- Email: [email protected]
The thermodynamics of imperfections in AgBr
J. Phys. Chem. Solids
Pergamon
Press 1965. Vol. 26, pp. 901-909.
THE THERMODYNAMICS
Printed in Great Britain.
OF IMPERFECTIONS
IN AgBr*
F. A. RRGGER Department
of Electrical
Engineering,
University
of Southern California, Los Angeles
(Received 28 May 1964) Abstract-Analysis of experimental data available in the literature leads to an evaluation of the enthalpies and entropies of formation of native atomic and electronic imperfections. Estimates are made for the free enthalpy of formation of neutral and charged Schottky defects, which show that bromine vacancies are always a minority species. INTRODUCTION
ON THE basis of experimental work on the density and lattice constants of pure AgBr by WAGNER and BEYER,~) and on the conductivity of cadmium doped crystals by KOCH and WAGNER,@) TELTOW@ and KURNICI&Q it is generally accepted that, at least at medium temperatures, AgBr has predominantly Frenkel type disorder of the silver ions, with interstitial silver ions as the main carriers of the ionic current. Plots of the logarithm of the conductivity, of the self diffusion coefficients, the expansion coefficient and the specific heat against l/T are slightly curved below, and more strongly curved above 375”C(s~s~7) : the enthalpy of formation of lattice defects increases, but the free enthalpy decreases. For this two explanations have been offered (1) Schottky disorder becomes predominant above 375”C.(4) (2) The crystal ‘loosens’, partly as a direct result of the increase in temperature (giving an increase in lattice vibrations), partly due to the presence of an increasing number of imperfections (which in turn affect the lattice vibrations).@**) Although the enthalpy of formation of defects, and the activation volume derived for Schottky disorder on this basis are reasonable, there are various reasons against the former explanation. As indicated by COMPTON and MAURER@) and
* Supported by the Technical Advisory Committee of the Joint Services Electronic Program, of the U.S. Army, Navy and the Air Force under Contract No.
AF-AFOSAR-496-65.
901
TANNHAUSER,t5)the appearance of bromine vacancies as dominant imperfections must result in a relative decrease of the conductance of interstitial silver; since these are the main current carriers, this should lead to a dip in the log cr- T curve. On the basis of similar considerations the selfdiffusion of bromine (if dependent on V&) should be expected to show an increased slope at T < 375%(s) Finally the thermoelectric power of pure AgBr which is positive for a Frenkel mechanism with interstitial silver more mobile than silver vacancies, should change to negative for a Schottky mechanism, the mobility of the silver vacancies exceeding that of bromine vacancies.(‘) Neither of these effects has been observed. Therefore, it seems that the loosening model has to be preferred. ZIETEN,(~@ on the basis of measurements of the coefficient of expansion as a function of T, assumed that Schottky disorder becomes important at low temperature. LAWSONt7) has shown, however, that Zieten’s results do not justify such an assumption. Thus, although possibly some Schottky disorder may occur to account for bromine self-diffusion, it is never predominant. This statement finds further support in the estimates made below. So far we have been dealing with essentially stoichiometric crystals. Let us now consider effects due to deviations from stoichiometry. Two such effects have been observed: (1) An excess of bromine, arising when AgBr is equilibrated under relatively high bromine pressures, gives rise to electronic (p-type) conducti+.U-1s)
(2) Polarization studies of AgBr between a reversible and a non-reversible electrode (Ag and Pt, or Pt and Pt, Bra, respectively) show electronic currents due to both electrons and holes,Q%IQ Analysis of these results along lines similar to those followed previously by WAGMZ~,(I*) MATEJEC(Ig) and RALEEN(~~) leads to the thermodynamic parameters of both atomic and electronic imperfections in AgBr which, as far as the latter are concerned, are consistent with values obtained from optical data.
As
ANALYSIS OF THE ELECTRONIC CONDUCTIVITY OF NON STOICHIOMETRIC A@ shown by WA~N~~,~I~~ LUCKEY et aLYG3)
S~A~OV~~II d &.,(I43 ~~~~~,~s~ and bLEIGE@) AgBr in equilibrium with a gas phase containing bromine at a partial pressure&j,, shows an excess conductivity due ta holes proportional to ~2:. Hanson’s data for T ,f 110°C can be represented by Ql. = 910~~~ exp(-O*53~~~~~~)
Q-I cm-I.
Shamovskii’s data show a slightly and can be represented by
(f)
larger slope
(2)
Hanson’s values at T > 110°C show a lower slope (N O-26 ev). A similar slope is found by &LEIGzx@~) from polarization measurements on the cell PtjAgBr/Pt, Bra. According to Hanson’s Hall measurements, the hole mobility at room temperature is approximately 2 cm2 V-I see-f. Raleigh showed that at higher temperatures it may be represented by vh = l-5 x lOI4 x T-s-5. Assuming this relation to hold up to temperatures well outside the measuring range, he found that the concentration of holes, p, can be described from room temperature to 300°C by the relation P = Q&r& =
Or, since there AgBr per ems,
ev/kT)
cm-a.
(3a)
are N = 2.07 x 102s molecules
P = 1*74x %0-s&~ exp(-0,439
1/2Bra(& + B&i-
Z&h.;
Hnr,~hr
(4)
the superscript cross, dash and dot indicating effective charges of respectively 0, -4 and -t-y. Application of the law of mass action gives
which shows that p cc pk$i as long as [Vi,] is independent of $nr,. If the AgBr is approximately pure, i.e. the concentrations of divalent impurities and of deviations from stoichiometry are smaller than the intrinsic native (Frenkel-type) disorder, [Yip]
w [Ag;] M K;;P’,
(0
Kb being the equilibrium constant of the Frenkel According to Raleigh this disorder reaction, applies to his data, to those of WAGNER,~I~I and also to Hanson’s data for T > 110°C. Combination of (3b), (S), and (6) using for Kh Teltow’s data(a) as interpreted by SCHMALZRIED@) : 3C.i = 3.13 x 105 exp( - 1.22 eV/KT)(mole
a~ = 50OO~~exp(-O.74e~~~T).
3.6 x lOss&~exp(-0.439
Formation of holes by incorporation of bromine from the vapor into the crystal can be formulated
fr.)2 (7)
we find &&,I% = 9.8 exp(-
1 *OSe~~~~} (mole fr-)2 (8)
The larger slopes of In bh :.1/T observed by Hanson at low temperature and by SHAMOVSKD et al. over the entire temperature range are no doubt due to the presence of divalent impurities in concentrations that are no longer small relative to &$‘a, which tends to make [V&] = constant = [&PI. The observed slopes are still smaller than the theoretical slope of l-074 eV, probably due to the fact that [I?&] is not yet quite constant in the temperature range concerned. Ilschner’s polarization experiments on the cell AgjAgBrjPt I give the electronic conductivity librium with silver:
of AgBr in equi-
et’j‘FzT) mole fr. (ix& = 4.35 x 10s exp(-1.7 (3b)
#j&T)
Q-I Cm-I. (9)
THE
THERMODYNAMICS
OF IMPERFECTIONS
According to YA~NA~ et aE.,@e)(~~)~~*oK= 70 cms V-1 set-1. From 25 to 35O”C, its temperature dependence is given by
84 exp(O.055 eV/kT) cm2 V-1 set-1.
