Band Orbital Mixing and Electronic Instability of Low-Dimensional Metals

Band Orbital Mixing and Electronic

Downloaded by GEORGE MASON UNIV on June 18, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch015

Instability of Low-Dimensional Metals Myung-Hwan Whangbo Department of Chemistry, North Carolina State University, Raleigh, NC 27695-8204

Electronic instability of a low-dimensional metal leading to its metal-insulator or metal-superconductor transition is discussedfrom the viewpoint of orbital mixing between filled and empty levels near the Fermi level of a normal metallic state. This chapter examines the concept of Fermi surface nesting, its relationship to a metal-insulator transition, the correlation between real and reciprocal space prop erties, and the difference in orbital mixing between insulating and superconducting states.

S O L I D - S T A T E M A T E R I A L S A R E O F T E N C L A S S I F I E D a c c o r d i n g to h o w

their

resistivities (p) v a r y as a function o f t e m p e r a t u r e (Γ). T h u s , metals a n d s e m i conductors are characterized b y positive a n d negative slopes i n t h e i r ρ v s . Τ plots. T h e stability o f a m e t a l l i c state d e p e n d s o n several factors, s u c h as t e m p e r a t u r e a n d pressure. W h e n t h e t e m p e r a t u r e is l o w e r e d , a m e t a l m a y b e c o m e a s e m i c o n d u c t o r (1) o r a s u p e r c o n d u c t o r (2).

τ

τ

1

2

T h e e l e c t r o n i c structure o f a s o l i d is d e s c r i b e d b y e n e r g y bands as s h o w n i n 3, w h e r e rectangular boxes represent a l l o w e d regions o f e n e r g y (i.e., 0065-2393/90/0226-0269$06.00/0 © 1990 American Chemical Society

Johnson et al.; Electron Transfer in Biology and the Solid State Advances in Chemistry; American Chemical Society: Washington, DC, 1989.

270

E L E C T R O N TRANSFER IN BIOLOGY A N D T H E SOLID STATE

• •


3 energy bands). A g i v e n b a n d consists of Ν discrete l e v e l s , w h e r e Ν is the total n u m b e r o f u n i t cells i n a solid. Because Ν —» °c for a l l practical purposes, all energy levels falling w i t h i n a b a n d are a l l o w e d . I n a o n e - e l e c t r o n b a n d p i c t u r e , e l e c t r o n - e l e c t r o n r e p u l s i o n is n e g l e c t e d so that each b a n d l e v e l can b e filled w i t h t w o electrons. I n this p i c t u r e , a s e m i c o n d u c t o r (or an insulator) contains o n l y c o m p l e t e l y filled a n d c o m p l e t e l y e m p t y bands, so that an e n e r g y gap (i.e., b a n d gap £ ) exists b e t w e e n the highest o c c u p i e d a n d the lowest u n o c c u p i e d b a n d levels (see 4a). ( A n insulator is a s e m i c o n d u c t o r g

±

4 a w i t h a large b a n d gap.) O n the o t h e r h a n d , a m e t a l has at least one partially filled b a n d (see 4b) so that no energy gap exists b e t w e e n the highest o c c u p i e d

4 b l e v e l (i.e., the F e r m i l e v e l e ) a n d the lowest u n o c c u p i e d b a n d l e v e l . T h e e l e c t r o n i c instability of a m e t a l that leads to e i t h e r a m e t a l - i n s u l a t o r or a m e t a l - s u p e r c o n d u c t o r transition m a y b e d e s c r i b e d o n the basis o f o r b i t a l m i x i n g b e t w e e n the o c c u p i e d a n d u n o c c u p i e d levels (1-3). A n e w e l e c t r o n i c state d e r i v e d f r o m the o r b i t a l m i x i n g m a y b e c o m e m o r e stable than the f


15.

271

Low-Dimensional Metals

WHANGBO

metallic state w h e n the energy gain r e s u l t i n g f r o m the interactions b e t w e e n the o c c u p i e d a n d u n o c c u p i e d levels outweighs the i n h e r e n t e n e r g y increase caused b y i n t r o d u c i n g h i g h e r - l y i n g , u n o c c u p i e d levels. Because the energy difference b e t w e e n the o c c u p i e d a n d u n o c c u p i e d levels a r o u n d the F e r m i l e v e l can b e v e r y s m a l l , the extent of this e n e r g y increase can be made v e r y small. C o n s e q u e n t l y , o r b i t a l m i x i n g a m o n g the b a n d levels i n the v i c i n i t y of the F e r m i l e v e l is c r u c i a l for a m e t a l - i n s u l a t o r or a m e t a l - s u p e r c o n d u c t o r transition. F r o m the v i e w p o i n t of one-electron b a n d theory, a m e t a l - i n s u l a t o r t r a n


sition occurs w h e n the F e r m i surface of a partially filled b a n d is w e l l n e s t e d . A m e t a l may have a chance to b e c o m e s u p e r c o n d u c t i n g o n l y w h e n it is free from electronic instability t o w a r d a m e t a l - i n s u l a t o r transition. T h i s chapter considers h o w the concept of F e r m i surface n e s t i n g comes about, w h y F e r m i surface nesting leads to an electronic i n s t a b i l i t y t o w a r d a m e t a l - i n s u l a t o r transition, h o w r e a l a n d reciprocal space properties are r e l a t e d , a n d finally h o w the o r b i t a l m i x i n g l e a d i n g to a s u p e r c o n d u c t i n g state differs from that l e a d i n g to an i n s u l a t i n g state.

