Bipolaron stability

Volume 95A, number 7

PHYSICS LETTERS

16 May 1983

BIPOLARON STABILITY K.A. CHAO Department of Physics and Measurement Technology, University of Link6ping, LinkOping, Sweden R. MICNAS, S. ROBASZKIEWICZ Institute of Physics, A. Mickiewicz University, Poznah, Poland and A.M. OLES and B. OLES Institute of Physics, Jagellonian University, Cracow, Poland Received 23 November 1982 Revised manuscript received 10 February 1983

The bipolaron stability is investigated in a finite system for which analytical solutions are derived. Whenever the bipolaron singlet can be well defined and is deeply bound, the bipolaron triplet is also bound. The presence of the bipolaron triplet between the bipolaron singlet and the free polaxon band plays a very important role in determining the thermodynamic properties of the bipolaron.

Anderson [1] has postulated that the ground state of a system of spin-l/2 centers consists of mobile singlet bands. This idea led to the conjecture that in amorphous chalcogenides one-center bipolarons (Anderson bipolarons) exist as a result of the local lattice deformation which produces centers of negative effective correlation energies [2]. Another type of bipolaron, the two-center bipolaron has been demonstrated by Lakkis et al. [3] as the ground state of T i 4 0 7. A similar ground state of vanadium bronzes was proposed later by Chakraverty et al. [4}. Recently, Alexandrov and Ranninger [5] have developed the theory of "small" two-center bipolarons under the condition that the bipolaron dissociation energy is much larger than the bipolaron bandwidth, In this letter the name bipolaron means the two-center bipolaron. The few existing theoretical investigations on bipolarons restrict themselves to a small region in the space of coupling constants. In this region the bipolaron singiet (BS) is deeply bound at zero temperature. While the experiments of ref. [4] (in order to explain specific-heat and magnetic-susceptibility measurements of vanadium bronzes) invoked the lower-lying bipolaron singlet, placing (without calculation) the bipolaron triplet (BT) above two unbound polarons, the detailed theoretical treatment of the bipolaron model in ref. [5] rectifies this error. In this letter we will examine the stability of both the BS and the BT in almost the entire space of the coupling constants. Based on the result of stability, some thermodynamic properties are then discussed. The central problem in the theory of small polarons or bipolarons is to treat the electronic and the vibronic behavior of the system in a coherent manner. As far as the electronic properties are concerned, a local deformation of the lattice will modify (i) the electronic energy levels of a given tight binding state, and (ii) the transfer integral between different sites due to a change of the interatomic distances. It is obvious that the equilibrium intramolecular distance will depend on whether two electrons sitting on such a molecule are in a singlet or in a triplet state. While singlet states have bonding character, the triplet states have antibonding character, hence leading to intramolecular distances bigger than for the singlet states. It is very difficult to represent such physical 0 031-9163/83/0000- 0000/$ 03.00 © 1983 North-Holland

391

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16 May 1983

picture via a solvable model hamiltonian. Neglecting the deformation dependence on the electron transfer integrals, these two states would give equal equilibrium intramolecular distances. Under this condition, the canonical transformation used in ref. [4] yields an effective hamiltonian r

n=tl

~ie( a

~ 1,~l',~ tijll'a 1, oaj, l ' ' o + U ~ ni,li , ' 1, o ai, 2, o +ai, 2, oai, 1,o) + i,j

l, ,ni, l. ~

t

+W1

~

i,~,o'

ni, l,oni, 2, o'+ ~ ~ WijlI'nj, l, onj, l' o'' i,] l,l',o,o' '

(1)