M
(10)
Hence ($1 = -
(%)I ve4N
= 1% x lo4 exp(- 1.75 ev/Rrr) mole fr.
(II)
The electrons originate from excess silver, incorporated at interstitial sites, Vf, by the reaction A&) + Vt” + Ag;fe’;
Wg;l
W
tAg( W,“l
HAM
(12)
n[Ag;] = K&r.
(13)
lirAg = 8.8 x 10s exp(2.36 eV/kT).
(14)
Combination of (8) and (14) with corresponding expressions for the evaporation of bromine and the formation of AgBr from silver and liquid bromine, using the relevant data from standard tables and remembering that Ag&+B& = AgBr(f), leads to the equilibrium parameters for the formation of electrons and holes by the reaction ET
(15)
with np = & = 3 x 105 exp( - 3.38 eV/kT) (mole fr.)2 (16) This is demonstrated in Table 1. As is easily verified, the parameters of Kh, though used in the calculation. dron out in the final result for Kf. An independent second expression for .& can be obtained from ILSCHNER'S work.(lsj The total current in these experiments was I
I
kT
J = -_(a-~) ql
ae =
(410
- exp( - E/kT))
(o&{exp(E/kT)-
(1W
11.
(17)
Wb)
The hole current was found to become important only at E > 0.45 V. Subtracting the saturation value of o at medium voltages (giving (G&) from the total conductivity at E = 0.45 V, and using (18b), we find (eh)~. Introducing again the temperature dependence of the hole mobility used before, we obtain three values for the hole concentrations at different temperatures which can be represented by (p)r = 263 exp(- 1.27 eV/kT) mole fr.
(19)
Combination with (11) leads to (fi)&)r
= np = Kg = 4.1 x 10s
x exp( - 3.02 eV~k~~( mole fr.)s
Since the experiments were carried out with a crystal of a purity similar to that used by Hanson, and at temperatures > lOO”C, the AgBr behaves as an essentially pure material and (6) holds. Combining (ll), (13) and (7) we find:
0 + e’+h*;
903
f being the thickness of the crystal;
crh =
we = 3*64x 105 T-312
IN AgBr
(20)
an expression not too different from (16). As indicated by RALEIGH~‘)and WAGNER,~~)equations (16) or (20) can be used to determine the band gap and its temperature dependence. According to semiconductor theory,far) G = gcg&&h exp(PP) exp(- J?$VPT)
(21)
in which gc and gv are the number of valleys in the conduction and valence band, respectively.
m* 3/s ‘J”siscm -3 = 4.84 x 1015 ( m1
(22)
with m*, the effective mass, equal to rnz for A, and to rnz for Aa. Eg = Ef-/3T
eV.
(23)
Accepting a band model similar to that proposed for AgCl,(ss) gc = 1, go = 12. Taking m,* = 0.27, as observed,(as) assuming arbitrarily rni = 1, and writing Ts = 1*88x 10s exp(-O-11 eV/kT) we find that between 25 and 350°C equation (16) can be interpreted in terms of (21), (22) and (23) with I$’ = 3.38 eV and j3 = 1.9x lo-seV/deg. The same analysis for (20) gives Ey = 3.02 eV and #I = 2.26 x 10-s eV/deg. This is to be compared
F.
904
A. KROGER
Table 1. Reactions involving structure elements of AgBr and the parameters K = K” exp( -H/W), with K” = exp(As/k)
Reaction
of their reaction constants
li;o
K
(mole fractions)
5)
X3
(cal per deg per mole)
As/k
1/2Brs(g) + Br&+ V&+-h’ Ag(s)SV,x+ Ag;-t-e’
KAg
1.05 2.36
Ag; + Vi, + Ag& -t F’: l/ZBrs(l) + 1/2Brs(g)
Ki-’ K$
-1.22 0.16
AgBr(s) C+ A&)
.KGX
1.03
3-67
1.3
Kt
C38
3 x105
12.64
KBraVh
+ 1/2Brs(l)
0 ;Ir. e’+h’
9.8 8.8 x 10s
+
2.28 16
4.5 31.6
3.2 x 10-s 3x10”
- 12.65 5.71
-25 11.3
with optically determined band gap data. BROWN has determined exciton energies as f( T) ; the energy gap might be expected to be somewhat larger. However, photoconductivity experiments by the same author show that the cut-off of photoconduction occurs at the same values. Therefore it seems that the values given represent the energy gap* Brown’s data and the slopes and intercepts calculated from (16) and (20) are shown in Fig. 1. Linear extrapolation of Brown’s curve (a) gives absolute values of E consistent with equation (16), but the temperature dependence dEjdT = 8 x 10e4 eV/deg is smaller than is indicated by (16). Curved extrapolation of Brown’s Curve (6) fits the (16)
I EQUATION ‘\C---
2+
\
1,
\
EQUATlON
100
+ AgBr+
Vt+h*,
(24)
with the reaction constant Km,m Br,,
Ag
=
--,
=
3.1 x 10-5 exp(+0.17
I
i arownj
0
*Brz(g)+Ag;
KIF
2-o c I.8
line reasonably well, in particular if Brown’s data would turn out to be exciton points after all, the bandgap being x O-1 eV larger; this can be the case if the photoconductivity observed by Brown involves center levels. Equation (20) does not fit the optical data; it is possibly in error due to the fact that part of the extra conductivity observed in Ilschner’s experiments at high voltages may have been due to decomposition. By combining equations and relations found so far, we can find expressions for other imperfection reactions. For instance combination of (4), (5) and the Frenkel reaction gives:
K
(16)
\
‘\
2.2
25 ~--
----_-
341
2.6
200
300
e~/~~)(mole
fr.) (25)
i.e. Hnrz~g = -0.17 this same quantity + and -0+4eV.
eV; (his
MATEJECN found for W,/2) values between
\ (2C$,
400
500
600
s; OK FIG. 1. Values of exciton energies according to BROWN, and values of the temperature dependence of the band gap according to equations (16) and (20).