Fermi Surface Nesting F o r s i m p l i c i t y , let us consider a t w o - d i m e n s i o n a l (2D) rectangular lattice 5, w i t h repeat distances a and b. T h e coordinate of a lattice site is g i v e n b y (ma, nb), w h e r e m a n d η are integers. I f one o r b i t a l represents each lattice site, the o r b i t a l located at the site (ma, nb) may d e n o t e d b y \ . I n this notation, the o r b i t a l at the coordinate o r i g i n is g i v e n b y χοο a n d those at its nearest-neighbor sites along the a- a n d h- directions b y χ and χ , re m n

1 0

0 1

spectively. T h e a l l o w e d energy levels of lattice 5 are d e s c r i b e d b y the b a n d h τ

a -w

·

·

,

,

b i.

»

5 orbitals φ ( * ,

k ),

β

b

Φ(Κ,

h)

=

N" 2 Σ exp(ifc ma) 1/2

a

· exp(ik nb) b

· \

m n

(1)

m η

w h e r e k a n d k are wave vectors along the a a n d b d i r e c t i o n s , r e s p e c t i v e l y , a n d the terms exp(ik ma) a n d exp(ik nb) are coefficients for the site o r b i t a l X . F o r convenience, the wave vectors k a n d k m a y b e confined to the a

h

a

m n

b

a

h


272

E L E C T R O N TRANSFER IN BIOLOGY A N D T H E S O L I D STATE

f o l l o w i n g values: - τ τ / α ^ k

< tt/a a n d -πlb

a

T h e nature of the b a n d energies E(k , k ) is usually e x a m i n e d b y p l o t t i n g E(k , k ) as a i u n c t i o n o f w a v e vectors (i.e., b a n d dispersions) a l o n g c e r t a i n lines of the first B r i l l o u i n zone (e.g., Γ—»X—»M—»Υ-»Γ i n F i g u r e 1). F i g u r e 2 shows three examples o f b a n d dispersions that illustrate h o w the ratio of the resonance integrals, β & / β , affects the shape o f b a n d d i s p e r s i o n . F i g u r e s a

a

b

b

α


15.

WHANGBO

273


2a, 2b, a n d 2c represent the 1, respectively. I n F i g u r e 2, the case i n w h i c h the b a n d is p e r site to c o n t r i b u t e to the

cases of β / β = 0, $ / $ < 1, a n d β ^ / β = each dashed l i n e refers to the F e r m i l e v e l for half-filled, w h i c h occurs i f t h e r e is one e l e c t r o n band. 6

α

b

a

α


F i g u r e 2 shows that wave vectors i n a certain r e g i o n of the first B r i l l o u i n zone l e a d to o c c u p i e d b a n d orbitals, a n d those i n the r e m a i n i n g r e g i o n to

\ Υ

Γ

X

M

Y

(a)

γ

r

x

M Y

(O Figure 2. Dispersion relation of eq 2 for (a) $bt&* — 0, (h) & (C) βΒ/β = 1.

< 1, and

&

u n o c c u p i e d b a n d levels. T h i s relationship is s h o w n i n F i g u r e s 3a, 3b, a n d 3c, w h i c h c o r r e s p o n d to the b a n d dispersions of F i g u r e s 2a, 2b, a n d 2c, respectively. I n F i g u r e 3, the wave vectors of the shaded regions are oc c u p i e d (i.e., t h e y lead to o c c u p i e d b a n d levels), a n d those of the u n s h a d e d region are u n o c c u p i e d . Because a l l wave vectors are e q u a l l y p r o b a b l e , the size of the o c c u p i e d wave vector region is p r o p o r t i o n a l to the b a n d filling. T h u s , for a half-filled b a n d , one-half of the first B r i l l o u i n z o n e is o c c u p i e d .



274


(a)

(b)

(c)

Figure 3. Fermi surfaces associated with the half-filled bands of Figure 2, (a) βι,/βα = 0, (b) βι,/βα < I, and(c) βι,/β = J. Filled wave vectors are indicated by hatching. 3

