the present position of theory and experiment for vo2

Mat. R e s . Bull . Vol. 5, pp. 691-70Z, 1970. P e r g a m o n P r e s s , Inc. P r i n t e d in the Uni ted S ta tes .

THE PRESENT POSITION OF THEORY AND EXPERIMENT FOR VO 2

William Paul Division of Engineering and Applied Physics Harvard University, Cambridge, Massachusetts

(Received June Z, 1970)

ABSTRACT

The most important electrical and optical properties of VOp below and above the transition temperature between insulatlng and metallic behavior are analyzed to provide a few basic conclusions regarding the band structure and the transport mechanism. An argument is presented that electron-phonon interaction is the basic mechanism underlying the phase transition.

Introduction

In this contribution, I shall first review briefly the most

important features of the physical measurements made on crystalline VO 2

above and below its transition temperature between insulator and metal, and

shall draw some conclusions from them regarding the band structure and the

transport mechanism. Drawing on thermodynamic arguments, I shall then assess

the probably driving force for the transition, and shall present a somewhat

speculative and intuitive microscopic mechanism for it. This will suggest

the experiments and theory deemed most useful for further progress in under-

standing.

Review of physical measurements

Above a transition temperature T of 339°K, VO 2 crystallizes C

691

692 VANADIUM OXIDE Vol. 5, No. 8

in a tetragonal rutile structure. The V 4+ ions are equally spaced

along the c-axis, and there are two V 4+ ions per unit cell in

identical surroundings with six nearest neighbor oxygen (0--) ions. Below

339°K there is a phase transition to monoclinic. The unit cell doubles in

size. There is a very small volume change at the transition, whose most

remarkable feature is a considerable distortion of the unit cell shape. The

V 4+ ions move together in pairs and shift so as to be no longer colinear.

Recently it has been demonstrated by Kennedy and Mackenzie(l) that amor-

phous VO 2 films deposited on substrates above T c are metallic and show no

discontinuity in conductivity o, and that amorphous films similarly deposited

on substrates held below T are semiconducting but also show no conductivity c

discontinuity. This work suggests that the amorphous films have short-range

order dictated by the temperature of the substrate, which thus determines the

magnitude of o and the sign of the conductivity change with temperature, but

that the structure change is suppressed, and with it, very significantly, the

conductivity discontinuity. Apparently a change in long range order of a

lattice is necessary for the abrupt change in o.

The stoichiometry, chemical purity and freedom from defects of the VO 2

hitherto measured have not been high, judging by the standards of germanium

technology. Typical results of conductivity versus temperature are shown in

Figure i. I shall address my analysis to the results from my own laboratory(2)

which are judged to be for higher quality crystals because of the larger con-

ductivity at T c, the larger conductivity discontinuity at T c, and the lower

conductivity just below Tc; the difference from other results is significant

for the deduction of gaps and masses (a) For T > Tc, o is about 104 ohm -I -

cm -I, and decreases linearly with increasing temperature. Hence the use of

the term "metallic phase." There is typically a hysteresis of about I°K at a

T of (339 ± I)°K. The value of T is not affected signi£icantly by changes c c

in the conductivity or its activation energy below T c.

Vol. 5, No. 8 VANADIUM OXIDE 693

101

i0 °

E 0

>'10-'

r1,-

10-2

10 -3

Sasaki & Watanabe [4]'x~."" / / Lade & Paul [ 2 4 ] ~ " q , ~ _

! .J" Everhart & ~ / UacChesney [14]

il, il 8ongers [6] + "+"

. : + . . - , 4 " " "

..j.:. r i i i

I ! i + ~ "

i: + ~ I d

i:: + I ~ x "1 ~ Morin [1] if' i~ I 'I

I I i i l Measurements made I I::1 along tetragonal c-axis I ":

I !~J I

-J ii',

- II

- I + I

I+

_ .~ r-'r...+-'_:;_.+_+_+ +.~-" T - __ +~+

I I I - i I t t t i

3 1000 4

FIG. 1

Conductivity v s temperature measurements by several investigators.


(b) For T < Tc, the most extensive measurements show that En @ changes

non-llnearly with i/T(2).Thls prevents simple deductions about energy gaps

and their change with temperature and pressure.

The Hall effect measurements from different laboratories substantially

agree. Below Tc, the use of the simple one-band formulae, o = ne~ and

R H = I/he, lead to an electronic Hall mobility of about 0.5 cm2/volt-sec,

which decreases slowly and linearly with increasing temperature, at approxi-

mately the same rate as the metallic conductivity above T c (3).