IMPERFECTiON CONCENTFUTIONS Using the parameters of the thermodynamic functions as given in Tables 1 and 2, the concentrations of imperfections in pure AgBr can be calculated as a function of temperature and the bromine or silver activity. From this, using the electron and hole mobifities as given and the mobilities
of Vi, and A& as determined by Teltow: v(Ag;) =f 0.058 exp(-0.145
eY/M’)
u(Y&) = 0.78 exp( -0.35 eV/kT)
(24) (27)
we can obtain the corresponding conductivities a$ and the transport numbers Q = crjfc, Figure 2 shows isobars for .p~r, = 0.23 atm. Note that although the lines of h‘ and Agi and Vis approach, they meet at such a low temperature and imperfection concentration that it is very unlikely that AgBr is ever pure enough to give rise to a range in which the neutrality condition is governed by [V&l = p. The temperature dependence of 4 as calculated is in reasonable agreement with the values observed by ~~AGN~R.(~~) Figure 3 shows isobars for AgBr in equilibrium with silver. Over the entire range there is a slight excess of electrons over holes. The electron conductivities range from 5 x lo-s1 0-x cm-1 at 25°C to 45 x 10-T a-1 cm-l at 300°C. Evidently tei = te+& is always < 1. Figure 4 shows isotherms for 25°C as a function of the silver activity (ads) and the bromine activity (&<), the two being related through ~~~~~ = ILQB~ = exp(-AGs~~~),
25 T-C 3.5
3
200
100
I 2.5 WffPK~
300 1 2
I.5
(of
(28)
AGO = 23.3 kcal/mde M 1 eV being the standard free enthalpy of the reaction AgBr(s) + Ag(s) + @h(g). The isotherms have only a meaning between the h&s a& = 1 (for AgBr in equilibrium with pure silver) and pur, = O-23 atm, the saturation pressure of liquid bromine at room temperature, corresponding to a& = 1*4x lo-17. We see again that in AgBr in contact with silver the concentration of electrons is somewhat larger than that of the holes. As is seen in Fig, 4(b), for Q& < 2 x 10-1s the electronic conduction, carried mainly by holes, exceeds the ionic conduction and thus leads to t,1 w tn M 1. Experimental points for tl determined by JAENICRE(~~) lie slightly above the theoretical line. This is no doubt due to the presence of divalent impurities in a concentration > X$‘s at 300°K, just as was the case for l%nson’s crystal.@@ ~~~
~~P~OF SCHOTI’KY IXSOl3JX3R
We have, so far, only been concerned with 6
25 3.5
TOC 1 3
i
I 200
100 I 2.9 103/ T(‘KI
300 I 2
1.5
(b1
FIG.2. Concentrations of dominant imperfections (a) and the corresponding conductivities and transport numbers (b) for pure AgBr in equilibrium with bromine vapor (PBrs = O-23 atm) Circles indicate experimental values of th determined by WAGNER.@~)Dashed lines: estimated values.
excess silver or bromine giving rise to ionized imperfections (Agi, Y&) and free electrons or holes. This does not imply that neutral centers do not also exist. In fact, such centers will be present in concentrations determined by reactions such as A&)+
Vf + Ag? or Ag; -I-&I$ Agf,
and corresponding ones for VA@ involving either single centers V& or pairs (VA&$ as found in the alkali Heidi. Additional, preferably optical,
16’” P i$
lo-7 tatm)
(0)
“K FIG. 3. Concentrations of imperfections in AgBr in equilibrium with silver. Dashed lines: estimated values.
experiments are necessary to reveal the presence of such centers and their concentrations. Such experiments have been carried out for AgCi with an excess of halogen, Hiere a square root dependence of the absorption strength on pcls indicated the presence of I’& centers.@?) For AgBr the presence of @a centers is shown by electron spin resonance.(ssf The position of the acceptor Ievel of V”Aprelative to the vafence band (or at Ieast a lower limit) may be estimated from:fss)
/
I
ldf7
lo’2
Pz2
lO-7 iotm) (b)
Taking the dielectric constant E = 13, the square of the refractive index & = 5, and the effective
Fro. 4. Isotherms for the concentraticms of imperfections in AgBr at WC, (a) and the corresponding conductivities and transport numbers ter = 4-k tat (b) Squares and circles indicate respectively experimental points of th = Itp and ts as determined by JAF.NX~EE.@~~
THE
THERMODYNAMICS
OF
IMPERFECTIONS
IN AgBr
907
Table 2
Reaction
Constant
AS (cal per deg. per mole)
KO (2)
4.5 - 12.6 5.6
1/2Brs(g) + Br&+V&+h* Br(g) + 1/2Brs(g) V&fh. + V&
KBQV~ -1/n KD K?
1.05 -1 -0.25
Br(g) + Br&+ V:s Ag(g) =+ A& + V$
KB~, v
KAg, v
-0.2 -o-2*
0.28 0*28*
-2.5 -2*5*
Br, v. v K-’ &Br KVBPBT~ l/n KD
-0.4 1.03 0.16 1.0 2.96
7.8 x lO+ 3.67 3 x 10s 5.75 x 10s 4.26 x lo2
-5.0 2.6 11.3 12.6 12.0
or
5.15 4.75
2.7 x 108 2.1 x 10’
38.5 33.5
+ Br(g) + AgBr(s) -I-V& + V& AgBr(s) + Ag(s) + 1/2Brs(l) 1/2Bra(l) $ 1/2Brz(g) 1/2Bra(g) + Br(g) i Ag(s) = Ag(g)
(a) : A&)
(b) AgBr(s) + Ag(g) + Br(g) (s)+(b): 0 f V&+V&
i%.
K -DAg KAL,
KS
9.8 1.74 x 10-s 16.4
* Estimated to be equal to the corresponding figures for Br.
mass (of holes) m* = rnz w m, we find Ea w 0.2 eV. For &, the distance of the donor level of Agr to the conduction band we find in a similar way & w 0.05 eV. The equilibrium constant of the ionization reaction Vj& + V&fh’(Ka) is now: K
=
a
w&J E=gV2
fi
exp( - Ea eV/kT)
= 6.1 x 10-2 exp(-0.25
eV/kT) mole fr. (30)
Similarly 4-%1 Kb = -
[&,x1
AC exp( 2
= gc -
& eV/kT)
= 7.1 x IO-4 exp( -0.1 eV/kT) mole fr.
for
the
formation
of
neutral
4.75 HSIHAg,Br=
(31)
Schottky
-
5.15
=
0.92
i.e. a value between O-5 and 1 as found in many crystals, but close to 1, which indicates that AgBr is largely ionic.(sO) We can derive from KS the parameters of the equilibrium constant K$ of the normal Schottky reaction : 0 +
Combining the reaction underlying (30) with other reactions, as carried out in Table 2, we find the enthalpy and entropy of formation of V& by the reaction of AgBr with atomic bromine. Assuming, tentatively, that the parameters for formation of V& by the reaction of AgBr with atomic silver are the same-which cannot be far wrong-we can find the parameters of KS, the equilibrium constant
defects (Table 2, bottom line). The enthalpy, Hs, may be compared with the enthalpy of evaporation of AgBr to single atoms (H&,Br)We find
Vig+
VI&,
with the aid of the relation K; = KS=; Kd here K, is the equilibrium constant of the ionization of V& with liberation of electrons into the conduction band. Assuming that Ka and Kc are given by expressions such as (30), taking for Ea and EC the values estimated from (29) (and thus inor __~~~__~ denendent of temoerature). ,, and using Y\(16)I for KS.-_
F.