T h e F e r m i surface of a partially filled b a n d is d e f i n e d as t h e b o u n d a r y that separates the o c c u p i e d wave vector r e g i o n from the u n o c c u p i e d w a v e vector r e g i o n (4). T h e F e r m i surface o f F i g u r e 3a o r 3 b consists of separated lines, a n d hence is said to be o p e n . F o r the electrons w i t h the crystal m o m e n t a hkl 2it a l o n g a c e r t a i n wave vector d i r e c t i o n , t h e r e exists n o b a n d e n e r g y gap i f the w a v e vector l i n e crosses a F e r m i surface. I n such a case, a partially filled b a n d system u n d e r consideration exhibits a m e t a l l i c c h a r acter along the wave vector d i r e c t i o n . W h e n a straight w a v e vector l i n e does not cross a F e r m i surface, w h i c h is possible i f the F e r m i surface is o p e n , the partially filled b a n d system behaves as a n o n m e t a l along that wave v e c t o r d i r e c t i o n . T h e presence of an o p e n F e r m i surface characterizes a o n e d i m e n s i o n a l ( I D ) m e t a l . T h u s , F i g u r e s 3a a n d 3b b o t h r e p r e s e n t I D m e t als. T h e w a v e - l i k e F e r m i surface o f F i g u r e 3 b reflects the fact that i n t e r a c tions along the b d i r e c t i o n are n o n v a n i s h i n g (i.e., β& Φ 0). T h e extent of h o w strongly the F e r m i surface deviates from the i d e a l I D F e r m i surface (i.e., the straight lines of F i g u r e 3a) is m e a s u r e d b y the wave vector ratio, AJfc/(ir/a). N o t e that àk = 0 for β * / β = 0 ( F i g u r e 3a), a n d àk = 1 for β /β = 1 ( F i g u r e 3c). T h u s , the àk/(ir/a) ratio scales w i t h t h e β * / β ratio. β

&

β

β

T h e F e r m i surface of F i g u r e 3c, o b t a i n e d for the case o f β & / β = 1, has a rectangular shape. A F e r m i surface w i t h a closed l o o p is said to b e closed. A 2 D m e t a l is characterized b y the presence of a closed F e r m i surface. F i g u r e s 4a a n d 4b show the F e r m i surfaces associated w i t h the b a n d of F i g u r e 2c for the cases i n w h i c h the b a n d is less a n d m o r e than half-filled, respectively. T h e F e r m i surface of F i g u r e 4a is a " d i s t o r t e d c i r c l e " c e n t e r e d at Γ. I n contrast, the F e r m i surface of F i g u r e 4b features " d i s t o r t e d c i r c l e s " c e n t e r e d at the four corners of the first B r i l l o u i n zone. (This b e c o m e s c l e a r e r α



15.

WHANGBO


275

w h e n the p a t t e r n of F i g u r e 4 b is r e p e a t e d b y translating it i n the k a n d k directions.) T h e F e r m i surface of F i g u r e 4a becomes m o r e c i r c u l a r as the b a n d filling is further r e d u c e d , a n d so does that of F i g u r e 4 b as the b a n d filling increases further. T h e F e r m i surfaces o f F i g u r e s 4a a n d 4 b b o t h represent 2 D metals. a

and the the and

b

A piece o f a F e r m i surface may b e translated b y a single wave v e c t o r q s u p e r i m p o s e d o n another piece of the F e r m i surface. I n s u c h a case, F e r m i surface is said to be nested b y the wave v e c t o r q. F o r e x a m p l e , F e r m i surfaces of F i g u r e 4 are not nested, b u t those of F i g u r e s 3a, 3b, 3c have the n e s t i n g vectors s h o w n i n F i g u r e s 5a, 5 b , a n d 5c, r e s p e c t i v e l y . Y

Y

Y

(a)

(b)

(c)

Figure 5. Nesting vectors associated with the Fermi surfaces of Figure 3. (a) βι,/β. = 0, (b) βΒ/βα < I, and (c) βι,/β. = I. I n F i g u r e 5b it is not i m m e d i a t e l y obvious to see w h a t the n e s t i n g vector does, because F i g u r e 5b shows o n l y the p o r t i o n of the F e r m i surface b e l o n g i n g to the first B r i l l o u i n zone. W i t h an e x t e n d e d zone r e p r e s e n t a t i o n o f the F e r m i surface, w h i c h is o b t a i n e d b y r e p e a t i n g the p a t t e r n of F i g u r e 5 b i n the two wave-vector directions, it b e c o m e s clear that the n e s t i n g v e c t o r


276


q superimposes the left-hand-side p i e c e of the F e r m i surface onto the r i g h t hand-side piece.


Electronic States Derived from a Normal Metallic State by Orbital Mixing A m e t a l l i c state p r e d i c t e d b y one-electron b a n d t h e o r y (i.e., a n o r m a l m e tallic state) is not stable w h e n its F e r m i surface is nested, a n d i t b e c o m e s susceptible to a m e t a l - i n s u l a t o r transition u n d e r a suitable p e r t u r b a t i o n . W e n o w examine the nature of the n o n m e t a l l i c e l e c t r o n i c states that are d e r i v e d from a n o r m a l m e t a l l i c state u p o n m i x i n g its o c c u p i e d a n d u n o c c u p i e d b a n d energy levels (1-3). F o r s i m p l i c i t y , w e w i l l discuss this o r b i t a l m i x i n g o n the basis of the i d e a l I D b a n d structure s h o w n i n F i g u r e 2a. T h i s b a n d is dispersive o n l y along the d i r e c t i o n p a r a l l e l to Γ—»X (i.e., the a d i r e c t i o n i n r e a l space). I n this situation, the energy E(k , k ) is i n d e p e n d e n t of k . T h u s , b y d r o p p i n g o u r reference to the w a v e vector k , eqs 1 a n d 2 are s i m p l i f i e d as a

b

b

b

= 0). T h e n the charge d i s t r i b u t i o n r e s u l t i n g from the u n m o d i f i e d o r b i t a l φ(£) (i.e., φ*(£)φ(&)) is given by 0

