Quantum effects on impurity pinning and depinning of Charge Density Waves

Physica 143B (1986) 143-145 North-Holland, Amsterdam

143

QUANTUM EFFECTS ON IMPURITY PINNING AND DEPINNING OF CHARGE DENSITY WAVES Tetsuro SASO and Yoshikazu SUZUMURA Department of Physics, Tohoku U n i v e r s i t y ,

Sendai, 980 Japan

Based on the previous quantum Monte Carlo study o f i m p u r i t y pinning of charge d e n s i t y waves by use o f Fukuyama-Lee model in one dimension, e f f e c t of a p p l i e d e l e c t r i c f i e l d is i n v e s t i g a t e d near but below t h r e s h o l d . The depinning takes place inhomogeniouslv at the s p a t i a l region where the pinning is r a t h e r weak and the quantum f l u c t u a t i o n is large,and the quantum hopping r a t e between metastable s t a t e s increases, Such metastable s t a t e s may c o n t r i b u t e to anomalous low frequency response of the system observed e x p e r i m e n t a l l y .

I,

ble.

INTRODUCTION Recent i n t e r e s t on the pinning of charge den-

S t a t i c c o n d u c t i v i t y measurement shows t h r e s h -

s i t y waves (CDW's) by i m p u r i t i e s and t h e i r non-

old behavior at the e l e c t r i c

l i n e a r response a g a i n s t a p p l i e d e l e c t r i c

p u t e r s i m u l a t i o n on c l a s s i c a l

field

has a t t r a c t e d many researchers to t h i s new f i e l d

field

Eth 7,

Com-

Fukuyama-Lee model 8

seems to well e x p l a i n the e l e c t r i c

field

depen-

of m a c r o s c o p i c a l l y condensed e l e c t r o n systems I .

dence of the s t a t i c

One of the c e n t r a l problems s t i l l

one takes i n t o account t h r e e d i m e n s i o n a l i t y 9,

unsolved seems

t o be whether the dynamics of CDW can be d e s c r i b -

It is,

ed by a p u r e l y c l a s s i c a l model, or quantum t u n -

l y c l a s s i c a l model can e x p l a i n the behavir close

n e l i n g process dominates.

to Eth.

Reagor and Gruner 2

however, s t i l l

c o n d u c t i v i t y above Eth i f ambiguous whether the pure-

Maki I0 pointed out r e c e n t l y t h a t the

showed r e c e n t l y t h a t at l e a s t f o r m i l l i m e t e r and

temperature dependence o f the t h r e s h o l d f i e l d

micrometer range of the wavelength in the dynami-

Eth(T) 7 can be explained by a c l a s s i c a l model i f

cal c o n d u c t i v i t y measurements of the a l l o y e d sys-

one includes thermal f l u c t u a t i o n

tem Tal_xNbxS3 the observed oCDW(m) is compati-

states.

ble w i t h t h e i r c l a s s i c a l over-damped s i n g l e p a r ticle

model and c o n f l i c t s

model 3.

w i t h the t u n n e l i n g

But both t h e o r i e s seem t o apply a t low

In a d d i t i o n t o t h i s ,

the observed low frequen-

cy c o n d u c t i v i t y o(~) y i e l d s power law behavior o(~) ~ ~

w i t h ~ < 14 .

Quantum mechanical models have been i n v e s t i gated t h e o r e t i c a l l y

Feigel'man and Vinokur 5

to c l a r i f y

the motion of

CDW, but the deformation of CDW pinned by impurities

frequencies.

o f the pinned

has not been taken i n t o account I I .

The

present paper s t a r t s from the study of the p i n ned s t a t e s of CDW and the e f f e c t s of i t s quantum f l u c t u a t i o n

around the c l a s s i c a l

states,

showed t h a t o(m) should behave as m2~n2~ f o r

where the s p a t i a l dependence of the pinned s t a t e s

some low frequency range in the weakly pinned

play c r u c i a l r o l e s .

limit

of CDW, which is the same as f o r n o n i n t e r -

As was pointed e a r l i e r 12, the pinning of CDW

a c t i n g e l e c t r o n s 6, and o(~) ~ e x p ( - c o n s t / ~ ) f o r

is e q u i v a l e n t t o Anderson l o c a l i z a t i o n

the low frequency l i m i t ,

a c t i n g e l e c t r o n s in one dimension.

The above power law

of i n t e r -

Thus i t

is.

behavior in experiment does not agree w i t h t h i s

also o f much i n t e r e s t to study the pinning of

theory.

