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T h e Peierls Transition in One-Dimensional Solids H . R.
ZELLER
a n d P.
BRÜESCH
B r o w n B o v e r i Research Center, C H - 5 4 0 1 B a d e n , S w i t z e r l a n d
In h i s book "Quantum Theory of Solids" (1) P e i e r l s shows that a one-dimensional metal i s i n h e r e n t l y unstable and will undergo a phase t r a n s i t i o n i n t o a semiconducting s t a t e . Almost simultaneously and independently the theory of t h i s phase transition was worked out by Fröhlich (2). Fröhlich showed that the phase t r a n s i t i o n may r e s u l t i n a low temperature s t a t e which i s not semiconducting but superconducting with t r a n s i t i o n temper a t u r e s not r e s t r i c t e d to the cryogenic range. Recently experimental systems have been studied which undergo a P e i e r l s t r a n s i t i o n and which are p o t e n t i a l candidates f o r the Fröhlich mechanism of s u p e r c o n d u c t i v i t y . In p a r t i c u l a r the t r a n s i t i o n was shown to occur i n K [Pt(CN) ] Br ·3(H O) (KCP) and r e l a t e d s a l t s (3). There is evidence that a P e i e r l s transition a l s o takes p l a c e i n TTF TCNQ (4). In the f o l l o w i n g we will restrict the d i s c u s s i o n to the best understood system, i . e . , K [Pt(CN) ] Br ·3(Η Ο). For a general i n t r o d u c t i o n the reader is r e f e r r e d to reference (3). Before we t u r n to the d i s c u s s i o n of the P e i e r l s instability, there is one p o i n t which should be made in connection with extended metal-metal i n t e r a c t i o n s . At first s i g h t it would seem that due to the small overlap a t i g h t binding model f o r the band s t r u c t u r e should be a very good approximation, i . e . , the c a r r i e r s would be holes i n a d band. I t came as a b i g s u r p r i s e when we discovered that the conduction band is d e f i n i t e l y not a s i n u s o i d a l t i g h t binding band but r a t h e r a p a r a b o l i c f r e e e l e c t r o n band with e f f e c t i v e mass m* = me. Although t h i s sounds h i g h l y implausible at first s i g h t , it can be explained as t y p i c a l d i m e n s i o n a l i t y e f f e c t . Nearly f r e e e l e c t r o n behaviour means n e a r l y constant e l e c t r o n d e n s i t y . As can be seen from F i g . 1 a n e a r l y constant e l e c t r o n d e n s i t y along the strand a x i s can be achieved a t a relatively modest overlap of the wave f u n c t i o n s ( i n t h i s case d -s o r b i t a l s ) . T h i s is not p o s s i b l e i n two or three dimensions. The e s s e n t i a l f e a t u r e s of the argument can a l s o be v i s u a l i s e d as f o l l o w s : In an a r r a y of spheres it is always p o s s i b l e to have them touch i n one l i n e but there is always empty space between the 2
2
4
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0
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z
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z
372 In Extended Interactions between Metal Ions; Interrante, L.; ACS Symposium Series; American Chemical Society: Washington, DC, 1974.
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spheres i n a plane or i n 3-d space. Thus as a f u n c t i o n of over lap, f r e e e l e c t r o n behaviour i s reached much e a r l i e r i n 1-d systems than i n 2-d and 3-d ones. Of course f r e e e l e c t r o n behaviour i s r e s t r i c t e d to the strand a x i s (5). In the f o l l o w i n g we w i l l d i s c u s s the P e i e r l s - F r o h l i c h t r a n s i t i o n and the p o s s i b i l i t y of high temperature superconductivity based on the F r o h l i c h mechanism. Below the phase t r a n s i t i o n a s i n u s o i d a l d i s t o r t i o n of the strands takes p l a c e such that an energy gap a t the Fermi energy i s created (6). For instance i n a quarter f i l l e d band t h i s d i s t o r t i o n w i l l have the period of four l a t t i c e spacings, s p l i t t i n g the conduction band i n t o a f i l l e d and three empty bands. Above the t r a n s i t i o n temperature a precursor shows up i n the form of a s o f t l a t t i c e v i b r a t i o n which corresponds to the low temperature s t a t i c d i s t o r t i o n . Due to the onedimensional nature of the system f l u c t u a t i o n s are extremely impor tant. There i s i n general no sharp t r a n s i t i o n but a very gradual transformation from a m e t a l l i c i n t o a semiconducting s t a t e . Associated with the s i n u s o i d a l d i s t o r t i o n i s a s i n u s o i d a l charge d e n s i t y wave (CDW). F r o h l i c h had r e a l i z e d that w i t h i n a continuum model the f r e e energy of the system does not depend on the phase of the CDW. This implies that the CDW can be s h i f t e d f r e e l y up and down the strands without any a c t i v a t i o n energy. As i n conventional superconductivity the presence of an energy gap e f f e c t i v e l y i n h i b i t s s c a t t e r i n g processes. Thus the system should behave as a superconductor w i t h the e l e c t r o n s s u r f i n g on the propagating l a t t i c e d i s t o r t i o n ( F i g . 2). The existence of a P e i e r l s d i s t o r t i o n i n KCP was c l e a r l y demonstrated by d i f f u s e x-ray s c a t t e r i n g (7) and i n e l a s t i c neutron s c a t t e r i n g (8) experiments. A l s o the P e i e r l s gap shows up i n the o p t i c a l spectra a t low temperatures a t about 0.2 eV (£). Next we t u r n to the c e n t r a l question whether the P e i e r l s F r o h l i c h t r a n s i t i o n r e a l l y leads to superconductivity. For a conventional BCS superconductor the c o n d u c t i v i t y σ (ω) i s represented by a ό-function a t ω * ο and a peak at energies c o r responding to the breaking of a Cooper p a i r . In an analogous f a s h i o n the i d e a l F r o h l i c h superconductor should e x h i b i t a 6f u n c t i o n a t ω « ο and a peak at ω * Eg, where Eg i s the P e i e r l s gap. F i g u r e 3 shows the experimental r e s u l t on KCP at 40 K obtained from Kramers Kronig a n a l y s i s of r e f l e c t i v i t y data. The peak a t ω * 1600 cm" « 0.2 eV corresponds to e x c i t a t i o n s across the P e i e r l s gap. But instead of a 6-function a t ω * ο F r o h l i c h c o l l e c t i v e mode produces a peak with f i n i t e width centered a t about 2 - 4 meV dependent on sample p e r f e c t i o n . At higher tempe r a t u r e s the peak gets broader and disappears around 200°K. In t h e i r fundamental paper Lee, R i c e and Anderson (10) have discussed why no true superconductivity based on the F r o h l i c h mode i s expected. Due to commensurability w i t h the l a t t i c e para meter, to random p o t e n t i a l s provided by i m p u r i t i e s or d i s o r d e r and to 3-d coupling the t r a n s l a t i o n a l i n v a r i a n c e i s broken and the CDW i s pinned. This corresponds to a spring constant and e
1
In Extended Interactions between Metal Ions; Interrante, L.; ACS Symposium Series; American Chemical Society: Washington, DC, 1974.