The reflectlvlty of crystals and the reflectlvlty and tramsmisslon of

films have been measured above T throughout the visible and the infrared. c

The results of different investigations substantially agree. At long wave-

lengths the optical constants are dominated by free-carrler-llke effects,

while in the visible there is an undramatic but distinct maximum in reflec-

tivity, and in the derived E2, near 3 eV. Below Tc, the. free-carrier-like

b e h a v i o r i s much r e d u c e d and s t r u c t u r e due t o phonon a b s o r p t i o n a p p e a r s . The

s t r u c t u r e a b o v e 2 eV i s s u b s t a n t i a l l y u n c h a n g e d b u t a new d i s t i n c t p e a k

a p p e a r s b e t w e e n 0 . 8 eV and 1 . 1 eV, d e p e n d i n g on t h e s a m p l e . T h i s 1 eV p e a k

c a n b e a t t r i b u t e d t o t h e o p e n i n g up o f a g a p i n t h e d - b a n d d e n s i t y o f s t a t e s

v e r s u s e n e r g y . T h e r e a r e no s h a r p p e a k s i n E 2 i n t h e v i s i b l e s u c h a s a r e

s e e n i n NiO and a t t r i b u t e d t h e r e t o i n t r a l o n l c d - t r a n s l t l o n s . The a b s o r p t i o n

n e a r 1 eV h a s b e e n i n v e s t i g a t e d i n c o n s i d e r a b l e d e t a i l ( 2 ) . F i g u r e 2 shows t h e

a b s o l u t e t r a n s m i s s i o n v e r s u s e n e r g y a t t e m p e r a t u r e s b e l o w T c . S i n c e t h e

e d g e s h a p e i s r o u g h l y t h e same a t a l l t e m p e r a t u r e s , a r e a s o n a b l e a p p r o x i m a -

t i o n t o an e n e r g y g a p c h a n g e w i t h t e m p e r a t u r e c a n b e o b t a i n e d f rom t h e s h i f t

o f t h e e n e r g y c o r r e s p o n d i n g t o T - 10 - 4 . T h i s i s shown on F i g u r e 3 : t h e

c h a n g e i n g a p i s l a r g e ( s t r o n g e l e c t r o n - p h o n o n i n t e r a c t i o n ? ) , t h e c u r v e i s

s l l g h t l y c o n c a v e t o t h e T a x i s , b u t t h e r e i s no u n d e n i a b l e e v i d e n c e o f an

a c c e l e r a t i n g d e c r e a s e o f g a p n e a r T c up t o 3 ° f rom t h e t r a n s i t i o n ( 2 ) .

The specific heat versus temperature (4) is shown on Fig. 4, The transl-

Vol. 5 , No. 8 VANADIUM OXIDE 695

1 . 0 -

1~1_-- C 0

(/~ , - -

E .~ 10 -2 --_ ~ -

1 0 - 3 _

10-4 _ . 2

I I I I I

.5 .4 .5 .6 .7 Energy (eV)

~IG. 2

Transmission vs energy near i eV at several temperatures below T c (2).

8

tion is first order, with a latent heat estimated at 1020 calories/mole of

VO 2. Unfortunately the data are sparse just above Tc, preventing determina-

tion of 0 D in the metallic phase, and preventing an accurate estimate of the

individual contributions of the electrons and the lattice vibrations to the

heat capacity above the transition. However, we note that the total heat

capacity above T c does not appear to have changed much.

The reported thermal conductivity is completely insensitive to tempera-

ture over the full temperature range including Tc, which it is hard to Jus-

tify without rather ad hoc assumptions about phonon scattering processes (4).

The magnetic susceptibility shows a large discontinuity at Tc, but is other-

wise constant with T both above and below Tc(4).Neutron diffraction, nmr ,

and susceptibility measurements have all failed to uncover any magnetic order-

696 VANADIUM OXIDE Vol. 5, No, 8

O.75

~ 0 . 7 0

,,5

0 . 6 5

I I I I I I I I I -120 -100 -80 -60 -40 -20 0 20 40 60

Temperature (°C)

FIG. 3

Variation of d-band energy gap with temperature, deduced from

the data of Figure 2.

Ing in the low temperature phase.

Conclusions regardln 8 band structure and transport

The optical data have been used to give us a general crude picture of the

band structure. Comparison with TiO 2 and V205, which are insulators with no d

electrons, and with SrTiO 3, very plausibly suggests that the oxygen 2p band

lies some 3 eV below the Fermi level in VO 2. Thus the structure in the optical

constants, for h~ > 2.5 eV in both phases, is caused by 2p-3d transitions.

The structure near 1 eV in the low temperature phase is then attributed to

transitions inside split 3d bands.