φ*(*)φ(*) = N~

Σ χ Λ , m

l

(6)

T h u s , a l l site orbitals have the same coefficient, 1 / N , so that a l l sites have an i d e n t i c a l density. T h e n e w orbitals, ψ ( - ^ 4- δ) a n d i|i(fcy 4- δ), l e a d to the following density distributions: ψ * ( - * / + δ)

+ δ) = I V " £ 1

[1 + 2 (1 Ύ

+ 7 )" 2

1

cos 2k ma] ·

*

f

Xm

Xm

m

(7a) ψ*(*/ + δ ) ψ ( ^ + δ) = Ν-

1

S U m

"

2 (1 Ύ

+ Υ ) " cos 2k ma] · 2

1

f

*

Xm

Xm

(7b)

E q u a t i o n 7 is i n d e p e n d e n t of the parameter δ, w h i c h measures h o w far away the orbitals φ ( - & / + δ) and φ ( ^ + δ) are from the F e r m i l e v e l . T h e coefficients of X * X i n e q 7 are not u n i f o r m , b u t t h e i r magnitudes vary i n a p e r i o d i c m a n n e r that is d e s c r i b e d b y the nesting vector 2fc^as 27(1 + 7 )" cos 2k ma. C o n s e q u e n t l y , the o r b i t a l mixings defined b y e q 5 i n t r o d u c e density waves w i t h respect to the metallic state. I f the absence of a density wave i n a chain is represented b y a straight l i n e , 6a, t h e n the presence of T O

m

2

f


1

278


it can be r e p r e s e n t e d b y a w a v y l i n e , 6b or 6c. I n 6b a n d 6c, e l e c t r o n gain


6 a

a n d loss w i t h respect to the density d i s t r i b u t i o n of 6a are r e p r e s e n t e d b y the crests a n d troughs of the waves, respectively. L i n e 6b shows d e n s i t y a c c u m u l a t i o n i n the region w h e r e 6c shows density d e p l e t i o n . T h u s , the density waves 6b a n d 6c are out of phase. T h e repeat distance, L , of 6b or 6c is increased b y a factor of ( 2 ι τ / α ) / 2 ^ w i t h respect to that of 6a, so that L = Ttlkf. F o r the I D b a n d g i v e n b y e q 4 b , the kf v a l u e for the b a n d o c c u p a n c y 1 / n (e.g., η = 2 for a half-filled band) is g i v e n b y kf = ττ/ηα, so that L = na. T h e density waves g i v e n b y the orbitals ψ(-&/ + δ) a n d i|/(fcy + δ) i n e q 7 are out of phase because i|f(-fy + δ) accumulates e l e c t r o n density o n the sites w h e r e ψ ( ^ + δ) depletes electron d e n s i t y , a n d vice versa. W h e n w e fill the orbitals ψ ( - £ / + δ) w i t h two electrons, l e a v i n g the orbitals $(kf + δ) e m p t y , a charge-density wave state ( C D W ) results. I n this state, each site has an e q u a l density of u p - s p i n a n d d o w n - s p i n electrons, b u t the total e l e c tron density at each site varies i n a p e r i o d i c m a n n e r . A n example of C D W d e r i v e d f r o m a half-filled m e t a l l i c b a n d of a I D c h a i n is s h o w n i n 7 (2),

_ 7

11 —·

η ·

u ··

« ·

w h e r e the l e n g t h of an u p w a r d or d o w n w a r d arrow at each lattice site r e p resents the m a g n i t u d e of the u p - s p i n or d o w n - s p i n e l e c t r o n d e n s i t y , r e spectively. I f w e fill the orbitals ψ(-&/ + δ) w i t h an u p - s p i n e l e c t r o n a n d the orbitals ψ ( ^ + δ) w i t h a d o w n - s p i n e l e c t r o n , there occurs a s p i n - d e n s i t y wave state ( S D W ) . I n this state, the total e l e c t r o n d e n s i t y at each site is i d e n t i c a l , b u t each site has u n e q u a l densities of u p - s p i n a n d d o w n - s p i n electrons that vary i n a p e r i o d i c m a n n e r . A n example of an S D W d e r i v e d from a half-filled b a n d of a I D c h a i n is d e p i c t e d i n 8 (2), w h e r e the net spins

ρ Ο

U

—·

4 ·

1k ·

4 ·


15.

279


WHANGBO

of the lattice sites have an antiferromagnetic o r d e r i n g . T h o u g h not s h o w n , w e o b t a i n results i d e n t i c a l to 8 b y m i x i n g the u n o c c u p i e d orbitals ψ(-&/ δ) w i t h the o c c u p i e d orbitals

- δ).