908
A.
KROGER
we find
Ka&IKt = 4 exp(
-P/k)
exp#‘-.&-
E&VIkT)
= 1.23 x 101“a~g exp(-2*26eV/kT)
M 6.3 x lo-11 exp(3.13 eV/kT) and K S ~1.3 x 10-S exp(-1.62
eV/kT)(mole
fr.)s (32)
The enthalpy contains uncertainties from two sources: (1) the assumption of an equal enthalpy of formation of V& and If&,, and (2) the estimate of j!?, and EC from (29). The first may introduce an error of t-O.1 eV; the second may give rise to a larger error, say +0*.50 eV, positive because (29) gives lower limits to Ea and EC. Thus H$ > 1.62 eV. The entropy term is almost certainly too small. It can be increased by allowing for a temperature dependence of Ea and EC, which will be the more likely (and the larger) the larger E, and EC. It is improbable, however, that K$ should become larger (7)]. Since Hi > Hk, our than K;c? [see analysis indicates that Schottky disorder will never become dominant, and bromine vacancies will always be a minority species.(sl) From (32) and (7) we obtain
[VL] = K& =3.3
x 10-s
F
x exp( - 1.0 eV/kT)(atom
fr.)
(33)
For 300°C this expression gives [V&l w 5 x 10-1s. This value is so low that it is very unlikely that the bromine self diffusion as observed@) takes place by the normal vacancy mechanism. Interstitial diffusion or diffusion via silver vacancies seems more likely. Using (30) and (31) we can find expressions for the concentrations of V_& and AgF as a function of T, per, and/or aAg : VZ,]
=
qi! a
= 159~:;
exp( - 0.8 eV/kT) atom fr.
= 1.75 x 10%;~
exp( - 1.99 eV/kT) atom fr. (34)
atom fr. (35)
These (estimated) concentrations are represented in Figs. 2(a), 3 and 4 (dashed lines). Note that Agr and VI, both become more important at higher temperatures, but only V& reaches an appreciable concentration, N 10~5-- lo-4 in brominated crystals at T > 300°C. The concentration of V_& present after cooling should be slightly larger, holes, trapped at V-is centres also giving rise to V& centers. This is under the provision that the cooling takes place sufficiently rapidly and to a sufficiently low temperature to prevent both precipitation of the bromine excess and the association of V& to ( VA~)~. Under the conditions of preparation applying in Hennig’s experiments, i.e. pnr, N” 5 atm and T = 39O”C@s), equation (3) predicts p = 1.75 x 10-s = 3.6 x 1017 cm-s and equation (34), [VAJ = 2.8 x 10-a = 5.9 x 101s cm-s, giving after cooling a total of [V&J = 6.3 x 1018 cm-s. This is considerably larger than the N 1017 cm-s observed by Hennig. This discrepancy may be explained in two ways. Firstly it is possible that our estimate of Ea is too large, which may be the case if wzi < 1. This might reduce the high temperature value of [V&l to below that of p, and Hennig’s observations would refer almost exclusively to the trapped holes. The other possibility is that in Hennig’s experiment it has been impossible to cool sufficiently quickly to prevent association and/ or precipitation. Further experiments are necessary to solve this point.
REFERENCES 1. WAGNER C. and BEYERJ., 2. phys. Chem. B32, 113 (1936). 2. KOCH E. and WAGNER C., 2. phys. Chem. B38, 295 (1937). 3. TELTOW5.. Ann. Phys. (Leipzi.) 5. 63, 71 (1949). 4. KURNICK-S.W., J. &em. Piys:iO; 218 (1952). 5. TANNHAUSERD. S., J. Phys. Chem. Solids 5, 224
(1958). 6. FRIAUFR. J., J. appl. Phys. 335,494 (1962). 7‘. LAWSONA. W., J. appl. Phys. 33s. 466 (1962). 8. SCHMALZRIED H., .%-phys.. Chem.. (Frankfurt) 22, 149 (1959).
THE
THERMODYNAMICS
OF
9. COMPTON W. D. snd MAR. J., I_ P&s. G-hem, Satids 1, 191 (1956). 10. ZIETEN W., 2. Phys. 145,125 (1956). 11. EBERT I. and ‘PELTOW J., &tn. Phys. (Leipzig) 15, 268 (195.5). 12. HANSON R. C., J. phys. C/rent. 16,2376 (1962). 13. LUCKY G. W. and WEST W.. J. &em. Phvs. 24. 879 (1956); LWXY G. XV., I&c. Faraday &e. 28; 113 (1959). 14. SHAMOVSKII L. M., DUNINA A. A. and G~ST~Z~A M. I., Sow. Phys. Dokl. 1 124 (1956); Sow. Phys. JETP 3, 511 (1956). 15. WAGNEB C., 2. phys. Chew. I1;32,447 (1936). 16. IUCHNER B,, J. them. Phys. 28, 1109 (1958). 17. RALEZGHD. O., J. Phys. Ckem. Solids 25,329 (1965’). 63, 1027 (1959). 18. WAGNSR C., 2. ~~~~~~~. 19. IMKTEJm R., Wrss~,scrr Z., P~at~~~aph~e~ ~kata~~~s. Pkotochmie 55, 121 (1961)s 20. YAMANAKA C., ITOH N. and SUITA S., cited in ref. (12). 21. See for instance GEBALLE T. H., Semiconductors, p. 331, N. B. HANNAY, ed., Reinhold, New York, Chapman and Hall, London (1959). (1962). 22. BROTH F. C., J. phys. Ckem 66,2368
IMPERFECTIONS
IN
AgBr
909
23. A~~&~Lx.x G. and BROWx F. C., P&x Ra. Lett. 9, 209 (1962). 24. JAENICKEW., Phys. Stat. Solidi 3, 31 (1963). 25. Both authors used crystals from the same source, Kodak Research Laboratories, Rochester. 26. See KR&XR F. A., The Chemistry of Imfierfect Crystals pp. 289, 526, North Hohand Pubj, Go., Amsterdam (19631: J&n Wilev, New York. 27. MOSER F., .J. a&. &zjs. 33S, 343.(1962). 28. HENN~C XI., Phys. Stat. Solidi 3, 91 (1963). 29. H~~~EN~TRAATEN W., P!&ps Res. Rept. 23, 658 (1958), Ref. (26), p. 25.5. 30. See Ref. (26), p. 446. 31. Note that, for Kurnick’s interpretation of the hightemperature data, ZZ$ * 2.2 eV > Hi. Nevertheless according to him at high temperatures KA > Ki because lu? > Kp. 32. TANNHA~JSER D., 1. Phys. Ckem. Sotids 5,224 (1958).
