Electronic transport properties of some low mobility solids under high pressure

JOURNALOF NON-CRYSTALLINESOLIDS4 (1970) 97-106 © North-Holland Publishing Co., Amsterdam

ELECTRONIC

TRANSPORT

LOW MOBILITY

PROPERTIES

OF SOME

SOLIDS UNDER HIGH PRESSURE

F. K. DOLEZALEK and W. E. SPEAR University of Dundee, Dundee, Scotland, U.K. The paper discusses results of drift mobility experiments in orthorhombic S, anthracene, trigonal and vitreous Se carried out under hydrostatic pressures p of up to 6 kbar. Some details of the experimental method are given. In the S crystals, where electrons propagate by an intermolecular hopping mechanism, the experiments show a remarkably large pressure dependence for the electron drift mobility (ae oc exp(ap)). It is concluded, on the basis of small polaron theory, that this is caused by the increase in overlap energy between hopping states on neighbouring molecules. In narrow band materials, a less pronounced pressure dependence of/z would be expected, which is borne out by Kepler's results for anthracene. In trigonal Se, (/zh IIc ~ 26 cm 2 sec-1 v - 0 we found that/~h was substantially independent of p up to 5 kbar. The same applies to ae and/zh in vitreous Se specimens and the implications of this result on the conduction mechanism are briefly discussed. It is also suggested that the pronounced pressure dependence of the steady dark- and photoconductivity in this material is primarily due to the effect of pressure on the injecting properties of the contacts.

1. Introduction The study o f the electrical p r o p e r t i e s o f s e m i c o n d u c t o r s as a function o f h y d r o s t a t i c p r e s s u r e 0 o r u n i a x i a l stress2), has c o n t r i b u t e d significantly to the detailed u n d e r s t a n d i n g o f the b a n d structure a n d the electronic t r a n s p o r t p r o p e r t i e s o f these materials. I n view o f this it a p p e a r e d o f s o m e interest to a p p l y high pressure techniques to the wide range o f low m o b i l i t y solids, b o t h crystalline a n d a m o r p h o u s , which are at present receiving an increasing a m o u n t o f attention. W e have recently s t a r t e d a research p r o g r a m m e a i m e d at a systematic investigation o f the electronic t r a n s p o r t p r o p e r t i e s o f low m o b i l i t y solids u n d e r h y d r o s t a t i c pressure. I n p a r t i c u l a r we have tried to gain s o m e unders t a n d i n g o f h o w pressure affects the t r a n s p o r t m e c h a n i s m s p r e d i c t e d by small p o l a r o n theory3-5). In the following we s h o u l d like to discuss s o m e results for o r t h o r h o m b i c S, a n t h r a c e n e a n d t r i g o n a l a n d vitreous Se.

2. Experimental method Fig. 1 shows a b l o c k d i a g r a m o f the a p p a r a t u s used. T h e high pressure system is o f a c o n v e n t i o n a l type, i n c o r p o r a t i n g an intensifier ( × 10) to a t t a i n 97

98

F. K. DOLEZALEKAND W.E. SPEAR

H.

E

V

OC Field ,

Suppty

EZI

]

T

S

INTENSIFIER

H T Supply

Fig. 1. Block diagram of the high pressure apparatus. Hi, H2 hydraulic handpumps; E expansion cylinder; V shut-off valve; MC manganin cell; SB specimen bomb; J cooling jacket; Sp pressurized spark gap.

a maximum hydrostatic pressure of about 7 kbar. This represents a comparatively modest value - experiments on semiconductors and other solids are generally carried out to very much higher pressures. The specimen is mounted in the cell SB, close to a small sapphire window. The temperature could be controlled in the range between 50°C and - 5 0 ° C by circulating water or methyl alcohol through the jacket J. Reliable mounting and contacting of the specimen, which is immersed in a suitable high pressure fluid (a light fraction oil or hexane), presented considerable difficulties in the initial experiments. These were overcome by sandwiching the material, in the form of a thin disc 9 mm in diameter, between a concentric guard-ring arrangement as the back electrode and a transparent Nesa front electrode. In the experiments, except those on trigonal Se, the drift mobility of photogenerated excess carriers was measured directly. Experimental techniques and other applications of drift mobility methods are discussed in a recent review article6). The charge carriers were produced close to the specimen front electrode by a light flash from a pressurised, high voltage spark gap unit Sp. We should like to emphasise that in our opinion direct drift mobility experiments are essential for a meaningful interpretation of transport data in low mobility, high resistivity materials. As compared to the well-known semiconductors, it is far more difficult with these materials to identify separately the changes in mobility and carrier density with pressure from measurements of the conductivity alone.