Here i labels the pair of atoms and 1 = 1, 2 specifies the atoms in the pair. a~lo,, all ° and nilo - a tilo a ilo are, respectively, the creation, destruction and number operators associated to the electronic state localized at the lth atom with spin o in the ith pair. tl, tifll, , U, W 1 and Wifl l, are the renormalized electronic energies. The system can be pictured as follows. Due to the e l e c t r o n - p h o n o n interaction atoms are paired in such a way that the intrapair hopping energy It 11 is larger than the interpair hopping energy Itifl l, [ and the effective interpair interaction energy IWijll, I is less than the effective intrapair interaction energy IW1 I. U is the intraatomic Coulomb energy. The bipolaron can be unambigouously defined at the pair limit where both tijll, and Wijll, are zero. Let us consider any two electrons at the pair limit. If the two electrons occupy two pairs of atoms and no other electrons occupy these two pairs of atoms, then they are in the state of free polaron with energy 2 t 1 . On the other hand, if the two electrons occupy the same pair of atoms, they form a bipolaron. If W1 is positive, the energy of the bipolaron triplet (BT) is always positive and so is unbound with respect to the free polaron (FP) state. Therefore, in this paper we will consider the case of W1 < 0, where the lowest energy of the bipolaron singlet (BS) is ½(U + W1 - [(U -- W1)2 + 16t2] 1/2} and the energy of the BT is simply W1. The stability condition for the BS and the BT is readily obtained. Away from the pair limit, only the situation of deeply bound BS with very small tiff l, has been treated. The problem with arbitrary values of tiff l, and Wifl l, is certainly very difficult. Consequently, it is worthwhile to study a finite system which can be solved exactly and so provides the important information for further analysis. This is the purpose of the present paper. For numerical investigation on the bipolaron stability in a finite system, we must solve (1) exactly. According to the above discussion, the analysis will be most convenient and unambiguous if the number of electrons is equal to the number of atom pairs. Since each bipolaron is made up with two electrons, for a finite system we must choose an even number of electrons. The smallest system will be two electrons and four atoms, and the next one will be four electrons and eight atoms. It is possible to derive analytical solutions for the case of two electrons and four atoms, but not for larger systems. Although the system is small, we will see from the result that it already exhibits the main features of the bipolaron stability. The inset in fig. 1 demonstrates the topological structure of the system we consider. The four atoms are located at the squares 1,2, 3 and 4. The hamiltonian (1) is then simplified as

• . t + a~qa3o+ a;oa2o) H= t I ~ ( a ~ o a 2 o +a~2oalo + aT3oa4a +a~oa3~)+ t 2 ~ ( a l o+a 4 o + a4oalo o

o

+ W1 ~ , (nlcrn2o, + n3on4o, ) + 14]2 ~ , (nlon4o, + n2on3o, ) + U ~ n i t n i $ . fro

O0

i

We first define the subspace for the triplets:

t,

~Jl=al*a2* iO>, if5

~2 = 2 - 1 /

=2-1/2~at "t~. 3~"4* +a~3ta~4~)10)'

392

at

1~ 2* ff6=a;+a?44[0),

t t ~7 = al ~a3, I0>,

(2)

Volume 95A, number 7

PHYSICS LETTERS

B

16 May 1983

F

~f 0.5

v~

w,I

\\\\

0.7S

\

~1 ZW1

¢8 = 2-1/2'at,l+"3*~t + a~,a;~)lO>, ¢11

=

Fig. 1. Bipolaron stability boundary for the special case t2 = 0. Heavy curves are for the bipolaron singlet and the thin curve is for the bipolaron triplet. The stable region for bipolaron is B while that for the free polaron is F. The inset illustrates the small system considered in this calculation.

1.5

¢9 =a~$at3+ i0>,

2-1/2(a ? a t t i" 25 4t +ct2ta4$)10),

? t ¢14 = 2 - 1 / 2 ( a t2~ a~ t + a2ta3~)lO>'

"~

¢10

=

t

a2ta4* [0>,

¢I 2 = a~at4~[0>,

¢13

t t Io>, ¢15 =a2~a3~

¢16 - al ta4t J0>,

¢18

=

t

_ t

t

t

al ~a4 ~ 1 0 ) .

If we express the eigenfunction of H as XiCi¢ i, then the hamiltonian matrix can be diagonalized analytically. The eigenenergies are E(BT0)--- W1,

E ( B T - ) = /W 1 - (W 2 + 16t22)1/2]/2.

which approach to the BT eigenenergies at the pair limit, and E(FT1)--- W2,

E ( F T 2 ) = [W 1 + ( W 2 + 16t22)1/2]/2,

E(FT-+) = [W2_+(W 2 + I6t2)1/2]/2,

which approach to the FP eigenenergies at the pair limit. All the eigensolutions are triply degenerate, and the nonzero Ci can be expressed as

