- Email: [email protected]
One-electron dispersion relations in metallic VO2
Volume 61A, number 6
PHYSICS LETI’ERS
13 June 1977
ONE-ELECTRON DISPERSION RELATIONS IN METALLIC V02 T. ALTANHAN and G.J. HYLAND Department of Physics, University of Warwick, Coventry CV4 7AL, UK Received 4 May 1977 One-electron dispersion relations are presented for those bands in V02 involved in the metallic conduction which chaxacterises the rutile phase. The density of states at the Fermi energy (0.56 eV) is found to be 2.89 eV’ per cation, and the metallic state to be stable against Mott-insulation.
This letter reports the results of a tight-binding calculation of the structure of the two bands, d~1and ir* which are generally considered to be electronically active in the rutile metallic phase of V02 above the temperature T~(~341 K) of the first order semiconductor-metal phase transition. Considerations of the quantum chemistry of V02 indicate (1) that the first of these bands is based on a pure 3d cation orbital d11, the dominant overlap amongst which occurs parallel to the rutile cr-axis, and the second on a higher energy hybridised orbital ~* which contains a significant admixture of oxygen 2p orbitals; from a ligand field-theoretical 4~ionsanalysis in TiO’, of the E.S.R. spectrum of isolated V [2], the separation of these levels is known to be 0.43 eV and the degree of covalency to be about 30%. In consequence of the 90 rotation about Cr of the anion octahedra which co-ordinate each cation the two cations per rutile unit cell are non-equivalent; VO, is then conveniently described in terms of a simple tetragonal Bravais lattice with a two-cation basis. The associated bonding and anti-bonding cell orbitals then give rise to Bloch functions of the form:
W(k; r~)=
N
2) ~eh1CR1
_________
+
ni
X [‘Pa (r—Rl)+fl”I’b(r—Rl)}
(1)
,
S denotes the relevant overlap integral. In addition to this set of Bloch functions for the d11 -band there is a set for the lr*.band identical in structure to (1) but with d11 in eq. (2) replaced by ir*. Neglecting all intra and inter basis cation overlap integrals (the superscripts (1) and (2) in eq. (2) serve to distinguish the two cations of a given basis) the following energy dispersion relation then follows in the standard way-having in the present case first minimized with respect to i~in order to achieve the appropriate admixture of cell bonding and antibonding orbitals upon neglecting 3-centre integrals and taming up to next-nearest neighbour interactions~ E(k)=e 0—cs—2j3coskc —
k~a kya k~c ±87 COS—COS—COS— 2 2 2
,
(3)
x andy are taken to be parallel to the rutile ar-axes and z parallel to Cr~0 and c~have their usual (cationic) interpretation, whilst ~ controls the decrease in kinetic energy achieved by delocalization via tunnelling to nearest neighbour cations located on the same cation sublattice along the Cr-axis at ±ci and ‘y that to the eight next-nearest located on the “other” neighbour sublattice. cations, Theequidistantly positive (negative) sign associated with the fourth term on the R.H.S. of eq. (3) corresponds to the antibonding (bonding)
where N is the number of unit cells, R 1 locates the bases at the simple tetragonal lattice sites, i~is a cornplex admixture parameter and ‘Pb and ‘Pa are the bonding and antibonding orbitals having 2~d 2)d the form ‘Pb = = (~d11 ( (2) + ~ 11 ‘~v12(l+S) ~f2(1 —S) ______
426
_______ —
,
branch of the Bloch band which, it should be noted, receives contributions from both bonding and antibonding cell orbitals. ~ and defined, y can be inrelated to theaswidths of the d11 andNow ir~bands each case, the difference between the maximum and minimum of (3). The width of the d 11-band in the r —z direction is, in turn, an im-
Volume 61, number 6
PHYSICS LETTERS
13 June 1977
24———-—-.—...._•
24
2.2
2.2
2•(
N
N.
2C
.
N
N
S.
