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COLOUR CENTRES, GENERAL.THE Vk CENTER
IN CESIUM HALIDES
T. Iida, R. Monnier
To cite this version:
Abstract. — The properties of the Vk center in cesium halides are investigated on the basis of the small polaron model. To observe the stability of the Vk center, relaxation energies are computed both for the hole localized on a single site and for the Vk center. The activation energies of the 180° jump and 90° jump are estimated using the simplest approximation. The static properties are successfully explained and the importance of acoustic mode interactions in the self-trapping is shown. For the jumping motion, the results indicate that more detailed studies are required.
Introduction. — The possibility for an electron in also estimate the relative change in the halogen-halogen a polar crystal to become self-trapped was first separation in the ground state of the center. Finally recognized by Landau [1] in 1933. Since then there we compute the activation energy for the hole's have been many investigations of this phenomenon, hopping motion, using the standard approach to non-focussing mainly on the interaction between the elec- radiative transitions t o lowest order in the hole-tron and the polar lattice vibration modes [2]. In 1961 p h o n o n interaction.
Toyozawa [3] showed that the interaction of the
par-ticle with the acoustic modes of the crystal also plays an 2. Model Hamiltonian. — W e consider a positive important role in the self-trapping process. Meanwhile, hole in a simple cubic cesium halide crystal, and sup-following the identification of the Vk center through its pose that the valence b a n d from which it originates is E P R spectrum by Kanzig [4], a number of theoretical composed of pure halogen p orbitals with well defined studies of the self-trapped hole appeared [5-9], most directions. According to Sewell's tight-binding treat-of them based on the so-called molecule in the crystal ment [10], we have for the total Hamiltonian treat-of the model. At present, this method, which has been brought system with the above approximation :
to a very high degree of numerical sophistication [9], is
restricted to the lighter halides for which self-consistent X. = Hh + HL + Hx. ( 2 . 1 ) solutions of the Schrodinger equation for the free mole- H e r e ^ i s t h e Hamiltonian for a free hole, given as
cular ion X2 exist.
I n this work we treat the Vk center as a small polaron Hh = — cc 2^ an,> an,; —
state, using a formalism originally due to Sewell [10]. >=*,?,*
In particular we assess the importance of the acoustic +
p h o n o n contribution to self-trapping, which, to our — *" X, an,ia«±n,/ + " n
knowledge, is the first application of Toyozawa's idea i=x,y,z
to real crystals. In our calculations we consider the w h e f e a j s t h e r o n.s i t e e n e f g y . , j s t h e ,a U i c e ions as point charges and introduce a single free c o n s t a n t • ] i s a u n i t v e c t o r i n t h e / d i r e c t i o n ; w, ,• a r e a parameter in order to mimic the true overlap matrix ^ a n d a b a n d i n d e t i v e l . T i s t h e a.iy elements. W e determine this parameter from the UV r • * i _. „< ,, , c t, •,. • *L
, t j .. . ,, , transfer integral a n d Hb the transfer Hamiltonian other
absorption spectrum a n d then compute the relaxation . . . r r . ,, . , . TT .., r c T , , ,• . t ... , t, , r -w than the one of a type. / / , is the lattice Hamiltonian energy for a hole localized at one site and t h a t tor a Vk . J r L
center in order to examine their relative stability. W e °
(*) Supported by the Fonds National Suisse de la Recherche HL = V froj (q) I bs+ bs -I— I
Scientifique. 5,q \ 2 /
COLOUR CENTRES, GENERAL
THE V
kCENTER IN CESIUM HALIDES
T. I I D A
Department of Physics, Osaka City University, Sumiyoshi-ku, Osaka, Japan
and
R. M O N N I E R ( * )
(*) N O R D I T A , Blegdamsvej 17, DK-2100 Copenhagen 0 , D e n m a r k
Résumé. — On étudie les propriétés du centre Vk dans les halogénures de césium à l'aide du modèle du petit polaron. On compare les énergies de relaxation pour le centre Vk et pour un trou localisé sur un seul site. On calcule également les énergies d'activation pour les sauts à 90° et 180° en appliquant la théorie des transitions non radiatives à l'ordre le plus bas dans l'interaction trou-phonon. Alors que les propriétés statiques du centre sont reproduites de façon satisfaisante, les calculs montrent qu'une étude plus détaillée est nécessaire pour expliquer la mobilité préférentielle.