It is notable that these data strongly suggest fairly wide 3d bands. In


24-

22-

20-

18-

6-

4-

2

0 0

_ 16 o

"' 14 ---I 0

12 l.J-I

.__i

• "~ 10

8

LATENT HEAT = VO 2

1020:*:5 CAL/MOLE

0

0

0

0 0

0 .,....= ="'~

f

-- RYDER ooooo COOK

THEORETICAL Cv FOR DEBYE TEMPERATURE OF 750°K

I I I I I I i I I I00 200 300 400 500 600 700 800 900

TEMPERATURE (OK)

FIG. 4

Specific heat vs temperature (4).

the first place, we note that Kahn and Leyendecker's calculations for SrTiO 3,

where the ionic separations are similar to those of VO2, suggest band widths of

the order of a volt on the basis of 2p-3d hybridization. Second, the

observed reflectivity peaks are broad, suggesting a band-wldth of perhaps

1.5 eV. Third, we find none of the sharp peaks in absorption in the visible

seen in transition metal oxides with more d-electrons, and almost certainly

associated with intraionic transitions. When we do see optical absorption

associated with the d-electrons, near i eV, the absorption coefficient


involved is 104 - 105 cm -I, in contrast to 500 cm -I for the sharp peaks in

NiO. As a matter of fact, there is a systematic trend, throughout all the

transltlon metal oxides, of a reduction in sharpness of the intralonic 3d

absorption as the number of 3d electrons is reduced. This suggests that for

the vanadium oxides there is very conslderable hybridization of the 3d cation

wave functions with the anion 2p functions, spreading the d levels into bands,

permitting the d-d optical transitions, and forcing a tendency toward metallic

band behavior.

Thus, above T c, we very likely have overlapplng d bands with a complex

Fermi surface corresponding to carriers of several masses, signs, and r~gimes

of transport. In this circumstance, the many estimates in the literature of

mobilltles, effective carrier densities and effective masses derived from

transport coefficients appllcable to parabolic bands and spherlcal energy

surfaces would appear to be questlonable.

The optical data for T < T strongly suggest that a gap in the d-electron c

state density opens up, corresponding to the changes due to the reduction in

symmetry and possible band spllttings at the new boundaries of the reduced

Brillouin zone. From the measured conductivity and Hall constant, the usual

Boltzmann transport expressions yield carrier densities of the order of

1017 - 1018 cm -3 and a Hall mobility of 0.5 cm2/volt-sec. It is easy to show

that a mean free path of one interatomic distance or 3A ° corresponds to a

mobility of 5~cm2/volt-sec, so that the measured mobility would require

a mass of i00 m 0 for a mean free path of even one interatomic distance. The

wide bands deduced from the optical data and reasonable estimates of the pola-

ron coupling constant make such a mass unlikely. We have to conclude that

the model of an itinerant electron suffering occasional scattering must break

down, since the low measured mobility corresponds to a mean free path less

than an interatomlc distance. It follows that a diffusion theory type of

calculation for both the conductivity and the Hall field are required before


we can confidently reach conclusions about the sign of carrier involved and

its density.

The insulator-to-metal transition

In considering the entropy changes at the transition it

seems appropriate to discard the possibility of magnetic

contributions (there being no identified change in order) and contributions

from vacancies and defects. We shall ,assume to begin with that the contribu-

tions of electrons and lattice vibrations can be considered separately. It

can then be straightforwardly argued that the electrons probably do not make

the main contribution to the change in entropy4). Although the deduction of

electron density below T c may be uncertain for the reasons discussed above,

a density of 1018 cm -3 would have negligibly small entropy. Above T the c

density of "free" electrons evidently cannot exceed i per V 4+ atom or

3.4 x 1022 cm -3. On a parabolic band model with a Fermi energy of i eV the

corresponding T c AS e = ~2k2Tc2/2 E F is about i00 calorles/mole. The maximum

value of T AS for an infinitely narrow band would be an order of magnitude c e

larger, close to the observed latent heat, but this would be inconsistent

with our earlier arguments for relatively wide bands, and would surely lead

to small polaron hopping transport. More appropriate formulae for the pos-

tulated complex "metallic" Fermi surface would provide a smaller entropy

increase. Thus we conclude that changes in purely electronic entropy would

be insufficient to explain the latent heat at the transition.

We therefore examine the possible lattice entropy change. Since we know

very little about the lattice vibration spectrum and since we know nothing

about how any of these mode frequencies change in the transition to the metal-

lic phase, we cannot make any sure statements about the lattice entropy.