Note added during correction : A recent estimate for Ea and En bv BUIMISTROVV. M.. Sow. Pkys. Solid State 5, 2387-(li)64) gives EB X I$
Pergamon
Press 1965. Vol. 26, pp. 901-909.
THE THERMODYNAMICS
Printed in Great Britain.
OF IMPERFECTIONS
IN AgBr*
F. A. RRGGER Department
of Electrical
Engineering,
University
of Southern California, Los Angeles
(Received 28 May 1964) Abstract-Analysis of experimental data available in the literature leads to an evaluation of the enthalpies and entropies of formation of native atomic and electronic imperfections. Estimates are made for the free enthalpy of formation of neutral and charged Schottky defects, which show that bromine vacancies are always a minority species. INTRODUCTION
ON THE basis of experimental work on the density and lattice constants of pure AgBr by WAGNER and BEYER,~) and on the conductivity of cadmium doped crystals by KOCH and WAGNER,@) TELTOW@ and KURNICI&Q it is generally accepted that, at least at medium temperatures, AgBr has predominantly Frenkel type disorder of the silver ions, with interstitial silver ions as the main carriers of the ionic current. Plots of the logarithm of the conductivity, of the self diffusion coefficients, the expansion coefficient and the specific heat against l/T are slightly curved below, and more strongly curved above 375”C(s~s~7) : the enthalpy of formation of lattice defects increases, but the free enthalpy decreases. For this two explanations have been offered (1) Schottky disorder becomes predominant above 375”C.(4) (2) The crystal ‘loosens’, partly as a direct result of the increase in temperature (giving an increase in lattice vibrations), partly due to the presence of an increasing number of imperfections (which in turn affect the lattice vibrations).@**) Although the enthalpy of formation of defects, and the activation volume derived for Schottky disorder on this basis are reasonable, there are various reasons against the former explanation. As indicated by COMPTON and MAURER@) and
* Supported by the Technical Advisory Committee of the Joint Services Electronic Program, of the U.S. Army, Navy and the Air Force under Contract No.
AF-AFOSAR-496-65.
901
TANNHAUSER,t5)the appearance of bromine vacancies as dominant imperfections must result in a relative decrease of the conductance of interstitial silver; since these are the main current carriers, this should lead to a dip in the log cr- T curve. On the basis of similar considerations the selfdiffusion of bromine (if dependent on V&) should be expected to show an increased slope at T < 375%(s) Finally the thermoelectric power of pure AgBr which is positive for a Frenkel mechanism with interstitial silver more mobile than silver vacancies, should change to negative for a Schottky mechanism, the mobility of the silver vacancies exceeding that of bromine vacancies.(‘) Neither of these effects has been observed. Therefore, it seems that the loosening model has to be preferred. ZIETEN,(~@ on the basis of measurements of the coefficient of expansion as a function of T, assumed that Schottky disorder becomes important at low temperature. LAWSONt7) has shown, however, that Zieten’s results do not justify such an assumption. Thus, although possibly some Schottky disorder may occur to account for bromine self-diffusion, it is never predominant. This statement finds further support in the estimates made below. So far we have been dealing with essentially stoichiometric crystals. Let us now consider effects due to deviations from stoichiometry. Two such effects have been observed: (1) An excess of bromine, arising when AgBr is equilibrated under relatively high bromine pressures, gives rise to electronic (p-type) conducti+.U-1s)
(2) Polarization studies of AgBr between a reversible and a non-reversible electrode (Ag and Pt, or Pt and Pt, Bra, respectively) show electronic currents due to both electrons and holes,Q%IQ Analysis of these results along lines similar to those followed previously by WAGMZ~,(I*) MATEJEC(Ig) and RALEEN(~~) leads to the thermodynamic parameters of both atomic and electronic imperfections in AgBr which, as far as the latter are concerned, are consistent with values obtained from optical data.
As
ANALYSIS OF THE ELECTRONIC CONDUCTIVITY OF NON STOICHIOMETRIC A@ shown by WA~N~~,~I~~ LUCKEY et aLYG3)
S~A~OV~~II d &.,(I43 ~~~~~,~s~ and bLEIGE@) AgBr in equilibrium with a gas phase containing bromine at a partial pressure&j,, shows an excess conductivity due ta holes proportional to ~2:. Hanson’s data for T ,f 110°C can be represented by Ql. = 910~~~ exp(-O*53~~~~~~)
Q-I cm-I.
Shamovskii’s data show a slightly and can be represented by
(f)
larger slope
(2)
Hanson’s values at T > 110°C show a lower slope (N O-26 ev). A similar slope is found by &LEIGzx@~) from polarization measurements on the cell PtjAgBr/Pt, Bra. According to Hanson’s Hall measurements, the hole mobility at room temperature is approximately 2 cm2 V-I see-f. Raleigh showed that at higher temperatures it may be represented by vh = l-5 x lOI4 x T-s-5. Assuming this relation to hold up to temperatures well outside the measuring range, he found that the concentration of holes, p, can be described from room temperature to 300°C by the relation P = Q&r& =
Or, since there AgBr per ems,
ev/kT)
cm-a.
(3a)
are N = 2.07 x 102s molecules
P = 1*74x %0-s&~ exp(-0,439
1/2Bra(& + B&i-
Z&h.;
Hnr,~hr
(4)
the superscript cross, dash and dot indicating effective charges of respectively 0, -4 and -t-y. Application of the law of mass action gives
which shows that p cc pk$i as long as [Vi,] is independent of $nr,. If the AgBr is approximately pure, i.e. the concentrations of divalent impurities and of deviations from stoichiometry are smaller than the intrinsic native (Frenkel-type) disorder, [Yip]
w [Ag;] M K;;P’,
(0
Kb being the equilibrium constant of the Frenkel According to Raleigh this disorder reaction, applies to his data, to those of WAGNER,~I~I and also to Hanson’s data for T > 110°C. Combination of (3b), (S), and (6) using for Kh Teltow’s data(a) as interpreted by SCHMALZRIED@) : 3C.i = 3.13 x 105 exp( - 1.22 eV/KT)(mole
a~ = 50OO~~exp(-O.74e~~~T).
3.6 x lOss&~exp(-0.439
Formation of holes by incorporation of bromine from the vapor into the crystal can be formulated
fr.)2 (7)
we find &&,I% = 9.8 exp(-
1 *OSe~~~~} (mole fr-)2 (8)
The larger slopes of In bh :.1/T observed by Hanson at low temperature and by SHAMOVSKD et al. over the entire temperature range are no doubt due to the presence of divalent impurities in concentrations that are no longer small relative to &$‘a, which tends to make [V&] = constant = [&PI. The observed slopes are still smaller than the theoretical slope of l-074 eV, probably due to the fact that [I?&] is not yet quite constant in the temperature range concerned. Ilschner’s polarization experiments on the cell AgjAgBrjPt I give the electronic conductivity librium with silver:
of AgBr in equi-
et’j‘FzT) mole fr. (ix& = 4.35 x 10s exp(-1.7 (3b)
#j&T)
Q-I Cm-I. (9)
THE
THERMODYNAMICS
OF IMPERFECTIONS
According to YA~NA~ et aE.,@e)(~~)~~*oK= 70 cms V-1 set-1. From 25 to 35O”C, its temperature dependence is given by
84 exp(O.055 eV/kT) cm2 V-1 set-1.
