so called d32 band (cut across by the Fermi level). This can be achieved very satisfactorily by heavily weighting the errors involving the ~ band in the total error function to be minimised for the fit [13] and is justified by the fact that all the other bands have been found to give a contribution to x which is almost independent of q and estimated ~ 10%. (ii) It has been assumed that the weak interactions between the sandwiches of the layer structure can be
—
~Ryd]
30 000
~
0
I I
10 I
I
neglected [14]. With this assumption E(k) becomes twodimensional (independent of k2) and it has been necessary to choose an horizontal plane in the BZ to be fitted. It has been taken the top face (k~= ir/c) since there the bands stick together [9] and the intersandwich interactions have only second order effects [15]. The resulting tight-binding hopping integrals are reported in Table 1 where a standard notation has been used and from them, in the way briefly outlined above, we have obtained EF = 0.620 Ry instead of the published value 0.629. Using a Gaussian quadrature procedure for both the k~and lc~,directions we have then computed x(q) including in the integrand only the d~2 band [see point (i)]. The result is shown in Fig. 1: as can be seen, although the new estimate of EF differs —
—
~ 0
0.1
0.2
0.3 0.333
0.4
0.5
q/g0
Fig. I. The calculated x(q) for 2H-NbSe2 for T = 1 K(o) and T = 300 K(+). q in a g0 direction.
from that used in reference [5] by only 1.5%, it sharpens up considerably the peak in x(q) about a wave-vector in very good agreement with that experimentally found to
can be simply obtained filling up such a d.o.s. with the number of available electrons. A calculation of d.os. of 2H-NbSe2 performed along these lines is presented in reference [12] where a more detailed description of the whole procedure can be found. For the present work we have used the same method (same orbitals, number of points to fit, etc.) with the following two modifications:
be associated with the PU). In Fig. 2 instead the Fermi line E(k) = EF is plotted (full line) and shifted by qc = ~g0 to show that this is just the shift producing the best possible nesting. The agreement between experimental evidence and the nest. ing theory of CDW as we have used it here seems better than that of reference [5]. This can either be due to a sheer question of consistency, in that the use of two
Vol. 22, No.5
CHARGE DENSITY WAVES IN NIOBIUM DISELENIDE
different fits (one to approximate E(k) and the other to determine E,) is not good enough for detailed calculations, or simply to the fact that the value of EF used in the present work is more realistic. In this case, finally, it would perhaps be worth pointing out how, although the recursion methods gives the d.o.s. of a finite cluster of atoms (which is then only an approximation of that
333
of the whole solid) together with tight-binding it works better than the purely mathematical Fourier fit, quite probably because of the physics embedded in it. Acknowledgements The author is very grateful to DJ. Titterington for his invaluable assistance and help with the computing of this work. —
REFERENCES 1. 2.
WILUAMS P.M., SCRUBY C.B. & TATLOCK G., Solid State Commun. 17, 1197(1975). MONCTON D.E., AXE J.D. & DI SALVO FJ.,Phys. Rev. Lett. 34,734, (1975).
3.
FRIEDEL J., The Physics ofMetals (Edited by ZIMAN J.M.), p. 340. Cambridge (1969).
4.
CHAN S.K. & HEINE V.,J. Phys. F3, 795 (1972).
5.
DORAN NJ., WEXLER G., HEINE V. & RIccO B. (in press), to appear in II Nuovo Clmento, 37B, 2(1977).
6. 7. 8.
RICE J.M. & SCOTT G.K.,Phys. Rev. Lett. 35, 120, (1975). TOSATTI E., Festkorperprobleme (Adv. Solid State Phys.) (Edited by QUEISSER HJ.), Vol. XV, p. 113. Pergamon/Wieweg Brauschweig (1975). OVERHAUSER A.W.,Phys. Rev. 167, 691, (1968).
9.
WEXLERG.&WOLLEYA.M.,J.Phys. C. 9,1185(1976).
10.
SLATER G.C. & KOSTER GJ., Phys. Rev. 94, 1498, (1954).
11.
HAYDOCK R., HEINE V. & KELLY M.J.,J. Phys. C. 5,2845 (1972).
12. 13.
RICCO B., Phys. Status Solidi (b) 77,287 (1976). RICCO B., Ph.D. Thesis, University of Cambridge (U.K.), (1975).
14.
YOFFE A.D., Festkorperprobleme XIII (Adv. SolidState Phys.) (Edited by QUEISSER HJ.) p. 1. Pergamon/ Vieurg ‘(1973).
15.
WEXLER G. (Private communication).