However we can rather easily find out the order of size of changes in the

spectrum needed. My student Larry Ladd has calculated that a drop in the

frequency of even one acoustical mode by 70% would provide 50% of the entropy

700 V A N A D I U M O X I D E V o l . 5, No. 8

change at Tc, or that a decrease in average frequency over the vibration

spectrum of less than 10% would accomplish the same increment in entropy.

Clearly the lattice entropy does not have to change by a large fraction

to explain the transition, and so it does not seem unreasonable to look to

the lattice as the drive for the transition.

In searching for a microscopic model for the transition we shall there-

fore confine ourselves to ones where a change in lattice entropy plays a

major role. For VO2, considered in isolation, it might be accepted as an

accident that two different crystal phases, which happened to have vastly

different mobile electronic densities, became equal in free energy. We must,

however, remember that many transition metal oxides and sulfides, and espe-

cially the oxides of vanadium, show insulator to metal transitions and so it

is worthwhile to examine possible reasons why they may be somehow special.

It appears that the reason for this may reside in the fact that the distri-

bution of d-electron charge near the vanadium ions is such that the spectrum

of lattice vibrational frequencies is vitally dependent on it, in such a way

that a rapidly increasing lattice entropy and decreasing free energy in the

metallic phase "drives" a phase transition from the insulating to the metal-

lic state as the temperature is increased. First, a phenomenological argu-

ment why vanadium compounds might be particularly susceptible to strong d-

electron-phonon interaction: we saw that the magnitude and freqdency band

width of intraionic d-electron optical absorption varied throughout the

series of transition metal oxides, being small and narrow in NiO and large

and wide in TiO. In VO 2 it was argued that the d-electrons were neither

tightly bound to the V 4+ (which would lead to sharp optical spectra) nor very

much spread out over neighboring oxygens and V 4+ cations (which would imply

quite wide bands). It follows that the hybridized d-electrons are critically

intermediate in character between localized and band electrons, bound in

their lowest energy states by less than a volt, so that excitation is easy

Vol . 5, No. 8 VANADIUM OXIDE 701

with changing temperature or stress at little cost in energy, and yet the

electrons are sufficiently localized that their excitation produces consider-

able local fluctuation in charge.

The strong d-electron-phonon interaction alters the total d-electron

energy in both phases, but, by second-order perturbation theory, will have a

greater effect in the metal because of the greater state degeneracy at the

Fermi level. However, the more important consequence is a softening of the

modes, and especially of the modes in the metallic phase, which will increase

as the temperature is raised. This means that with increasing temperature

the lattice free energy decreases more rapidly for the "metallic" configura-

tion and eventually the total free energy becomes lower for this phase.

In this argument the charge distribution of electrons near the Fermi lev-

el is important in determining the lattice vibrational spectrum. In VO 2 these

electrons are probably in states described by orbitals along the c-axis or

pseudo c-axls. It is notable that stress applied along this axis shifts the

transition temperature by a much greater amount than basal plane stress or

hydrostatic pressure2). It is clear that the main effect described is not

connected with the Mort transition, although the possibility of correlation

and polaronic effects must be included.

Our analysis of the available data and speculative model for the transi-

tion mechanism suggest the following experiments and calculations be tried

(i) an extension of the heat capacity measurements to higher temperatures to

firmly establish the thermodynamic data (2) measurements of sound velocity,

infrared absorption spectra, Raman scattering spectra, Debye-Waller factor,

x-ray thermal diffuse scattering and neutron diffraction to give the spectrum

of lattice vibrations on both sides of the transition and to examine any con-

tinuous mode softening in either phase, particularly near T (3) fundamental c

calculations on the transport coefficients in the regime of low mobility and

on infrared absorption caused by internal transitions in narrow bands, and a

70Z VANADIUM OXIDE Vol. 5, No. 8

closer look at the shielding of lattice vibrations by electrons in narrow bands

which distribute their charge in hybridized and yet still highly directed

orbitals.

A cknowledsment

It is a pleasure to acknowledge help on the subject of

this review given me by Larry Ladd, John Fan and William Rosevear. I am

especially indebted to Clifford Hearn for many valuable discussions on the

possible mechanism for the phase transition.

1.

2.

3.

4.

5.

References

T. N. Kennedy and J. D. Mackenzie, J. Non-tryst. Solids l, 326 (1969).

L. Ladd and W. Paul, Solid State Communications ~, 425 (1969).

W. Rosevear and W. Paul, Bull. Am. Phys. Soc., Ser. I~, 15, 316 (1970).

C. N. Berglund and H. J. Guggenheim, Phys. Rev. 185, 1029 (1969).

D. Adler and H. Brooks, Phys. Rev. 155, 826 (1967).

the present position of theory and experiment for vo2

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