M
(10)
Hence ($1 = -
(%)I ve4N
= 1% x lo4 exp(- 1.75 ev/Rrr) mole fr.
(II)
The electrons originate from excess silver, incorporated at interstitial sites, Vf, by the reaction A&) + Vt” + Ag;fe’;
Wg;l
W
tAg( W,“l
HAM
(12)
n[Ag;] = K&r.
(13)
lirAg = 8.8 x 10s exp(2.36 eV/kT).
(14)
Combination of (8) and (14) with corresponding expressions for the evaporation of bromine and the formation of AgBr from silver and liquid bromine, using the relevant data from standard tables and remembering that Ag&+B& = AgBr(f), leads to the equilibrium parameters for the formation of electrons and holes by the reaction ET
(15)
with np = & = 3 x 105 exp( - 3.38 eV/kT) (mole fr.)2 (16) This is demonstrated in Table 1. As is easily verified, the parameters of Kh, though used in the calculation. dron out in the final result for Kf. An independent second expression for .& can be obtained from ILSCHNER'S work.(lsj The total current in these experiments was I
I
kT
J = -_(a-~) ql
ae =
(410
- exp( - E/kT))
(o&{exp(E/kT)-
(1W
11.
(17)
Wb)
The hole current was found to become important only at E > 0.45 V. Subtracting the saturation value of o at medium voltages (giving (G&) from the total conductivity at E = 0.45 V, and using (18b), we find (eh)~. Introducing again the temperature dependence of the hole mobility used before, we obtain three values for the hole concentrations at different temperatures which can be represented by (p)r = 263 exp(- 1.27 eV/kT) mole fr.
(19)
Combination with (11) leads to (fi)&)r
= np = Kg = 4.1 x 10s
x exp( - 3.02 eV~k~~( mole fr.)s
Since the experiments were carried out with a crystal of a purity similar to that used by Hanson, and at temperatures > lOO”C, the AgBr behaves as an essentially pure material and (6) holds. Combining (ll), (13) and (7) we find:
0 + e’+h*;
903
f being the thickness of the crystal;
crh =
we = 3*64x 105 T-312
IN AgBr
(20)
an expression not too different from (16). As indicated by RALEIGH~‘)and WAGNER,~~)equations (16) or (20) can be used to determine the band gap and its temperature dependence. According to semiconductor theory,far) G = gcg&&h exp(PP) exp(- J?$VPT)
(21)
in which gc and gv are the number of valleys in the conduction and valence band, respectively.
m* 3/s ‘J”siscm -3 = 4.84 x 1015 ( m1
(22)
with m*, the effective mass, equal to rnz for A, and to rnz for Aa. Eg = Ef-/3T
eV.
(23)
Accepting a band model similar to that proposed for AgCl,(ss) gc = 1, go = 12. Taking m,* = 0.27, as observed,(as) assuming arbitrarily rni = 1, and writing Ts = 1*88x 10s exp(-O-11 eV/kT) we find that between 25 and 350°C equation (16) can be interpreted in terms of (21), (22) and (23) with I$’ = 3.38 eV and j3 = 1.9x lo-seV/deg. The same analysis for (20) gives Ey = 3.02 eV and #I = 2.26 x 10-s eV/deg. This is to be compared
F.
904
A. KROGER
Table 1. Reactions involving structure elements of AgBr and the parameters K = K” exp( -H/W), with K” = exp(As/k)
Reaction
of their reaction constants
li;o
K
(mole fractions)
5)
X3
(cal per deg per mole)
As/k
1/2Brs(g) + Br&+ V&+-h’ Ag(s)SV,x+ Ag;-t-e’
KAg
1.05 2.36
Ag; + Vi, + Ag& -t F’: l/ZBrs(l) + 1/2Brs(g)
Ki-’ K$
-1.22 0.16
AgBr(s) C+ A&)
.KGX
1.03
3-67
1.3
Kt
C38
3 x105
12.64
KBraVh
+ 1/2Brs(l)
0 ;Ir. e’+h’
9.8 8.8 x 10s
+
2.28 16
4.5 31.6
3.2 x 10-s 3x10”
- 12.65 5.71
-25 11.3
with optically determined band gap data. BROWN has determined exciton energies as f( T) ; the energy gap might be expected to be somewhat larger. However, photoconductivity experiments by the same author show that the cut-off of photoconduction occurs at the same values. Therefore it seems that the values given represent the energy gap* Brown’s data and the slopes and intercepts calculated from (16) and (20) are shown in Fig. 1. Linear extrapolation of Brown’s curve (a) gives absolute values of E consistent with equation (16), but the temperature dependence dEjdT = 8 x 10e4 eV/deg is smaller than is indicated by (16). Curved extrapolation of Brown’s Curve (6) fits the (16)
I EQUATION ‘\C---
2+
\
1,
\
EQUATlON
100
+ AgBr+
Vt+h*,
(24)
with the reaction constant Km,m Br,,
Ag
=
--,
=
3.1 x 10-5 exp(+0.17
I
i arownj
0
*Brz(g)+Ag;
KIF
2-o c I.8
line reasonably well, in particular if Brown’s data would turn out to be exciton points after all, the bandgap being x O-1 eV larger; this can be the case if the photoconductivity observed by Brown involves center levels. Equation (20) does not fit the optical data; it is possibly in error due to the fact that part of the extra conductivity observed in Ilschner’s experiments at high voltages may have been due to decomposition. By combining equations and relations found so far, we can find expressions for other imperfection reactions. For instance combination of (4), (5) and the Frenkel reaction gives:
K
(16)
\
‘\
2.2
25 ~--
----_-
341
2.6
200
300
e~/~~)(mole
fr.) (25)
i.e. Hnrz~g = -0.17 this same quantity + and -0+4eV.
eV; (his
MATEJECN found for W,/2) values between
\ (2C$,
400
500
600
s; OK FIG. 1. Values of exciton energies according to BROWN, and values of the temperature dependence of the band gap according to equations (16) and (20).
