Pseudogap and Anomalous Dispersion of Expanded Liquid Mercury

The ex is tence o f a pseudo~ap i n the densi ty o f s ta tes o f expanded l i q u i d mercury i s asc e r t a i ned from f i r s t p r i n c i p l e s . The d ispers ion of expanded f l u i d mercury i s ca lcu la ted both by the r e l a t i v i s t i c KKR ~ e t h o d f o r a m u f f i n t i n system and by an extended Hiickel model. The theory due t o M a t t i s and Yonezawa concerning anonlalous dispersions and l o c a l i z a t i o n i s employed t o i d e n t i f y t h e pseudogap o r the m o b i l i t y gap. 1. INT2ODUCTIOM The concept o f a 'pseudogap' has been in t roduced by l.iott1 t o account f o r e l e c t r o n i c s ta tes i n sucit 60 e~;erS;y reg inn 9 f d i s ordered mate r ia l s t h a t the t a i l s o f a conduct ion and valence band overlap. Very of ten, t h e word 'pseudocjap' i s used more o r l e s s as a synonym o f a ' m o b i l i t y gap'. I n t n i s case, the dens i t y o f s ta t e s w i t h i n a pseudogap i s nonzero w h i l e the corresponding s ta tes are l o c a l i z e d . I n expanded l i q u i d mercury, f o r instance, the degree o f over lap can be var ied by changing t h e dens i t y o f inercury, and a wide varaety o f e l e c t r o n i c p roper t ies can be explained c o n s i s t e n t l y when the ex is tence o f a pseudogap i s assumed. 1-3 Although the idea o f a pseudogap has o r i g i n a l l y been proposed as a na tu ra l extension o f the Anderson l o c a l i z a t i o n 4 i n band t a i l s , no t h e o r e t i c a l l y f i r m p roo f from f i r s t q r i n c i p l e s has been g iven f o r the ex is tence o f l o c a l i z e d s ta tes i n the reg ion o f the band over lap. The whole d i f f i c u l t y i s p a r t l y ascr ibed t o the d i f f i c u l t y o f eva lua t ing re1 i a b l e dens i t i es o f s ta tes i n t h i s energy reg ion and p a r t l y t o the d i f f i c u l t y o f judging whether a given s t a t e i s l o c a l i z e d o r not . The problem o f the Anderson l o c a l i z a t i o n i n band t a i l s ' i s d i f f i c u l t by i t s e l f even f o r one-band cases. #hen two bands(or raore than two Liands) overlap, the crossings o f energy l e v e l s may occur. Therefore, t i e cowbinat ion o f t h e l e v e l n i x i n g and the l o c a l i z a t i o n nlechanism makes t h e problela harder al though t h e s e l f s a r , ~ reason makes t h e physical s i t u a t i o n more i n t e r e s t i n g . The purpose o f t h e present a r t i c l e i s t o ca lcu la t e a r e l i a b l e d ispers ion o f expanded 1 i q u i d mercury from f i r s t p r i n c i p l e s and t o show t h a t the obta ined d i s p e r s i 0 n . i ~ r e l a t e d t o the l o c a l i z e d s tates. As f o r the dens i t y o f states, we reported, a t the l a s t conference i n 2 r i s to1 , the r e s u l t s o f o u r 5 nuner ica l ca lcu la t ion ; we showed t h a t the d i p i n the t a i l reg ion becomes niore and more marked when the dens i t y o f rnercury i s decreased and t h a t the band opens up i n t h i s reg ion when the densi ty o f I@ i s low enough. As an extension of t h e prev ious work, we c a l c u l a t e t h e d ispers ion curve by means o f tne KKR method f o r a muff i n t i n system, as we1 1 as by ineans of the extended i i i c k e l model. The ca lcul a t e d d ispers ion shows an anomalous behaviour i n some energy reg ion as i l l u s t r a t e d i n sect ion 2 . 