0 - 8 0 T. IIDA AND R. MONNIER
where we consider only longitudinal acoustic (S = ac) ing that the hole at site n interacts only with the and optical (S = op) modes of wave vector q. For the nearest-neighbour halide and cesium ions :
frequency spectrum, we assume that W,, = W, and
W,, = vq, v being the longitudinal sound velocity. On H , =
H!''
+
H,'?the basis of a simple point ion model we may write the Here the first term is the band-diagonal type interaction hole-phonon interaction in the following form, assum- given as
H ( ' ) =
I
C C
2 Xj(s, q) ex~(iqRn) $',('I) an,j+
1
2
C
Yj,a(s, 4) ex~(iqRn) $s(q) a l j an+,,j,jn , j s,q 6 = n , j s,q
with
Xj(op, q) = Kop[gtj
+
(1 - g ) V ~ x P ( - iqz,)] Xj(ac, q ) = Kac[t - tl ~ x P ( - iqzl)],
Yj.*(op, 9) =
+
i ~ , , gAfl(G$ (ell)'. [l-
~ x P ( + ilq j)] Yj,*(ac, g) =+
X,, ( e l l )'.
[l-
exp(+ ilqj)] , @,(Q) = bs,q+
b,t-, 9where
K,, = (h12 NM,, w,)"~
and
K,, = (h12 NM,, vq)'I2 ,
N being the number of unit cells and M,, and M,, being respectively the reduced and total mass of a pair consisting of a halide and a cesium ion ;
t j
=(;;D
sin (1qj) (ejl) andthe zk7s being the vectors shown in figure 1. A, is a free parameter whose sign is easily seen to be negative.
H ~ ( ~ ' is the band-off-diagonal type interaction, for which we do not give the explicit form since we do not need it in the subsequent discussion.
FIG, 1.
-
Geometry of the vectors zk, The halide is at the centerof the cube ; the corners are occupied by ccsiurn ions.
3. Self-trapped state and relaxation energy. -
Due to the strength of the hole-phonon interaction in the systems we consider, the bandwidth for the coupled hole-phonon (E polaron) state is extremely
small , and it is therefore reasonable to consider the polaron a s localized.
Let us calculate the relaxation energies for a hole localized on a single site and for the V, center. At this stage, there is no practical method for finding exact solutions to our problem. We therefore make the following Anzatz for the states :
I
$j(n)>
= a; I 0>
I
X j>
(3.1) for the hole localized on a single site n and1 +
(a,,,
k
+ a: pLj,j)/
0>
1
X ; ( ~ Z ~ P)>
(3.2) for the V, center where the hole is shared between two adjacent sites n and n+
=+ or
-). Here I0>
is the hole vacuum state, and
I
xj
>
andI
p)>
are lattice wave-functions to be determined variatio- nally. In (3.2), the upper and the lower sign corres- pond respectively to the state y = U (or ' 2 : ) and y = g (or 2,Z:). We minimize the expectation values of the total Hamiltonian with respect to (3.1) and (3.2) to determine the lattice wave functions and the relaxation energies. With this procedure, we find the stationary state having the energy
for the hole localized on a single site, where the relaxa- tion energy is given by
THE V K CENTER IN CESLUM HALIDES
having the energy
where
with
and
4. Discussion.
-
We, first of all, determine the value of A, by comparing the observed UV absorption spectrum with the calculated one. The absorption spectrum due to the transition from 'C: to '.Zg+ is calculated to be a Gaussianwhere d,, is the matrix element of the dipole operator between the initial and the final state. The peak position is given by
E = 2
K
+
(E;, - E;,)+
C
I
C,, I 2
fZ~,(q)S,,
and the width by
D = (8 In 2)"' W = (8 In 2)"'
( E
1
f S ,1'
(2 p,,+
I) [hoS(q)l2)
'l2where
Using (4.2), we determine A , from the observed UV
absorption band-widths. Input data are listed in table I. The results for A , are listed in table
TT.