IMPERFECTiON CONCENTFUTIONS Using the parameters of the thermodynamic functions as given in Tables 1 and 2, the concentrations of imperfections in pure AgBr can be calculated as a function of temperature and the bromine or silver activity. From this, using the electron and hole mobifities as given and the mobilities
of Vi, and A& as determined by Teltow: v(Ag;) =f 0.058 exp(-0.145
eY/M’)
u(Y&) = 0.78 exp( -0.35 eV/kT)
(24) (27)
we can obtain the corresponding conductivities a$ and the transport numbers Q = crjfc, Figure 2 shows isobars for .p~r, = 0.23 atm. Note that although the lines of h‘ and Agi and Vis approach, they meet at such a low temperature and imperfection concentration that it is very unlikely that AgBr is ever pure enough to give rise to a range in which the neutrality condition is governed by [V&l = p. The temperature dependence of 4 as calculated is in reasonable agreement with the values observed by ~~AGN~R.(~~) Figure 3 shows isobars for AgBr in equilibrium with silver. Over the entire range there is a slight excess of electrons over holes. The electron conductivities range from 5 x lo-s1 0-x cm-1 at 25°C to 45 x 10-T a-1 cm-l at 300°C. Evidently tei = te+& is always < 1. Figure 4 shows isotherms for 25°C as a function of the silver activity (ads) and the bromine activity (&<), the two being related through ~~~~~ = ILQB~ = exp(-AGs~~~),
25 T-C 3.5
3
200
100
I 2.5 WffPK~
300 1 2
I.5
(of
(28)
AGO = 23.3 kcal/mde M 1 eV being the standard free enthalpy of the reaction AgBr(s) + Ag(s) + @h(g). The isotherms have only a meaning between the h&s a& = 1 (for AgBr in equilibrium with pure silver) and pur, = O-23 atm, the saturation pressure of liquid bromine at room temperature, corresponding to a& = 1*4x lo-17. We see again that in AgBr in contact with silver the concentration of electrons is somewhat larger than that of the holes. As is seen in Fig, 4(b), for Q& < 2 x 10-1s the electronic conduction, carried mainly by holes, exceeds the ionic conduction and thus leads to t,1 w tn M 1. Experimental points for tl determined by JAENICRE(~~) lie slightly above the theoretical line. This is no doubt due to the presence of divalent impurities in a concentration > X$‘s at 300°K, just as was the case for l%nson’s crystal.@@ ~~~
~~P~OF SCHOTI’KY IXSOl3JX3R
We have, so far, only been concerned with 6
25 3.5
TOC 1 3
i
I 200
100 I 2.9 103/ T(‘KI
300 I 2
1.5
(b1
FIG.2. Concentrations of dominant imperfections (a) and the corresponding conductivities and transport numbers (b) for pure AgBr in equilibrium with bromine vapor (PBrs = O-23 atm) Circles indicate experimental values of th determined by WAGNER.@~)Dashed lines: estimated values.
excess silver or bromine giving rise to ionized imperfections (Agi, Y&) and free electrons or holes. This does not imply that neutral centers do not also exist. In fact, such centers will be present in concentrations determined by reactions such as A&)+
Vf + Ag? or Ag; -I-&I$ Agf,
and corresponding ones for VA@ involving either single centers V& or pairs (VA&$ as found in the alkali Heidi. Additional, preferably optical,
16’” P i$
lo-7 tatm)
(0)
“K FIG. 3. Concentrations of imperfections in AgBr in equilibrium with silver. Dashed lines: estimated values.
experiments are necessary to reveal the presence of such centers and their concentrations. Such experiments have been carried out for AgCi with an excess of halogen, Hiere a square root dependence of the absorption strength on pcls indicated the presence of I’& centers.@?) For AgBr the presence of @a centers is shown by electron spin resonance.(ssf The position of the acceptor Ievel of V”Aprelative to the vafence band (or at Ieast a lower limit) may be estimated from:fss)
/
I
ldf7
lo’2
Pz2
lO-7 iotm) (b)
Taking the dielectric constant E = 13, the square of the refractive index & = 5, and the effective
Fro. 4. Isotherms for the concentraticms of imperfections in AgBr at WC, (a) and the corresponding conductivities and transport numbers ter = 4-k tat (b) Squares and circles indicate respectively experimental points of th = Itp and ts as determined by JAF.NX~EE.@~~
THE
THERMODYNAMICS
OF
IMPERFECTIONS
IN AgBr
907
Table 2
Reaction
Constant
AS (cal per deg. per mole)
KO (2)
4.5 - 12.6 5.6
1/2Brs(g) + Br&+V&+h* Br(g) + 1/2Brs(g) V&fh. + V&
KBQV~ -1/n KD K?
1.05 -1 -0.25
Br(g) + Br&+ V:s Ag(g) =+ A& + V$
KB~, v
KAg, v
-0.2 -o-2*
0.28 0*28*
-2.5 -2*5*
Br, v. v K-’ &Br KVBPBT~ l/n KD
-0.4 1.03 0.16 1.0 2.96
7.8 x lO+ 3.67 3 x 10s 5.75 x 10s 4.26 x lo2
-5.0 2.6 11.3 12.6 12.0
or
5.15 4.75
2.7 x 108 2.1 x 10’
38.5 33.5
+ Br(g) + AgBr(s) -I-V& + V& AgBr(s) + Ag(s) + 1/2Brs(l) 1/2Bra(l) $ 1/2Brz(g) 1/2Bra(g) + Br(g) i Ag(s) = Ag(g)
(a) : A&)
(b) AgBr(s) + Ag(g) + Br(g) (s)+(b): 0 f V&+V&
i%.
K -DAg KAL,
KS
9.8 1.74 x 10-s 16.4
* Estimated to be equal to the corresponding figures for Br.
mass (of holes) m* = rnz w m, we find Ea w 0.2 eV. For &, the distance of the donor level of Agr to the conduction band we find in a similar way & w 0.05 eV. The equilibrium constant of the ionization reaction Vj& + V&fh’(Ka) is now: K
=
a
w&J E=gV2
fi
exp( - Ea eV/kT)
= 6.1 x 10-2 exp(-0.25
eV/kT) mole fr. (30)
Similarly 4-%1 Kb = -
[&,x1
AC exp( 2
= gc -
& eV/kT)
= 7.1 x IO-4 exp( -0.1 eV/kT) mole fr.
for
the
formation
of
neutral
4.75 HSIHAg,Br=
(31)
Schottky
-
5.15
=
0.92
i.e. a value between O-5 and 1 as found in many crystals, but close to 1, which indicates that AgBr is largely ionic.(sO) We can derive from KS the parameters of the equilibrium constant K$ of the normal Schottky reaction : 0 +
Combining the reaction underlying (30) with other reactions, as carried out in Table 2, we find the enthalpy and entropy of formation of V& by the reaction of AgBr with atomic bromine. Assuming, tentatively, that the parameters for formation of V& by the reaction of AgBr with atomic silver are the same-which cannot be far wrong-we can find the parameters of KS, the equilibrium constant
defects (Table 2, bottom line). The enthalpy, Hs, may be compared with the enthalpy of evaporation of AgBr to single atoms (H&,Br)We find
Vig+
VI&,
with the aid of the relation K; = KS=; Kd here K, is the equilibrium constant of the ionization of V& with liberation of electrons into the conduction band. Assuming that Ka and Kc are given by expressions such as (30), taking for Ea and EC the values estimated from (29) (and thus inor __~~~__~ denendent of temoerature). ,, and using Y\(16)I for KS.-_
F.