CDW f o r wide range of parameters from t h a t p o i n t

I t i s argued t h a t the e x i s t e n c e of meta-

s t a b l e s t a t e s are r e s p o n s i b l e t o t h i s behavior 4, Thus a d e t a i l e d study of such s t a t e s are d e s i r a 0378 - 4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation

of view 13

T. Saso and I~ Suzumura / Quantum e.ff~'cts on impurity pinning attd depinning

144

2. FORMULATION We s t a r t

the quantum f l u c t u a t l o n

from t h e d i s c r e t i z e d

v e r s i o n o f the

ever,

Fukuyama-Lee model in one d i m e n s i o n 14

p o i n t e d out 13 t h a t

ble spatial yields

N

H = ~ [ A(@i+l-@i)2+ i=l and map i t

CP#- V c o s ( e i - ~ i )

t o the e q u i v a l e n t

classical

+ FO i ] (I)

It

there exists

was, howconsidera

dependence of f l u c t u a t i o n s ,

whlch

quantum hopping between m e t a s t a b ] e s t a t e s

In the f o l l o w i n g s ,

Hamilton-

f o r T-~ ~ O.

we d e m o n s t r a t e how these

states

behave under the a p p l i c a t i o n

static

electric

o f small

field.

ian in two d i m e n s i o n s by use o f S u z u k i - T r o t t e r f o r m u l a 15

3. RESULTS AND DISCUSSIONS 1

H2(T) = ~

M

For t h i s

N

~ ~ [ A(Oi+I(J)-@i(J)) j=li=l

+ Ay(Oi(J+I)-6i(j))2-V

2

0.5,

and d e t e r m i n e d a q u a s i - g r o u n d

cos(@i(j)-~ i)

+ FOi(J)

-(NMT/2)Ln(~MT/(4w2rI2/A)) where t h e a d d i t i o n a l degree o f freedom.

j-axis

boundary c o n d i t i o n

i s assumed in both d i r e c t i o n s . t h e magnitude o f d e f o r m a t i o n @i and t h a t

o f the c o u p l i n g

(2)

expresses the quantum

Periodic

A and V denotes e n e r g y o f CDW phase

to impurities

are r e p r e s e n t e d by t h e random v a r i a b l e s denotes e l e c t r i c qA) o r i g i n a t e s

field.

which ~i"

F

Ay = p2 w i t h p = MT/(4~I

from the k i n e t i c

e n e r g y CP# in (1)

and the p a r a m e t e r s q = ( C / A ) I / 2 / 2 ~

and e = V/4~A

r e p r e s e n t t h e degree o f quantumness and o f r a n domness, r e s p e c t i v e l y .

"Classical"

p a r t o f @i

M

t o H2(T),

j .= ~lei(J)

>>

(3)

<<,°,>> is t a k e n w i t h r e s p e c t

The quantum f l u c t u a t i o n

by the o r d i -

for classical around 0~ w i l l

models. be es-

t i m a t e d from

(e~)211/~"

ly.

For each step o f the

tric

field,

state without

i n c r e a s e o f the e l e c

we seek f o r a new e q u i l i b r i u m

the m o t i o n o f the c l a s s i c a l process.

Note t h a t

elec

was a p p l i e d g r a d u a l

( t h o u g h sometimes m e t a s t a b l e ) .

state

Fig.l(a)

shows

p a r t @~ t h r o u g h t h i s

the phase U# a t the s i t e

5 moves d r a s t i c a l l y ,

since this

v e r y w e a k l y a t the b i g i n n i n g

site

i =

was pinned

o f the process and

had l a r g e quantum f l u c t u a t i o n

(Fig. l(b)).

Th~

m o t i o n o f 0~ a t i = 5 is not monotonous ( F i g . 2 (a))

as the f i e l d

flected

increases,

in the v a r i a t i o n

(Fig.2(b),

solid

line),

o f the i n t e r n a l

total

which is a l s o r e -

o f the i n t e r n a l while

slower.

cancel each o t h e r ,

energy (defined

as the i n t e r n a l

as a f u n c t i o n

s i n c e the f l u c t u a t i o n transient

aver-

As the v a r i a

e n e r g y and the c o u p l i n g

partly

most q u a d r a t i c a l l y

energy

the s p a t l a l

plus t h e c o u p l i n g e n e r g y t o f i e l d )

the

energy

decreases a l o f F.