The computed peak positions are listed in table 111: and are in fair agreement with experiment, which gives us some confidence in our values for A,.The computed relaxation energies for the hole localized on a single site and for the V, center are shown in table IV arid table V, respectively. In both cases, we can see that the interaction with acoustic phonons plays an important role in the self-trapping of the hole, as pointed out by Toyozawa [3]. Using the above estimates we may examine whether i t is more favourable for a hole to be the ground state of a V, center rather than to move freely in the crystal or to be
I ~ f p u t data
Lattice Longitud.
M o p
(‘'1
M,,C')
const. [l 1 ] sound vel. o ~ o ( r > , T, (")amu amu (4 K) (A) 105 cm S - ' l o i 3 S - eV
.-
-
- -CsCI 27.99 168.36 4.068 3.08 [l21 3.08 [l21 0.43
CsBr 49.90 21 2.82 4.236 2.58 [l 31 2.14 [l31 0.45
Csl 64.92 259.81 4.502 2.22 [l41 1.64 [l41 0.40
((0 Computed with thc data in reference [ l l].
T. IIDA AND R. MONNIER
Detemination of A,
CsCl CS Br CS I
- -
One-site relaxation energy
E,R(~P) eV E1,(ac) eV E,, eV
- - D eV 0.22
1
A,1
0.161
A,1
0.131
A,I
CSCI - 0.12-
0.79 - 0.91-
0.46v)
-
0.40 ( h ) 0.35 [l81 CSBr D,,, eV - 0.20 - 0.82-
1.02 A,-
2.09-
2.50 - 2.69 CSI-
0.33 - 0.80 - 1.13(") Preliminary measurements by J.-P. Pellaux and T. Sidler show the existence of V* band but its width could not yet been determined. The quoted value is obtained by assuming the cesium halides follow the same trend as potassium halides.
( b ) J.-P. Pellaux and T. Sidler, private communication.
TABLE V
Two-site relaxation energy
-
- -Peak position of UV band
CsCl = - 0.31 - 0.79
-
1.10 CsCl CsBr CsI y = g -0.56 - 0.67-
1.23 - --
7 = U-
0.36 - 0.90 - 1.26E
e~ 2.94 3.29 3.16 CsBr y = g -0.51 - 0.75 - 1.26Etx,
eV 3.25 (") 3.18 (") [l71 3.02 [l81 y = U-
0.39 - 0.95-
1.34 Csl(") J.-P. Pellaux and T. Sidler, private communication. y = g -0.50 - 0.79
-
1.09localized on a single site. Supposing that the free hole is created at the
r
point, the relevant energies for the three cases areand
Comparing the magnitudes of these energies, for instance E,,,,
-
- a - 0.8eV, E,-
a-
1.1 eV, and E; N-
ci-
1.7 eV for CsT, we find that the V,formation appears to be most favourable.
The relative difference between the halogen-halogen separation in the ground state of the V, center and the one in the undistorted crystal is given by
the corresponding numbers are
-
0.16,-
0.15 and-
0.13 for CsCI, CsBr and CsI, respectively. These are in fair agreement with the observed values (10%
-
30 %reduction).Finally we consider the hopping motion of the V, center. The hopping motion in CsI has been observed by Sidler et al. 1181 in the measurement of the thermo- luminescence. They found that two types of jumps are possible, that is, the 1800 jumps in which the V, orien- tation is not altered and the 900 jumps in which the initial and final orientations are at right angle. Accord- ing to their experiment the 180° jump has a lower activation energy than the 900jump. On the basis of our simple polaron model, the jumping transition proba- bility is given by
where t,bi and $, are an initial and a final state with energies E, and E, ; pi is the thermal distribution in the initial state ; the summation is carried out over phonon states. The matrix elements are given in the form
<
H i )1
H-
Ei/
$(f)>
= J l<
{ PS,,I
~ x P [ - L:(- 1, +)] exp[Lt(O, +)]I(
P:,1
> +
for the 1800 jump, where J , is the band-diagonal type transfer matrix and
v,,,
is a function of the band-diagonal type hole-phonon interactions. For the 90° jump, the matrix element is written as<
$(i)1
H - EiI
$(f)>
= J 2<
{ P,,,1
(
exp[- LXO,+)l
exp[Lt(O,+)I
I
{P:,,
1
>
-
1
-
-<
( P,,,1
l
exp[- L W ,+)l
exp[L:(O,3-11
C
CtS,,
+
5;
b,f,I
1
( P:,,1
>
9 (4.5)2 S.,
THE VK CENTER IN CESIUM HALIDES C7-83
If we ignore these terms even though there is no reason in our model, the transition probabilities are calculated in high temperature range as
I
"
cosech [hos(q),2 k ~ ] [ho,(q)]'where R = 1 and 2 for the 1800 jump and 900jump, respectively and
The effective activation energy is defined as
a
E,,, = k ~ ~{In Wir - ) .