T h i s o r b i t a l m i x i n g cannot occur unless a certain p e r t u r b a t i o n , H', is i n t r o d u c e d to a m e t a l l i c system so that a n e w effective H a m i l t o n i a n b e c o m e s Ijeff , j ψ(_^ ) j eigenfunctions +

of H

eS

H

T

h

e

o

r

b

i

t

a

s

+

δ

a n (

+

a r e

n

o

t

+ H\ so the interaction b e t w e e n φ ( - & / + δ) a n d φ ( ^ + δ) u n d e r

the p e r t u r b a t i o n H

f

(i.e., < φ ( - & / + δ) | Η' \ $(kf

+ δ ) > ) does not neces

sarily v a n i s h . T h e p e r t u r b a t i o n causing the C D W state associated w i t h a


F e r m i surface n e s t i n g vector, 2kf, is a lattice v i b r a t i o n (i.e., phonon) w i t h the w a v e vector 2kf. T h e C D W is r e f e r r e d to as c o m m e n s u r a t e w h e n the ratio (itld)l2kf is an integer (e.g., 2, 3, . . .) a n d i n c o m m e n s u r a t e o t h e r wise. A c o m m e n s u r a t e C D W state is susceptible to a p e r m a n e n t lattice distortion as a result of e l e c t r o n - p h o n o n interactions. T h e p e r t u r b a t i o n caus i n g an S D W state is an on-site r e p u l s i o n , U (2). G i v e n that the total d e n s i t y at each site is i d e n t i c a l , the total energy of a system can b e l o w e r e d b y i n t r o d u c i n g s p i n p o l a r i z a t i o n o n each site because it reduces the extent o f on-site repulsion.

Fermi Surface Nesting and Electronic Instability W e n o w consider w h y F e r m i surface n e s t i n g plays a p r o m i n e n t role i n i n d u c i n g a C D W o r S D W state from a n o r m a l m e t a l l i c state. T o s i m p l i f y o u r notation, an o c c u p i e d w a v e vector (k , k ) m a y b e r e p r e s e n t e d b y k, a n d a n u n o c c u p i e d wave vector (k \ k ) b y k'. O r b i t a l m i x i n g b e t w e e n an o c c u p i e d o r b i t a l φ ( φ a n d an u n o c c u p i e d o r b i t a l φ ( ^ ) leads to n e w orbitals φ(1ζ) + 7 φ ( ^ ) a n d α - 7 φ ( £ ) + φ ^ ' ) . T h e s e n e w orbitals i n t r o d u c e electron-density waves d e s c r i b e d b y the wave vector q = k - k', w h i c h are out of phase i n t h e i r density d i s t r i b u t i o n s . W h e n a F e r m i surface has a n e s t i n g vector q, this o r b i t a l m i x i n g can b e p e r f o r m e d for a l l the wave vectors i n the nested r e g i o n of the first B r i l l o u i n zone, t h e r e b y l e a d i n g to the sets o f n e w orbitals {*|*(k)} a n d {i|*(k')} differing i n t h e i r w a v e vectors b y q = k - k'. A C D W state is o b t a i n e d w h e n the orbitals i|/(k) are each d o u b l y o c c u p i e d . A n S D W state is o b t a i n e d w h e n the orbitals and are each singly o c c u p i e d b y u p - s p i n and d o w n - s p i n electrons, respectively. a

a

i|i(k) oc

b

b

i|i(k')

i|i(k)

i|i(k')

{i|i(k)}

W i t h a n e s t e d F e r m i surface, the sets of orbitals a n d {i|/(k')} i n c l u d e those d e r i v e d from m i x i n g the o c c u p i e d a n d u n o c c u p i e d levels [ φ ( φ a n d φ ( ^ ) , respectively] i n the v i c i n i t y of the F e r m i l e v e l . T h e e n e r g y difference b e t w e e n such orbitals φ ^ ) a n d φ ( ^ ) is s m a l l , so that the o r b i t a l m i x i n g b e t w e e n t h e m is significant, a n d so is the i n t e r a c t i o n e n e r g y < φ ( φ | H ' | φ ^ ' ) > . I n a d d i t i o n , the energy l o w e r i n g that results from such an i n t e r a c t i o n matrix e l e m e n t can be gained from a l l the wave vectors k a n d k' r e l a t e d b y the n e s t i n g vector q = k - k' i n the nested r e g i o n of the first B r i l l o u i n zone.


280


T h i s is w h y the electronic instability toward a C D W or an S D W state is strong w h e n a n o r m a l metallic state possesses a c o m p l e t e F e r m i surface nesting. A s the extent of F e r m i surface nesting d i m i n i s h e s , that of the e l e c tronic instability decreases.