6 Using the asser t ion due t o M a t t i s and Yonezawa , we s i ~ o n i n sect ion 3 t h a t the s ta tes corresponding t o t n i s anomalous d ispers ion a re loca l i zed . A d i s cussion i s given i n sec t ion 4 i n which we suggest t h a t the i n d i c a t i o n s of anomalous d ispers ions may be more f r u i t f u l than we can see now. 2. DISPERSION RELATION OF EXPANDED LIQUID Hg For the purpose of s tudy ing e l e c t r o n i c p roper t ies o f expanded 1 i q u i d mercury, we ca lcu la te t h e average one-electron Green funct ion where the average is ' taken over a l l poss ib le conf igurat ions o f atoms in . the 7 i qu id . The one-electron Green func t ion f o r a m u f f i n t i n system i s def ined. by where U(P) consis ts o f an a r b i t r a r y assembly o f i d e n t i c a l non-over1 apping muff i n t i n p o t e n t i a l s centred around each atomic s i t e R. I n o u r ca lcu la t i o n , we have s t a r t e d w i t h the r e l a t i v i s t i c HartreeFock-Slater atomic p o t e n t i a l and determined t h e muff i n t i n p o t e n t i a l b y the Xa method w i t h a = 2 / 3 i n Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980822 c8-82 JOURNAL DE PHYSIQUE FCC Hg. In the tight-binding(T3) scheme, the Green function i s defined by The TB Hamil tonian H i s expressed as follows; where i , j designate disordered atomic s i t e s and t, t ' denote orbi ta ls ; E i t i s the s i t e energy associated with orbital t a t s i t e i while v: : ' i s a transf e r energy from orbital t ' a t s i t e j to orbital t a t s i t e i . For our calculation, we take into accou n t sspx, py and p orbi ta ls . iJe have determined parameters E i t antd;y1 in an extended ~Gckel approximation a t various densit ies sucn that the band structure of the model agrees well, up t o several eV above the Fermi level , with tha t obtained from the r e l a t iv i s t i c KKR calculation. Assuming that these parameters make 'sense even in the corresponding liquid s t a t e s , we have calculated the Green function. In both cases of a muffin-tin model and a TB nodel , the density of s t a t e s i s obtained from < ~ ( r , r , ; ;E)) while the dispersions are derived from the Fourier transform of ( G ( r , r ' ; ~ ) ) which we denote by Gk(E). The angular bracket indicates the ensemble average. According to the generally accepted procedure, 7 we approximate mu1 t iion distr ibution functions necessary for the ensemble average by the surns and products of pair distr ibution functions g(R). In our calculation, we use the Percus-Yevi c solution for g(R) since the experinental data for expanded l iquid Hg i s not avialable. For some l iquid nonsinlpl e m t a l s , the Percus-Yevic solution i s shown to serve as a good approximation, and t h i s i s one of the grounds why we use the Percus-Yevic model. Ue deternine the hard-core diameter by f i t t i n g the Percus-Yevic structure factor to the observed structure fac tor of Hg under the normal condition with 3 density 13.6g/cm . de use the same hard-core diameter fo r expanded Hg with reduced densities. Therefore, in our model, the system i s expanded i n such a way tha t the nearest-neighbour(nn) atomic distance i s kept fixed while the effective coordination number' is reduced according to the decrease of the density. This way of expanding a system i s different from tha t of a unifom expansion where the coordination number i s fixed while the nn atoraic distance i s enlarged. Fig.1. The dispersion E vs Re k fo r expanded 1 iquid tig with various densit ies. In most cases, the average cannot be evaluated exactly even when we employ approximated multi-ion distr ibution functions as described in the above. In our calculation, we use the effective medium ap8 proximation(Ef4A) which has been proved to be the best sing1 e-si t e approximation. The Et:A has been examined and shown to be superior to other approximation also from the viewpoint of analytic propert i e s , although the actual Eli4 calculation i s extremely d i f f i cu l t especially when nore than one orbit a l s are taken into account. The densit ies of s ta tes calculated from O . Assuming spheri,cal symmetry in k, we have: where w=E-Er We a1 so expand k about kg: av -1 = kg+(=) (Vk-VkO) = k O k k +z /v which serves to define vo=ak/aV 1 and zi=Vk-VkOTaking slowly varying factors outside the k integra l , we have Y(r,t) = e-jEOt DO(VkO) [ ~ ( l k ~ r l ~ {du fiw) x ,-iwt [I1 (w)+12(w)] (3.5)