dT
By numerical computation, we obtain activation energies which are almost temperature independent in the range 6 0 K W 9 0 K :
E,,, = E3,,(op)
+
Enc,(ac) = 0.18+ 0.34eV
for the 1800 jumpE,,, = E,,,(op)
+
E,,,(ac) = 0.07+
0.32 eVfor the 900 jump. Note that the difference in the activation energies mainly comes from the interaction of the hole with the optical phonons. Unfortunately, the computation yields a lower activation energy for the 900 jump than the 1800 jump, which contradicts the experimental result. This observation forces us to go to more detailed investigation in order to overcome the discrepancy. We shall give a more detailed discussion of this pro- blem in a later work.
References
[l] LANDAU, L., Phys. 2. Sowjefunion 3 (1933) 664.
[2] For instance, FROHLICH, H., Adv. Phys. 3 (1954) 325. [3] TOYOZAWA, Y., Prog. Theor. Phys. 26(1961) 29.
[4] KANZIG, W., Phys. Rev. 99 (1955) 1890.
CASTNER, T. G. and KANZIG, W., J. Phys. & Chem. Solirls
3 (1957) 178.
[S] DAS, T. P., JETTE, A. N. and KNOX, R. S., Phys. Rev. 134
(1 964) A 1079.
[6] JEITE, A. N., GILBERT, T. L. and DAS, T. P., Phys. Rev.
184 (1 969) 884.
[7] SONG, K. S., J. Phys. Soc. Japan 26 (1969) 1 13 1.
[8] JEITE, A. N. and DAS, T. P., Phys. Rev. 186 (1969) 919. [9] NORGEIT, M. J. and STONEHAM, A. M., J. Phys. C 6 (1973)
229.
I101 SEWELL, G. L., Phil. May. 3 (1958) 1361.
[l11 MAHLER. G. and ENGELHAKDT, P., Phys. Status Solidi ( b )
45 (1971) 543.
(121 AHMAD, A. A., SMITH, H. G., WAKABAYASHI, N., and WLLKINSON, M. K., Phys. Rev. B 6 (1972) 3956.
[l31 DAUBERT, J., Thesis, J. W. Goethe Univ., Frankfurt am Main (1973).
[l41 BUHKER, W. and HALG, W., Phys. Slat~is Solidi ( b ) 46
(197 1) 679.
[IS] P o o ~ e , R. T., JENKIN, J. G., LIFSEGANG, J. and LECKEY, R. C. G., Phys. Rev. B 11 (1975) 5179.
[l61 KUNZ, A. B., Phys. REV. 159 (1965) 738.
1171 CHOWDARI, B. V. R. and ITOH, N., J. Phys. & Chem. Solids
33 (1972) 1773.
[l81 SIDLFR, T., PELLAUX, J.-P., NOUAILHAT, A. and AEGER- TER, M., Solid State Commun. 13 (1973) 479.
DISCUSSION
A. B. LIPIARD.
-
Since the usual model of the V,-centre as a halogen molecule-ion embedded in the crystal serves the subject quite well, it appears to me that one valuable outcome of your calculations is the prediction that hole localisation onto two centres is energetically preferable to localisation on only one.R. MONNIER. - Your comment about the molecule
in the crystal approach is quite correct. We wanted
to give a unified treatment of all simple cubic Cs- halides, with a special emphasis on CsI, for which we had data available from Sidler et al. in Neuchgtel. To our knowledge, the number of electrons in the 12- molecule is too large to be treated with the present
molecule in tlze crystal programs, so that we had to