B o t h C D W a n d S D W states are i n s u l a t i n g i n nature because a b a n d e n e r g y gap is created at the F e r m i l e v e l as a result of o r b i t a l m i x i n g (2). A n i n s u l a t i n g state has no F e r m i surface, b y d e f i n i t i o n , because its highest o c c u p i e d a n d lowest u n o c c u p i e d levels are not degenerate. T h u s , for a m e t a l - i n s u l a t o r phase transition arising from a C D W o r an S D W formation, the o r b i t a l m i x i n g i n d u c e d b y the F e r m i surface nesting is said to destroy the F e r m i surface. A slightly i n c o m p l e t e F e r m i surface n e s t i n g occurs w h e n the pieces of the F e r m i surface i n v o l v e d have slightly different curvatures i n certain w a v e v e c t o r regions. F o r such a case as w e l l , t h e r e exists an e l e c t r o n i c instability r e s u l t i n g from the n e s t e d portions o f the F e r m i surface. T h e latter are d e s t r o y e d u p o n the associated o r b i t a l m i x i n g . A f t e r a phase transition r e s u l t i n g from this o r b i t a l m i x i n g , the s m a l l u n n e s t e d portions of the F e r m i surface give rise to s m a l l pocketlike n e w F e r m i surfaces a r o u n d t h e m , so that the r e s u l t i n g n e w e l e c t r o n i c state is still m e t a l l i c (5). T h a t is, an i n c o m p l e t e F e r m i surface nesting leads to a n i n c o m p l e t e destruction o f the F e r m i surface, w h i c h is responsible for m e t a l - m e t a l transitions. So far w e have c o n s i d e r e d that a b a n d l e v e l b e l o w e is c o m p l e t e l y filled (i.e., occupancy of 1); that above e is c o m p l e t e l y e m p t y (i.e., o c c u p a n c y of 0). T h i s p i c t u r e is v a l i d for a l l levels w h e n Τ = 0, b u t o n l y for the levels l y i n g outside the v i c i n i t y of the F e r m i l e v e l (e.g., e < e - 4k T a n d e > e + 4k T, w h e r e k is the B o l t z m a n n constant) w h e n Τ > 0. F o r the levels l y i n g close to the F e r m i l e v e l (e.g., e - 4k T < e < e + 4fc T), w h o s e o r b i t a l m i x i n g plays a c r u c i a l role i n l o w e r i n g the e n e r g y of a I D m e t a l , t h e r e b y l e a d i n g to a m e t a l - i n s u l a t o r transition, t h e i r o r b i t a l o c c u p a n c y f(e) at n o n z e r o t e m p e r a t u r e is g i v e n b y the F e r m i - D i r a e d i s t r i b u t i o n f u n c t i o n , f(e) = {1 + expf(e - e )/k T]}-\ T h u s , f(e) < 1 for e < e a n d f(e) > 0 for e > e . F o r e x a m p l e , i n F i g u r e 6, the o c c u p a n c y of φ ( - & / + 8) is less than 1, a n d that of φ(&/ + δ) is larger than 0 (at Τ > 0). C o n s e q u e n t l y , the e n e r g y gain r e s u l t i n g from the o r b i t a l m i x i n g b e t w e e n φ ( - & / + δ) a n d $(k 4- δ) is m a x i m u m at Γ = 0 a n d decreases as Τ is raised. T h u s , o n l y w h e n Τ is l o w e r e d b e l o w a certain t e m p e r a t u r e does the e n e r g y gain associated w i t h the o r b i t a l m i x i n g b e c o m e substantial e n o u g h to cause a m e t a l - i n s u l a t o r transition. f

f

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f

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f

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b

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CDW Instability and Real vs. Reciprocal Space Correlations M e t a l - i n s u l a t o r transition arising from a C D W instability is not abrupt, b u t t y p i c a l l y undergoes a series o f steps (6). T h i s process can be i l l u s t r a t e d b y c o n s i d e r i n g a I D m e t a l as c o m p o s e d of w e a k l y interacting chains. A t a h i g h temperature, each c h a i n has no t e n d e n c y for C D W formation, so a l l chains have u n i f o r m density distributions, as illustrated i n F i g u r e 7a. B e l o w a


15.

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Figure 7. Schematic representation of what happens to a ID metal with CDW instability as a function of temperature T . (a) Uniform electron-density dis tribution at Τ > TJD, (7?) ID fluctuation at T < Τ < T , (cj ZaferaZ ordering at T < Τ < T , and (dj long-range order at Τ < T . X

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certain t e m p e r a t u r e T , each c h a i n has a t e n d e n c y for C D W formation. A s s h o w n i n F i g u r e 7b, a C D W is f o r m e d a n d d e s t r o y e d d y n a m i c a l l y at m a n y parts o f each c h a i n , a n d C D W formation i n one c h a i n is i n d e p e n d e n t o f those i n other chains. T h e average l e n g t h o f a C D W segment (i.e., coherent length) is ξ . B e l o w a c e r t a i n t e m p e r a t u r e T ( < T ) , C D W segments a m o n g different chains b e g i n to o r d e r along the i n t e r c h a i n d i r e c t i o n as s h o w n i n F i g u r e 7c, w h e r e ξ& is the coherent l e n g t h along the i n t e r c h a i n d i r e c t i o n . B e l o w a certain t e m p e r a t u r e T ( < T ) , C D W formation i n each c h a i n is c o m p l e t e a n d C D W s a m o n g different chains are o r d e r e d , as s h o w n i n F i g u r e 7 d . T h u s , a long-range o r d e r sets i n . T h e coherent lengths ξ a n d ξ& increase gradually u p o n l o w e r i n g the temperature as the extents o f long-range o r d e r 1 D