1.8
1.8
.“
/
1.6 14 E~)
/
1.2
/
/
/
/ 1.6
/ 14 Elk) /
~-
12
/
/
1.0 //
+
0.8
//
1.0
/
08
0.6
/
0.6
/
0.4
0.4
—.—
0.2
0.2
0
0
.11 2
iT
-~
2
k
5c
Fig. la. One-electron energy dispersion lion in rutileVO2. d11-band; —
along the r— Z direc= 7r*~band.
—• —. —
portant parameter (along with the energy separation of the bottoms of the two bands, 0.2 eV) in Hearn’s model [3] for the semiconductor-metal phase transition based on a phonon softening mechanism, a value of 2 eV being used in this work. Further details of his analysis indicate that in order to ensure the stability of the crystal through the phase transition a minimum value of 2/3 eV must be assigned to the width of this same band in the basal plane e.g. in the 1’ X direction in the Brillouin zone. Taking the zero of energy to coincide with the bottom of the d11 -band the following d band structure then results: —
~-
Fig.
lb. One electron energy dispersion along the F—I tion in rutile V02. —= d11-band; — — = n*band
direc-
sumption that the 1r*~bandis isotropic (/3 = ~); it can then be shown that E~~(k)= 1.59 —0.28 cosk2c ±1.11
cos k~a k~a k2c 2 cos 2 cos 2
lk~l, Ik~l~ ir/a, Ik~I~ ir/c. (5) The band structures (4) and (5) are shown in fig. la for the r Z direction (parallel to the cr-axis) of the Brillouin zone the positive (negative) signs identifying the antibonding (bonding) branches and in fig. lb for the F X direction (basal plane). The occurrence of an accidental degeneracy at the Fermi energy for the two directions already investigated could prove to be of particular importance in connection with the —
—
—
—
E11(k)
1.16—0.83 cosk2c k~a k~a k2c ±0.33 COS cos 2 COS 2’ =
k~I~I
I ~ ir/a, 1k2 I ~ ir/c.
(4)
The structure of the 1r*~bandcan be obtained from the above information, together with the fact that (e0 a)~* (0 a)11 = 0.43 eV, upon making the conventionally adopted [e.g. 1] as—
—
—
occurrence of the phase transition to the semiconducting state and warrants further investigation via a construelion of the whole Fermi surface; it should, however, be pointed out that the occurEence and location of the degeneracy is rather sensitive to the width of the r”band whose value cannot be considered to be more than 10% accurate. 427
Volume 61A, number 6
PHYSICS LETTERS
~*~bandand the (unoccupied) antibonding d11-band, varying between 0.46 eV at the F-point to 0.55 eV at the Fermi level in the F—Z direction, values which are not far removed from those found experimentally
-~
N(E),eV per catOn
via optical reflectivity [6] and photoemission [7] studies. (In the vicinity of 0.7 eV.) Finally in view of problems well known to be associated with the validity of the band model in transition metal compounds the dispersion relations (4) and (5)
8’
I 6
were used to compare the average energy associated with the delocalized band states with that of the corresponding Mott-insulating state in which the delectrons are localized, one per cation. Owing to the increase in Coulomb potential energy associated with doubly occupied cations which the band model, through
]
5 6
~
~ 2
L1
1
0.4
08
I 1.2 E,eV
16
2.0
2.4
Fig. 2. Joint density of states in rutile VO2 calculated numerically from eqs. (4), (5); the shaded area denotes occupied states (EF= 0.56 eV).