374
EXTENDED
INTERACTIONS
BETWEEN
METAL
IONS
OgGD80D8GD2O 2
Iψ I
Φ
f (z) 11^
Downloaded by UNIV MASSACHUSETTS AMHERST on October 18, 2013 | http://pubs.acs.org Publication Date: June 1, 1974 | doi: 10.1021/bk-1974-0005.ch025
iff ( l )
" FREE
9
e
ELECTRONS"
Figure 1. Even at a relatively small overlap hybridized d,2-s orbitals ηαϋβ an electron density which does nof depend on z, and tence the sys tem is free electron like in the z-direction. The same is true for higher dimensions.
Figure 2. Electron transport by the Frohlich collective mode. As in any in sulator the electrons are bound to a periodic potential. In the Peierls Frohlich state the periodic potential is not fixed in space but propagating and able to carry an electric current along the strands.
In Extended Interactions between Metal Ions; Interrante, L.; ACS Symposium Series; American Chemical Society: Washington, DC, 1974.
Downloaded by UNIV MASSACHUSETTS AMHERST on October 18, 2013 | http://pubs.acs.org Publication Date: June 1, 1974 | doi: 10.1021/bk-1974-0005.ch025
25.
ZELLER
A N D BRUESCH
Peierls
Transition
375
hence the c o n d u c t i v i t y peak w i l l be centered not a t ω « ο but a t a small (compared w i t h Eg) but f i n i t e frequency* Lifetime effects which a r e p a r t i c u l a r l y Important a t higher temperature cause a f i n i t e width o f the peak i n s t e a d o f a (S-function. As a con sequence i n the r e g i o n o f the t r a n s i t i o n temperature the F r o h l i c h mode may c o n t r i b u t e t o o r even dominate the dc c o n d u c t i v i t y . Whether t h i s i s the case o r not can most e a s i l y and d i r e c t l y be determined from measurements o f R (ω) i n the microwave and f a r i n f r a r e d r e g i o n . Due to the l a r g e e f f e c t i v e mass o f the F r o h l i c h mode i t s o s c i l l a t o r strength i s small and i t can a t most form a narrow peak i n σ (ω) superimposed on the very broad s i n g l e p a r t i c l e conductivity. From what we have learned on the model system KCP i t seems f e a s i b l e t o s y n t h e s i z e systems which e x h i b i t a s u f f i c i e n t l y small pinning f o r c e such that high dc c o n d u c t i v i t i e s based on the F r o h l i c h mode can be achieved. A l a r g e p a r t o f the work on KCP described i n t h i s paper was c a r r i e d out i n c o l l a b o r a t i o n w i t h D. Kuse, H.J. R i c e and S. S t r a s s l e r . We a l s o wish to acknowledge s t i m u l a t i n g d i s c u s s i o n s with P. Fulde and T.M. R i c e . F i g . 2 i s due to L. Niemeyer. References 1. 2. 3. 4.
5. 6. 7. 8.
9. 10.
P e i e r l s R.E., "Quantum Theory o f S o l i d s " (Oxford U n i v e r s i t y Press, London (1955). Fröhlich, Η., Proc. Roy.Soc.,London (1954) A 223, 296. Z e l l e r , H.R., Advances in S o l i d State P h y s i c s , (1973) 13, 31. Coleman, L.B., Cohen, M.J., Sandman, D.J., Yamagishi, F.G., G a r i t o , A.F. and Heeger, A.J., S o l i d State Comm. (1973) 12, 1125. The above arguments a r e due to P. Fulde and S. Strässler. R i c e , M.J. and Strässler, S., S o l i d State Comm. (1973) 13, 125. Comès, R., Lambert, Μ., Launois H. and Z e l l e r , H.R., Phys. Rev. (1973) B8, 571. Renker, B., R i e t s c h e l , H., P i n t s c h o v i u s , L., Gläser, W., Brüesch, P., Kuse, D. and R i c e , M.J., Phys. Rev. Letters(1973) 30, 1144. Brüesch, P. and Z e l l e r , H.R., t o be p u b l i s h e d . Lee, P., R i c e , T.M. and Anderson, P.W., p r e p r i n t .
In Extended Interactions between Metal Ions; Interrante, L.; ACS Symposium Series; American Chemical Society: Washington, DC, 1974.