[Ci' Ci+3, Ci+6, Ci+9, C i + l

2,

Ci+151

= [ 2 - 1/2, _ 2-1/2, 0, 0, 0, 01,

i = 1, 2, 3

for E(BT0) ,

= [-2t2/{W I - E ( B T - ) } , - 2 t z / { W 1 - E ( B T - ) } , 0, 0, - 1 , 1],

i = 1, 2, 3

for E ( B T - ) ,

= [0, 0, 2 -1/2 , - 2 -1/2 , O, 0],

i= 1, 2, 3

forE(FT1),

[-2t2/(w 1 -E(FT2)},-2t2/{W 1 -E(FT2)}, 0, 0 , - 1 , 1],

i= 1,2, 3

forE(FT2),

= [0, 0, - 2q/{W 2 -E(FT-+)}, - 2 q / ( W 2 -E(FT-+)), 1, 1},

i= 1, 2, 3

forE(VT-+),

=

except for a possible normalization constant. Each value o f i corresponds to a different eigenfunction. Next we define the subspace for the singlets: 393

Volume 95A, number 7

+1= 2-I/2

PHYSICS LETTERS

,4,-

~4 = 2-1/2 (a34a41" "p 5"

t + 02 = ai tai ~[0>,

_a~3,i,a~,)lO),

05 =a3ta3,10),

03

= a~

16 May 1983 t

ta2 ~10>,

2-,/2

06=a4ta4~[O),

qb9 = 2-1/2 (a2,a3, + + -- a2ta3 ~- -~+)10),

q510 = 2-1/2

,4,-4,4,)i0>, .~

(a],!a{t

.t.

--a~ ta~l,!,)lO).

If again we express the eigenfunctions as EiDfPi, then six eigensolutions can be derived explicitly as /E(BS1) = U , 1

1

1

[D 1, D2, D3, D4, D5, D6, D7, D8, D9, D1 o] = [ 0, ~-, - g , 0, - g , ~, 0, 0, 0, 0] , E(BSI')=[U+W 2+((U

W2)2+162}1/2]/2 ,

[D I , D 2 , D 3 , D 4 , D S , D 6 , D 7 , D a , D 9 , D l o ] = [0, 1, -1, 0, 1 , - 1 , 23/2t2/(W 2 - E ( B S I ' ) } , --23/2t2i{W 2 --E(BSI')}, 0, 0] , E(BS+) = [U+ Wl)2 + { ( U - W1)2 + 16t2}1/21/2 , [D1, D 2 , D 3 ' D 4 ' D s , D 6 , D 7 , D8' D9'D10] = [23/2t1/(W1 _E(BS+)}, 1, 1, - 2 3 / 2 t l / ( W 1 _E(BS+)},

-1,-1,0,

0,0, 0] ,

which approach to the BS eigensolutions at the pair limit, and E(FSl) -- 0 ,

[ D 1 , D 2 , D 3 , D 4 , D s , D 6 , D 7 , D 8 , D 9 , D l o ] = [0, O, 0, 0, 0, 0, 0, 0, 2 -I/2, -2-1/2 ], E(FS2)= [U+ W2 - ( ( U - W2)2 + 16t2}1/2]/2, [D 1,D 2,D 3,D 4 , D 5 , D 6,DT,D 8,D 9,DlO ] = [0, 1 , - 1 , 0 , 1 , - 1 , 2 3 / 2 t 2 / { W 2 - E ( F S 2 ) } ,

-23/2t2/ {W 2 - E(FS2)}, O, O] , which approach to the FP eigensolutions at the pair limit. The four rest eigenfunctions obey the symmetry D 1 = D4, D 2 = D 3 = D 5 = D6, D 7 = D 8 and D 9 = D 10, and are the solutions of the coupled equations (W1 - E ) D 1 - 2 1 / 2 t ] D 2 + 2 t 2 D 9 = O ,

- 2 1 / 2 t 2 D 2 + (W2 - E ) D 7 + 2 t l D 9 = 0,

-23/2tlD 1 +(U-E)D 2-23/2t2D 7=0, 2t2D 1 + 2 t l D 7 - E D 9 = 0.