Calculation of the joint density of states associated with eqs. (4) and (5) by Monte Carlo method yields the results shown in fig. 2 from which the Fermi energy is found to lie at 0.56eV above the bottom of the d11band, at which point the density of states is 2.89 eV~ per cation. It is gratifying to note that our value of EF is identical to that found from the A.P.W. calculation of Chatterjee et al. [4] and only 0.02 eV below that obtained by Caruthers et al. [5]. Other associated quantities of interest are (a) the fractional electron occupancy of the d11 -band which turns out to be 0.88, indicating a 12% spillage*i into the i~*~band, (b) the density of states effective mass at the Fermi level whose directionally averaged values for the d11 and ir -bands are found to be 4.1 m and l.3m, respectively, and (c) the occurrence of a vertical gap between the (occupied) portion of the bonding *
This value falls in the range of possible values (of n) admitted by Hearn [3], indicating a certain stability of his parameters against change from his assumed band structure to that given above, eqs. (4) and (5).
428
13 June 1977
its neglect of positional correlations between electrons of antiparallel spin, allows to occur with a binomial probability distribution it can be shown that the average itinerant energy associated with eq. (1) is lower than that of the Mott-insulating state by only 0.2 eV; it is to be noted that this stability of the metallic state is contingent upon the d11-band being overlapped by the 7r*~band,for otherwise the d11 -band is half filled in which case it can be shown that the average energy of the itinerant state exceeds that of the Mott-insulating state by 0.4 eV. Further details of the calculatio~sreported above will shortly be published elsewhere. T.A. wishes to thank the Turkish Government for the award of a scholarship during the tenure of which this work was performed, and the British Council for an Overseas Fees Award (O.S.F.A.S.).
References Eli J.B. Goodenough, Bull. Soc. Chim. Fr. 4(1965)1200. [2] T. Shimizu, J. Phys. Soc. Japan 23 (1967) 848. [3] C.J. Hearn, J. Phys. CS (1972) 1317. [4] S. Chatterjee, T.K. Mitra and G.J. Hyland, Phys. Letts. 42A (1972) 56. [Si E. Caruthers, L. Kleinman and H.I. Zhang, Phys. Rev. B7 (1973) 3753. [6] V.G. Mokerov and A.V. Rakov, Soy. Phys. Solid St. 11 (1969) 150. [7] R.J. Powell, C.N. Berglund and W.E. Spicer, Phys. Rev. 178 (1969) 1410.
PHYSICS LETI’ERS
13 June 1977
ONE-ELECTRON DISPERSION RELATIONS IN METALLIC V02 T. ALTANHAN and G.J. HYLAND Department of Physics, University of Warwick, Coventry CV4 7AL, UK Received 4 May 1977 One-electron dispersion relations are presented for those bands in V02 involved in the metallic conduction which chaxacterises the rutile phase. The density of states at the Fermi energy (0.56 eV) is found to be 2.89 eV’ per cation, and the metallic state to be stable against Mott-insulation.
This letter reports the results of a tight-binding calculation of the structure of the two bands, d~1and ir* which are generally considered to be electronically active in the rutile metallic phase of V02 above the temperature T~(~341 K) of the first order semiconductor-metal phase transition. Considerations of the quantum chemistry of V02 indicate (1) that the first of these bands is based on a pure 3d cation orbital d11, the dominant overlap amongst which occurs parallel to the rutile cr-axis, and the second on a higher energy hybridised orbital ~* which contains a significant admixture of oxygen 2p orbitals; from a ligand field-theoretical 4~ionsanalysis in TiO’, of the E.S.R. spectrum of isolated V [2], the separation of these levels is known to be 0.43 eV and the degree of covalency to be about 30%. In consequence of the 90 rotation about Cr of the anion octahedra which co-ordinate each cation the two cations per rutile unit cell are non-equivalent; VO, is then conveniently described in terms of a simple tetragonal Bravais lattice with a two-cation basis. The associated bonding and anti-bonding cell orbitals then give rise to Bloch functions of the form:
W(k; r~)=
N
2) ~eh1CR1
_________
+
ni
X [‘Pa (r—Rl)+fl”I’b(r—Rl)}
(1)
,
S denotes the relevant overlap integral. In addition to this set of Bloch functions for the d11 -band there is a set for the lr*.band identical in structure to (1) but with d11 in eq. (2) replaced by ir*. Neglecting all intra and inter basis cation overlap integrals (the superscripts (1) and (2) in eq. (2) serve to distinguish the two cations of a given basis) the following energy dispersion relation then follows in the standard way-having in the present case first minimized with respect to i~in order to achieve the appropriate admixture of cell bonding and antibonding orbitals upon neglecting 3-centre integrals and taming up to next-nearest neighbour interactions~ E(k)=e 0—cs—2j3coskc —
k~a kya k~c ±87 COS—COS—COS— 2 2 2
,
(3)
x andy are taken to be parallel to the rutile ar-axes and z parallel to Cr~0 and c~have their usual (cationic) interpretation, whilst ~ controls the decrease in kinetic energy achieved by delocalization via tunnelling to nearest neighbour cations located on the same cation sublattice along the Cr-axis at ±ci and ‘y that to the eight next-nearest located on the “other” neighbour sublattice. cations, Theequidistantly positive (negative) sign associated with the fourth term on the R.H.S. of eq. (3) corresponds to the antibonding (bonding)
where N is the number of unit cells, R 1 locates the bases at the simple tetragonal lattice sites, i~is a cornplex admixture parameter and ‘Pb and ‘Pa are the bonding and antibonding orbitals having 2~d 2)d the form ‘Pb = = (~d11 ( (2) + ~ 11 ‘~v12(l+S) ~f2(1 —S) ______
426
_______ —
,
branch of the Bloch band which, it should be noted, receives contributions from both bonding and antibonding cell orbitals. ~ and defined, y can be inrelated to theaswidths of the d11 andNow ir~bands each case, the difference between the maximum and minimum of (3). The width of the d 11-band in the r —z direction is, in turn, an im-
Volume 61, number 6
PHYSICS LETTERS
13 June 1977
24———-—-.—...._•
24
2.2
2.2
2•(
N
N.
2C
.
N
N
S.
1.8
1.8
.“
/
1.6 14 E~)
/
1.2
/
/
/
/ 1.6
/ 14 Elk) /
~-
12
/
/
1.0 //
+
0.8
//
1.0
/
08
0.6
/
0.6
/
0.4
0.4
—.—
0.2
0.2
0
0
.11 2
iT
-~
2
k
5c
Fig. la. One-electron energy dispersion lion in rutileVO2. d11-band; —
along the r— Z direc= 7r*~band.
—• —. —
portant parameter (along with the energy separation of the bottoms of the two bands, 0.2 eV) in Hearn’s model [3] for the semiconductor-metal phase transition based on a phonon softening mechanism, a value of 2 eV being used in this work. Further details of his analysis indicate that in order to ensure the stability of the crystal through the phase transition a minimum value of 2/3 eV must be assigned to the width of this same band in the basal plane e.g. in the 1’ X direction in the Brillouin zone. Taking the zero of energy to coincide with the bottom of the d11 -band the following d band structure then results: —
~-
Fig.
lb. One electron energy dispersion along the F—I tion in rutile V02. —= d11-band; — — = n*band
direc-
sumption that the 1r*~bandis isotropic (/3 = ~); it can then be shown that E~~(k)= 1.59 —0.28 cosk2c ±1.11
cos k~a k~a k2c 2 cos 2 cos 2
lk~l, Ik~l~ ir/a, Ik~I~ ir/c. (5) The band structures (4) and (5) are shown in fig. la for the r Z direction (parallel to the cr-axis) of the Brillouin zone the positive (negative) signs identifying the antibonding (bonding) branches and in fig. lb for the F X direction (basal plane). The occurrence of an accidental degeneracy at the Fermi energy for the two directions already investigated could prove to be of particular importance in connection with the —
—
—
—
E11(k)
1.16—0.83 cosk2c k~a k~a k2c ±0.33 COS cos 2 COS 2’ =
k~I~I
I ~ ir/a, 1k2 I ~ ir/c.