ELECTRONIC TRANSPORT PROPERTIES

99

3. Experimental results and their discussion We should now like to discuss some of our results and their possible interpretation in the light of small polaron theory. 3.1. H O P P I N G TRANSPORT

There has in recent years been a growing amount of evidence for this transport mechanism, predicted by small polaron theory. In the case of molecular solids, fairly conclusive evidence is available for an intermolecular hopping transport of electrons in orthorhombic S crystals. This is based on drift mobility results in both the solid 7) and liquid s) states, and supported by optical datag). Fig. 2 shows the pressure dependence of the electron hopping mobility at 20 °C. The results, which are reproducible from specimen to specimen, show a remarkably large exponential pressure dependence of Pe. It doubles its value in about 2.15 kbar on the average. This should be compared with mobility changes under pressure in a wide band semiconductor such as Ge 10), where at 5 kbar the electron mobility has decreased by about 5% and the increase in hole mobility amounts to less than 1%. However, in the impurity conduction range, below 6°K, the transport depends critically on the wavefunction overlap between donor states, and large changes in resistivity have been observed under uniaxial stress2). In fig. 3 the temperature dependence of/~¢ for S has been plotted at two values of p: 0.8 kbar and 4.9 kbar. It can be seen that within the limits of

"~

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//"

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~5~ 4

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/

g

1

2

3 4 5 6 PRESSURE (kbar)

Pressure dependence of the electron drift mobility in ortborhombJc sulphur.

100

F.K. DOLEZALEK AND W.E. SPEAR

Orfhorhombic Sulphur S124 d: 355 t.rn [11~dire c tion

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c~

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~x'~4.9 kbor

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Fig. 3. Temperature dependence of the electron drift mobility in orthorhombic sulphur at two pressures. the experimental accuracy the thermal activation energy associated with the mobility remains independent of pressure. The large increase in #e shown in fig. 2 can certainly not be attributed to a change in activation energy. To interpret these results, let us briefly look at the relevant expressions of small polaron theory. The carrier mobility is related to the hopping probability W by ea 2

= kT W,

(1)

where a denotes the average spacing between sites. In the non-adiabatic case, the high-temperature approximation 3) gives

w=h where Eb is the polaron binding energy and J the overlap energy between states on neighbouring sites associated with the conduction process. The non-adiabatic condition applies as long as the overlap is sufficiently weak, so that the chance of the carrier tunneling from one molecule to the next during an excited state of the whole system is small. This requires that J < hcgo, the energy of the predominant vibrational mode. We believe that the electron transport in orthorhombic S is a non-adiabatic process, although an analysis of previous results 7) indicated that J~-ho~ o.

ELECTRONIC TRANSPORT PROPERTIES

l 01

On the basis of eqs. (1) and (2) and the results of fig. 3, we conclude that the large observed pressure dependence is due to an increase in the preexponential product a2J 2. It is useful in the discussion to assume the following simple analytical from to describe the weak overlap: J (p) ~ I exp ( - a (p)/2),

(3)

where 2 is a decay parameter (2 ~ a). With an isotropic compressibility K, it follows from eqs. (l), (2) and (3) that I-q

~ Kp ,

(4)

where Pl and al denote the values of these quantities at 1 bar. For S, K ,-~ 10 -2 (kbar)-l, so that the first term can be neglected and/2 (p) should vary exponentially with p, in agreement with the experimental results. Before leaving the subject of hopping transport, it may be worth noting that in the adiabatic case (J>ho2o), the pressure dependence of W is determined only by its activation energy, which is 5) approximately (½Eb--J). Measurements such as those shown in fig. 3 should in principle make it possible to distinguish between these two forms of hopping transport. 3 .2 . TRANSPORT IN A NARROW BAND