x

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1 D

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282


along the i n t r a - a n d i n t e r c h a i n directions increase. T h e s e coherent lengths b e c o m e infinite w h e n a long-range o r d e r is c o m p l e t e i n b o t h d i r e c t i o n s . C r y s t a l structures are largely d e t e r m i n e d b y single-crystal X - r a y dif fraction measurements, whose diffraction patterns are r e c o r d e d i n r e c i p r o c a l space. T h e t e m p e r a t u r e d e p e n d e n c e of the diffraction p a t t e r n of a I D m e t a l is closely related to w h a t happens to it i n r e a l space, as discussed i n c o n n e c t i o n w i t h F i g u r e 7. A t Τ > T , the diffraction p a t t e r n of a I D m e t a l shows o n l y B r a g g peaks, as shown i n F i g u r e 8a. A t T < Τ < T , the diffraction pattern shows difiuse lines (or sheets i n 3 D representation) p e r p e n d i c u l a r to the k d i r e c t i o n , w h i c h are located at ± 2kf from the rows of the B r a g g peaks, as shown i n F i g u r e 8b. T h e thickness of the difiuse l i n e is g i v e n b y ξ ~ . A t T < Τ < T , the difiuse lines are transformed i n t o difiuse spots (or rods i n 3 D representation) c e n t e r e d at (2k , q ) a n d its e q u i v a l e n t positions, as s h o w n i n F i g u r e 8c. T h e thickness of a difiuse spot a l o n g the k a n d k directions is g i v e n b y ξ a n d ξ " \ respectively. B e l o w T , a l o n g range o r d e r sets i n so that the difiuse spots are c o n v e r t e d i n t o superlattice spots, as s h o w n i n F i g u r e 8 d . T h e difiuse spot thicknesses along the i n t r a a n d i n t e r c h a i n directions are i n v e r s e l y p r o p o r t i o n a l to the coherent lengths ξ and ξ respectively. Therefore, the diffuse spots b e c o m e s m a l l e r as the extent of long-range o r d e r along the two directions increases. T h e y e v e n tually b e c o m e superlattice spots after a long-range o r d e r sets i n along b o t h directions. 1 D

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Superconductivity and Electronic Instability W h e n the t e m p e r a t u r e is l o w e r e d , a m e t a l m a y b e c o m e susceptible to a n o t h e r t y p e of electronic instability, formation o f a s u p e r c o n d u c t i n g state. F o r a m e t a l to b e s u p e r c o n d u c t i n g , i t s h o u l d a v o i d the e l e c t r o n i c i n s t a b i l i t y t o w a r d a m e t a l - i n s u l a t o r transition l e a d i n g to a C D W or an S D W state associated w i t h a good F e r m i surface nesting. I n general, the F e r m i surface of a I D m e t a l is w e l l - n e s t e d , so a I D m e t a l rarely undergoes a m e t a l superconductor transition. A s seen from F i g u r e s 3c a n d 4, the F e r m i surface n e s t i n g of a c e r t a i n 2 D m e t a l l i c system can b e r e m o v e d b y partial oxidation or partial r e d u c t i o n . F o r example, w h e n h a l f - f i l l e d , the C u O l a y e r x - y b a n d of a h i g h - t e m p e r a t u r e c o p p e r oxide superconductor has a w e l l - n e s t e d F e r m i surface (as i n F i g u r e 3c) (7). I n s u c h a case, the C u O l a y e r is no longer a m e t a l b u t exhibits an antiferromagnetic state (i.e., an example of an S D W state). T h e C u 0 layer shows s u p e r c o n d u c t i v i t y o n l y w h e n some electrons are r e m o v e d from (8, 9) or a d d e d to (10, 11) the x - y b a n d so that its F e r m i surface loses n e s t i n g character (as i n F i g u r e 4). 2

a

2

a

2

2

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F r o m the v i e w p o i n t of one-electron b a n d theory, a s u p e r c o n d u c t i n g state also involves o r b i t a l m i x i n g a m o n g the b a n d levels above a n d b e l o w the F e r m i l e v e l . H o w e v e r , the way this o r b i t a l m i x i n g comes about differs considerably f r o m that discussed for C D W a n d S D W states. C h a r g e carriers


WHANGBO Low-Dimensional Metals

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Figure 8. Schematic representation of diffraction patterns of a ID metal with CDW. (a) Only Bragg peaks are present atT > T , (b) diffuse lines occur at T < Τ < T , (c) diffuse lines become diffuse spots at T < Τ < T , and (d) difiuse spots become superlattice spots at Τ < T . w

x

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of a s u p e r c o n d u c t i n g state are not i n d i v i d u a l electrons, b u t pairs of electrons (called C o o p e r pairs) h a v i n g opposite m o m e n t a (i.e., opposite wave vectors) (12-15). T h u s , C o o p e r pairs are n o t d e s c r i b e d b y i n d i v i d u a l b a n d orbitals

(k) and (k'), but by product functions (k)(-k) and (-k')

(i.e., that cause a s u p e r c o n d u c t i n g state (5,19). W h e n t h e relative stabilities o f C D W , S D W , a n d s u p e r c o n d u c t i n g states are s i m i l a r , preference of one state o v e r the o t h e r is delicately b a l a n c e d b y a change i n t e m p e r a t u r e and pressure.