908
A.
KROGER
we find
Ka&IKt = 4 exp(
-P/k)
exp#‘-.&-
E&VIkT)
= 1.23 x 101“a~g exp(-2*26eV/kT)
M 6.3 x lo-11 exp(3.13 eV/kT) and K S ~1.3 x 10-S exp(-1.62
eV/kT)(mole
fr.)s (32)
The enthalpy contains uncertainties from two sources: (1) the assumption of an equal enthalpy of formation of V& and If&,, and (2) the estimate of j!?, and EC from (29). The first may introduce an error of t-O.1 eV; the second may give rise to a larger error, say +0*.50 eV, positive because (29) gives lower limits to Ea and EC. Thus H$ > 1.62 eV. The entropy term is almost certainly too small. It can be increased by allowing for a temperature dependence of Ea and EC, which will be the more likely (and the larger) the larger E, and EC. It is improbable, however, that K$ should become larger (7)]. Since Hi > Hk, our than K;c? [see analysis indicates that Schottky disorder will never become dominant, and bromine vacancies will always be a minority species.(sl) From (32) and (7) we obtain
[VL] = K& =3.3
x 10-s
F
x exp( - 1.0 eV/kT)(atom
fr.)
(33)
For 300°C this expression gives [V&l w 5 x 10-1s. This value is so low that it is very unlikely that the bromine self diffusion as observed@) takes place by the normal vacancy mechanism. Interstitial diffusion or diffusion via silver vacancies seems more likely. Using (30) and (31) we can find expressions for the concentrations of V_& and AgF as a function of T, per, and/or aAg : VZ,]
=
qi! a
= 159~:;
exp( - 0.8 eV/kT) atom fr.
= 1.75 x 10%;~
exp( - 1.99 eV/kT) atom fr. (34)
atom fr. (35)
These (estimated) concentrations are represented in Figs. 2(a), 3 and 4 (dashed lines). Note that Agr and VI, both become more important at higher temperatures, but only V& reaches an appreciable concentration, N 10~5-- lo-4 in brominated crystals at T > 300°C. The concentration of V_& present after cooling should be slightly larger, holes, trapped at V-is centres also giving rise to V& centers. This is under the provision that the cooling takes place sufficiently rapidly and to a sufficiently low temperature to prevent both precipitation of the bromine excess and the association of V& to ( VA~)~. Under the conditions of preparation applying in Hennig’s experiments, i.e. pnr, N” 5 atm and T = 39O”C@s), equation (3) predicts p = 1.75 x 10-s = 3.6 x 1017 cm-s and equation (34), [VAJ = 2.8 x 10-a = 5.9 x 101s cm-s, giving after cooling a total of [V&J = 6.3 x 1018 cm-s. This is considerably larger than the N 1017 cm-s observed by Hennig. This discrepancy may be explained in two ways. Firstly it is possible that our estimate of Ea is too large, which may be the case if wzi < 1. This might reduce the high temperature value of [V&l to below that of p, and Hennig’s observations would refer almost exclusively to the trapped holes. The other possibility is that in Hennig’s experiment it has been impossible to cool sufficiently quickly to prevent association and/ or precipitation. Further experiments are necessary to solve this point.
REFERENCES 1. WAGNER C. and BEYERJ., 2. phys. Chem. B32, 113 (1936). 2. KOCH E. and WAGNER C., 2. phys. Chem. B38, 295 (1937). 3. TELTOW5.. Ann. Phys. (Leipzi.) 5. 63, 71 (1949). 4. KURNICK-S.W., J. &em. Piys:iO; 218 (1952). 5. TANNHAUSERD. S., J. Phys. Chem. Solids 5, 224
(1958). 6. FRIAUFR. J., J. appl. Phys. 335,494 (1962). 7‘. LAWSONA. W., J. appl. Phys. 33s. 466 (1962). 8. SCHMALZRIED H., .%-phys.. Chem.. (Frankfurt) 22, 149 (1959).
THE
THERMODYNAMICS
OF
9. COMPTON W. D. snd MAR. J., I_ P&s. G-hem, Satids 1, 191 (1956). 10. ZIETEN W., 2. Phys. 145,125 (1956). 11. EBERT I. and ‘PELTOW J., &tn. Phys. (Leipzig) 15, 268 (195.5). 12. HANSON R. C., J. phys. C/rent. 16,2376 (1962). 13. LUCKY G. W. and WEST W.. J. &em. Phvs. 24. 879 (1956); LWXY G. XV., I&c. Faraday &e. 28; 113 (1959). 14. SHAMOVSKII L. M., DUNINA A. A. and G~ST~Z~A M. I., Sow. Phys. Dokl. 1 124 (1956); Sow. Phys. JETP 3, 511 (1956). 15. WAGNEB C., 2. phys. Chew. I1;32,447 (1936). 16. IUCHNER B,, J. them. Phys. 28, 1109 (1958). 17. RALEZGHD. O., J. Phys. Ckem. Solids 25,329 (1965’). 63, 1027 (1959). 18. WAGNSR C., 2. ~~~~~~~. 19. IMKTEJm R., Wrss~,scrr Z., P~at~~~aph~e~ ~kata~~~s. Pkotochmie 55, 121 (1961)s 20. YAMANAKA C., ITOH N. and SUITA S., cited in ref. (12). 21. See for instance GEBALLE T. H., Semiconductors, p. 331, N. B. HANNAY, ed., Reinhold, New York, Chapman and Hall, London (1959). (1962). 22. BROTH F. C., J. phys. Ckem 66,2368
IMPERFECTIONS
IN
AgBr
909
23. A~~&~Lx.x G. and BROWx F. C., P&x Ra. Lett. 9, 209 (1962). 24. JAENICKEW., Phys. Stat. Solidi 3, 31 (1963). 25. Both authors used crystals from the same source, Kodak Research Laboratories, Rochester. 26. See KR&XR F. A., The Chemistry of Imfierfect Crystals pp. 289, 526, North Hohand Pubj, Go., Amsterdam (19631: J&n Wilev, New York. 27. MOSER F., .J. a&. &zjs. 33S, 343.(1962). 28. HENN~C XI., Phys. Stat. Solidi 3, 91 (1963). 29. H~~~EN~TRAATEN W., P!&ps Res. Rept. 23, 658 (1958), Ref. (26), p. 25.5. 30. See Ref. (26), p. 446. 31. Note that, for Kurnick’s interpretation of the hightemperature data, ZZ$ * 2.2 eV > Hi. Nevertheless according to him at high temperatures KA > Ki because lu? > Kp. 32. TANNHA~JSER D., 1. Phys. Ckem. Sotids 5,224 (1958).
Note added during correction : A recent estimate for Ea and En bv BUIMISTROVV. M.. Sow. Pkys. Solid State 5, 2387-(li)64) gives EB X I$