Analytical solutions of these coupled equations can be obtained with the standard method. For the special case t z = 0, the four eigenenergies are E(BS+-) and E(FS+) = E(FT+_). As mentioned above, the bipolaron can be unambiguously defined only at the pair limit. However, for the general case away from the pair limit, we can define the BS branch (or the BT branch or the FP branch) composed of those eigensolutions approaching to the BS (or BT or FP) states at the pair limit. We can further define the BS (or BT) binding energy A s (or At) as the energy difference between the bottom of the BS (or BT) branch and the bottom of the FP branch. If A s (or At) is positive, the BS (or BT) is said to be stable. If we normalize the electronic energies with respect to the intraatomic Coulomb energy U, the bipolaron stability boundaries are shown in fig. 1 for the special case t 2 = 0. The heavy curves are for the BS with various values of W1. However, in the W2/W 1 - 2tl/W 1 plane all the boundaries for the BT are scaled into a single curve, regard394

Volume 95A, number 7 0.25

O.S

PHYSICS LETTERS 2tJWl=.75

!

1.25

0.1

-0,1

.

.

.

16 May 1983

0.25 0 . 3 5 : - -

0.5

2tl/W~ =25

1

f. 25

.

....

-0,1

-0.1

-0.35"

!o

;'o t2/tl

!o

, o

Fig. 2, The binding energies of BS (heavy curves) and BT (thin curves) for W1 = -0.1. Numbers next to the curves are the corresponding values of W2/W1.

--,o

~ ~o

t2/t~

I o

0

~-03s ]ro ,

Fig. 3. Same as in fig. 2 but for 1¢1 = -0.3.

less the values of W1 . This universal boundary is indicated by the thin curve. In fig. 1 the region for stable bipolaron is marked as B, while the region of stable free polaron is marked as F. The binding energies A s and A t as functions of t2/t 1 are shown in fig. 2 for W1 = - 0 . 1 and in fig. 3 for W1 = - 0 . 3 . Each figure contains five columns for different values of 2 t l / W 1 indicated at the top. In each column the higher curve corresponds to the lower value of W2/W1, and the values of W2/W 1 are given in the column to the left side. The heavy curves are for the BS while the thin curves are for the BT. These results clearly indicate that whenever both the BS and the BT are bound, the BT always lies between the BS and the bottom of the FP band. At the pair limit, 41, 42, ~3, ~04, 45 and ~6 are the BT states, 4)1, q~2, q~3, ~4, ~5 and q~6 are the BS states, and the other ¢i and Oi belong to the FP states. In order to clarify the bipolaron properties, the bipolaron singlet component ~6= 1 [Dil2 in the ground state is plotted in fig. 4 with a similar arrangement as in figs. 2 and 3. Heavy curves are for the cases that the BS is bound, while thin curves are for the cases that the BS is not bound. From fig. 4, there is no doubt that a bipolaron can only be unambiguously defined if t2/t I is very small. We have demonstrated that whenever the BS can be well defined and is deeply bound, the BT is also bound but lies between the BS and the b o t t o m of the FP band. Moreover, in this case the binding energy of the BT is comparable to that of the BS. The change of the magnetic susceptibility into the Curie-Weiss type with increasing temperature has its origin in the BS to BT transition. The theory of bipolaron transport is based on the assumption that t 2 is small enough to be treated as a perturbation. The presence of the BT as the first excited state when the ground state is the BS will certainly affect drastical-

o.s 0

, I

i~o

;o Fig. 4. The bipolaron singlet component contained in the ground state for 1+'1 = -0.1. 395

Volume 95A, number 7

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16 May 1983

ly the bipolaron transport properties. Recently, Alexandrov and Ranninger [6] have investigated the possibility of superconductivity due to the bipolaron singlet. The possibility of having bipolaron triplet superconductivity and its impact on the general theory of bipolaron superconductivity remain as open questions. In this paper we have considered the case of W2 < O. From fig. 1 it is clear that when W2 becomes positive the bipolaron phase becomes more stable. However, one important feature of the effect of W2 in bulk system cannot be investigated in the present small system. In a bulk system, if W2 is positive and sufficiently large, there is a tendency of formation of a bipolaron lattice.

References [1] [2] [3] [4] [51 [6]

396

P.W. Anderson, Mater. Res. Bull. 8 (1973) 153. P.W. Anderson, Phys. Rev. Lett. 34 (1975) 953. S. Lakkis, C. Schlenkar, B.K. Chakraverty, R. Buder and M. Marezio, Phys. Rev. B14 (1976) 1429. B.K. Chakraverty, M.J, Sienko and J. Bonnerot, Phys. Rev. B17 (1978) 3781. A. Alexandrov and J. Ranninger, Phys. Rev. B23 (1981) 1796. A. Alexandrov and J. Ranninger, Phys. Rev. B24 (1981) 1164.