But

becomes l a r g e t h r o u g h the

m o t i o n f o r 0.2 ~ F < 0.6 ( f l u c t u a t i o n

reaches maximum a t F = 0.3 [ F i g . l ( b ) ] ) , states

(4) As was r e p o t e d in our p r e v i o u s papers 14, the

U

Then the f i e l d

example

05 can

be r e g a r d e d as hopping between two m e t a s t a b l e

<~2>i/~ [ <> 1 mj=l i results

field.

energy to field

and e v a l u a t e d n u m e r i c a l l y

n a r y Monte C a r l o a l g o r i t h m

numerical

tric

tion M

where the a v e r a g e

J

age ~c E N-l#:i6~ v a r i e s

is d e f i n e d as 9c _ << 1

purpose, we chose N = lO, M = 20, { =

q =0.3 and T = l as an i l l u s t r a t i v e

for e.g.

t h e degree o f p i n n i n g

e N-IZ~=lcoincides_

by Fukuyama and Lee 8 f o r c l a s s i c a l and w i t h the s e l f - c o n s i s t e n t

w i t h the t h e o r y limit

rl ~ 0

phonon t r e a t m e n t

of

quantum m e c h a n i c a l l y .

s t a t e s may a f f e c t s i d e r a b l y 4, est for tion

d ( ~ ) a t low f r e q u e n c i e s

Therefore,

future

E x i s t e n c e o f such

it

will

study to investigate

of these states

numerically.

con-

be o f much i n t e r the d i s t r i b u -

T. Saso and Y. Suzumura / Quan turn effects on impurity pinning and depinning

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REFERENCES I. For a review, see Charge Density Waves in Solids, eds. Gy. Hutiray and J. Solyom (Springer Verlag, 1985). 2. D. Reagor and G. Gruner, Phys. Rev. Lett. 56 (1986) 659. 3. J. Bardeen, Phys. Rev. Lett. 42 (1979) 1478, i b i d 45 (1980) 1978. 4. Wei-Yu Wu, L. Mihaly-, G. Mozurkewich and G. Gruner, Phys. Rev. Lett. 52 (1984) 2382, and ref. l , p.311. 5. M.V. Feigel'man and V.M. Vinokur, Phys. Lett. 87A (1981) 53.

i

i

F

i

I

I

FIGURE 2 (a) Field dependence of the classical part of the phase e~ at the s i t e i = 5(0) and the spat i a l average ~c(e).

FIGURE 1 (a) Spatial v a r i a t i o n of the classical part of CDW phase for e l e c t r i c f i e l d F = 0(o and dashed l i n e ) , 0. I ( ~ ) , 0.2([]), 0.3(~), 0.4(~), 0.5(e), 0.6(A)° O.8(m) and l . O ( v and s o l i d l i n e ) . Not a l l the symbols are drawn since they j u s t l i e between F : 0 and 1.0. The crosses indicate ~ i t s . (b) Spatial dependence of quantum f l u c t u a t i o n of the phase of CDW for F : 0(o) and O.3(v).

i

(b) Variation of the i n t e r -

nal energy E(O) and the t o t a l energy E + FZie~(e ) as a function of e l e c t r i c f i e l d F.

Lager and Z.Z. Wang, in ref. l , p.279. 8. H. Fukuyama and P.A. Lee, Phys. Rev. BI7 (1978) 535. 9. H. Matsukawa and H. Takayama, Solid State Commun. 50 (1984) 283, and in t h i s volume. I0. K. Maki, Phys. Rev. B33 (1986) 2852. I I . Ko Hida, Z. Phys. B61 (1985) 223, and r e f e r ences therein. 12. Y. Suzumura and H. Fukuyama, J. Phys. Soc. Jpn. 52 (1983) 2870, i b i d 53 (1984) 3918, ibid 54 (1984) 2077, see also, T. Saso, Y. Suzumura and H. Fukuyama, Prog. Theor. Phys. Suppl. 84 (1985) 269. 13. T. Saso and Y. Suzumura, J. Phys. Soc. Jpn. 55 (1986) 25, and submitted to J. Phys. Soc. Jpn.

6. N.F. Mort, P h i l . Mag. 17 (1968) 1259, V.L. Berezinskii, Zh. Eksp. Teor. Fiz. 65 (1973) 1251 [ Sov. Phys. JETP 38 (1974) 620 ].

14. T. Saso, Y. Suzumura and H. Fukuyama, Proc. Int. Conf. 17th Low Temp. Phys., Karlsruhe 1984 (North Holland Pub. Co., 1984) p. 1345, T. Saso and Y. Suzumura, ref. l , p..531.

7. P. Monceau, M. Renard, J. Richard. M.C. Saint-

15. M. Suzuki. Prog. Theor. Phys. 56 (1976) 1454.