(4)
The structure of the 1r*~bandcan be obtained from the above information, together with the fact that (e0 a)~* (0 a)11 = 0.43 eV, upon making the conventionally adopted [e.g. 1] as—
—
—
occurrence of the phase transition to the semiconducting state and warrants further investigation via a construelion of the whole Fermi surface; it should, however, be pointed out that the occurEence and location of the degeneracy is rather sensitive to the width of the r”band whose value cannot be considered to be more than 10% accurate. 427
Volume 61A, number 6
PHYSICS LETTERS
~*~bandand the (unoccupied) antibonding d11-band, varying between 0.46 eV at the F-point to 0.55 eV at the Fermi level in the F—Z direction, values which are not far removed from those found experimentally
-~
N(E),eV per catOn
via optical reflectivity [6] and photoemission [7] studies. (In the vicinity of 0.7 eV.) Finally in view of problems well known to be associated with the validity of the band model in transition metal compounds the dispersion relations (4) and (5)
8’
I 6
were used to compare the average energy associated with the delocalized band states with that of the corresponding Mott-insulating state in which the delectrons are localized, one per cation. Owing to the increase in Coulomb potential energy associated with doubly occupied cations which the band model, through
]
5 6
~
~ 2
L1
1
0.4
08
I 1.2 E,eV
16
2.0
2.4
Fig. 2. Joint density of states in rutile VO2 calculated numerically from eqs. (4), (5); the shaded area denotes occupied states (EF= 0.56 eV).
Calculation of the joint density of states associated with eqs. (4) and (5) by Monte Carlo method yields the results shown in fig. 2 from which the Fermi energy is found to lie at 0.56eV above the bottom of the d11band, at which point the density of states is 2.89 eV~ per cation. It is gratifying to note that our value of EF is identical to that found from the A.P.W. calculation of Chatterjee et al. [4] and only 0.02 eV below that obtained by Caruthers et al. [5]. Other associated quantities of interest are (a) the fractional electron occupancy of the d11 -band which turns out to be 0.88, indicating a 12% spillage*i into the i~*~band, (b) the density of states effective mass at the Fermi level whose directionally averaged values for the d11 and ir -bands are found to be 4.1 m and l.3m, respectively, and (c) the occurrence of a vertical gap between the (occupied) portion of the bonding *
This value falls in the range of possible values (of n) admitted by Hearn [3], indicating a certain stability of his parameters against change from his assumed band structure to that given above, eqs. (4) and (5).
428
13 June 1977
its neglect of positional correlations between electrons of antiparallel spin, allows to occur with a binomial probability distribution it can be shown that the average itinerant energy associated with eq. (1) is lower than that of the Mott-insulating state by only 0.2 eV; it is to be noted that this stability of the metallic state is contingent upon the d11-band being overlapped by the 7r*~band,for otherwise the d11 -band is half filled in which case it can be shown that the average energy of the itinerant state exceeds that of the Mott-insulating state by 0.4 eV. Further details of the calculatio~sreported above will shortly be published elsewhere. T.A. wishes to thank the Turkish Government for the award of a scholarship during the tenure of which this work was performed, and the British Council for an Overseas Fees Award (O.S.F.A.S.).
References Eli J.B. Goodenough, Bull. Soc. Chim. Fr. 4(1965)1200. [2] T. Shimizu, J. Phys. Soc. Japan 23 (1967) 848. [3] C.J. Hearn, J. Phys. CS (1972) 1317. [4] S. Chatterjee, T.K. Mitra and G.J. Hyland, Phys. Letts. 42A (1972) 56. [Si E. Caruthers, L. Kleinman and H.I. Zhang, Phys. Rev. B7 (1973) 3753. [6] V.G. Mokerov and A.V. Rakov, Soy. Phys. Solid St. 11 (1969) 150. [7] R.J. Powell, C.N. Berglund and W.E. Spicer, Phys. Rev. 178 (1969) 1410.