Concluding Remarks T h e e l e c t r o n i c structure o f a C D W , a n S D W , o r a s u p e r c o n d u c t i n g state m a y b e d e s c r i b e d i n terms o f o r b i t a l m i x i n g b e t w e e n filled a n d e m p t y levels o f a n o r m a l m e t a l l i c state. I n contrast to t h e case o f C D W a n d S D W states, o r b i t a l m i x i n g i n a s u p e r c o n d u c t i n g state occurs i n d i r e c t l y v i a interactions b e t w e e n filled a n d e m p t y p a i r functions. Because i n c o r p o r a t i o n o f u n o c c u p i e d o r b i t a l character raises energy, o n l y those levels i n t h e v i c i n i t y o f the F e r m i l e v e l are c r u c i a l for a m e t a l - i n s u l a t o r o r m e t a l - s u p e r c o n d u c t o r t r a n sition. T h e p e r t u r b a t i o n s causing C D W a n d S D W states are p h o n o n a n d on-site r e p u l s i o n , r e s p e c t i v e l y , a n d o c c u r r e n c e o f these t w o states r e q u i r e F e r m i surface n e s t i n g . O u r discussion is l i m i t e d to those e l e c t r o n i c states that originate from a n o r m a l m e t a l l i c state, a n d is appropriate w h e n t h e o n site r e p u l s i o n U is small c o m p a r e d w i t h t h e b a n d w i d t h W (e.g., W = 4 β

β

i n t h e I D b a n d s h o w n i n F i g u r e 6) ( I , 2, 20-22). W h e n U is greater than W , electrons a r e l o c a l i z e d o n lattice sites. It is difficult to d e s c r i b e t h e l o w l y i n g excited states o f such a l o c a l i z e d state b y a b a n d e l e c t r o n i c s t r u c t u r e t h e o r y because t h e latter is based u p o n t h e a s s u m p t i o n that electrons are


286


d e l o c a l i z e d throughout the lattice. L o c a l i z e d electronic systems are t y p i c a l l y e x a m i n e d i n terms o f m o d e l H a m i l t o n i a n s (e.g., s p i n a n d H u b b a r d H a m i l tonians) d e s i g n e d to study t h e i r l o w - l y i n g excited states (21).

Acknowledgment T h i s w o r k was s u p p o r t e d b y the U . S . D e p a r t m e n t o f E n e r g y , Office o f B a s i c E n e r g y Sciences, D i v i s i o n o f M a t e r i a l s Sciences u n d e r G r a n t D E - F G 0 5 -


86ER45259.

References 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Whangbo, M . - H . J. Chem. Phys. 1979, 70, 4963. Whangbo, M . - H . J. Chem. Phys. 1980, 73, 3854. Whangbo, M . - H . J. Chem. Phys. 1981, 75, 4983. Jones, H . Theory of Brillouin Zones and Electronic States in Crystals; 2nd ed.; North-Holland: Amsterdam, Netherlands, 1975; pp 43-49. Whangbo, M . - H . ; Canadell, E . Acc. Chem. Res. 1989, 22, 375. Pouget, J. P. In Low Dimensional Electronic Properties of Molybdenum Bronzes and Oxides; Schlenker, C., E d . ; Reidel: Dordrecht, Netherlands, 1989; Chap ter 3. Whangbo, M . - H . ; Evain, M . ; Beno, Μ. Α.; Williams, J. M . Inorg. Chem. 1987, 26, 1829, 1831, 1832. Sleight, A. W. Science 1988, 242, 1519. Sleight, A. W.; Subramanian, Μ. Α.; Torardi, C. C. Mater. Res. Soc. Bull. 1989, XIV, 45. Tokura, Y.; Takagi, H.; Uchida, S. Nature 1989, 337, 345. James, A. C. W. P.; Zahurak, S. M . ; Murphy, D. W. Nature 1989, 338, 240. Bardeen, J . ; Cooper, L. N . ; Schrieffer, J. R. Phys. Rev. 1957, 108, 1175. Grassie, A. D. C. The Superconducting State; Sussex University Press: London, 1975; Chapter 2. Solymar, L . ; Walsh, D. Lectures on the Electrical Properties of Materials; 4th ed.; Oxford University Press: Oxford, England, 1988; Chapter 14. McMillan, W. L. Phys. Rev. 1968, 167, 331. Whangbo, M . - H . ; Williams, J. M . ; Schultz, A. J . ; Emge, T. J . ; Beno, M . A. J. Am. Chem. Soc. 1987, 109, 90. Whangbo, M . - H . ; Kang, D. B.; Torardi, C. C. Physica C, 1989, 158, 371. Wang, E.; Tarascon, J . - M . ; Greene, L. H . ; Hull, G. W. Phys. Rev. B., in press. Whangbo, M . - H . ; Canadell, E . J. Am. Chem. Soc. 1988, 110, 358. Whangbo, M . - H . Acc. Chem. Res. 1983, 16, 95. Brandow, Β. H . Adv. Phys. 1977, 26, 651. Mott, Ν. F. Metal Insulator Transitions; Barnes and Noble: New York, 1977.

RECEIVED for review June 6, 1989. A C C E P T E D revised manuscript August 7, 1989.


Band Orbital Mixing and Electronic Instability of Low-Dimensional Metals

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