There are quite a number of solids with carrier mobilities p ~ l cm 2 sec-1 V-1 in which the drift mobility decreases with increasing temperature a typical feature of band conduction. Small polaron formation is likely to play a significant part in these narrow bands as long a s E b > J. Expressions for the drift mobility have been given by several authors n) and most of these show a dependence on J. We would, however, expect larger values of J in narrow band conduction than in a non-adiabatic hopping transport. This means that the parameter 2 in eq. (3) will now be larger, so that the fractional change in overlap energy, -

AJ(p) dl

I al Kp , 3 2

(5)

should be smaller for compression to the same final value ofp. Narrow band transport should thus lead to less pronounced changes in p (P)/Pl than those found for hopping conduction. The only published data are for anthracene lZ). In fig. 4 we have re-plotted Kepler's mobility values on a logarithmic scale. The largest mobility changes amount to about 459/0 at the maximum pressure of 3 kbar, appreciably less than for hopping transport in S. Rice and Jortner la) have recently calculated the

F. K. DOLEZALEKAND W.E. SPEAR

102

[inobple~/I

=-FI / C ,I

I

7 I _j,.~erpend to ab pl~ r---t

I

1.5

I

-

2o ,o.

1,0

o,,,

2"0

1,5

/ . / "

Fig. 4.

ELECTRONS 1

I 1.o 2 3 4 PRESSURE (kbar)

Pressure dependence of the electron and hole drift mobilities in anthracene, as determined by R. G. Kepler12).

changes with pressure in the overlap integrals for anthracene. The results predict an increase in carrier mobility, but not as large as is found experimentally. Another solid in this category, of particular interest in the present discussion, is trigonai Se. Mort 14) has determined the hole mobility in good single crystals from measurements of the acoustoelectric interaction. He found a value of about 26 cm 2 sec- 1 V - 1 for the hole transport along the aligned Se-chains (c-direction). We have recently used the same technique to investigate /th(llc) as a function of pressure. The results show that any mobility changes lie within the limits of accuracy of the experimental method and are unlikely to exceed 20% at a pressure of 5 kbar. This would suggest, that even with such a relatively low carrier mobility, one is already approaching the behaviour of wider band semiconductors under pressure. 3.3. VITREOUSSe Where does the amorphous material, vitreous Se, fit into this scheme? The temperature dependence of electron and hole drift mobilities in thin evaporated specimens were investigated by Spear 15,16) and by Hartke 17). Both Pe and Ph show an activated temperature dependence; at room temperature ~ h = 0 . 1 4 c m 2 sec -1 V -1 and/~e=5.2 X 10 -3 cm 2 sec -1 V -1. There has been considerable uncertainty as to the conduction mechanism in this material. Although #h(T) and p c ( T ) possess the typical temperature dependence associated with hopping transport, the magnitude of/~h, in particular, lies rather close to the estimated upper limit (/~<1 cm 2 see -~ V -1) for this transport mechanism.

ELECTRONIC TRANSPORT PROPERTIES

103

In an attempt to clarify the interpretation we have looked at the transport in vitreous Se as a function of pressure. Fig. 5 shows results for the temperature dependence of/~e and Ph at 0.8 kbar and 4.2 kbar. For experimental reasons these results were obtained on thick, self-supported specimens evaporated on a substrate at 50 °C to increase the carrier lifetimes. The hole mobility shows a curvature at higher T, not found in previous work on thinner specimens. This appears to be a feature of the modified preparation technique and has also been observed by Grunwald and Blakney 18). The important point, brought out in fig. 5, is, however, the absence of any measurable pressure dependence in Pe and Ph- In our opinion this fact rules out a hopping transport. The type of conduction mechanism, originally suggested in reference 16, appears still the most likely interpretation; it is also in keeping with more recent ideas on the electronic structure of noncrystalline solids19). It is suggested that the hole transport in vitreous Se takes place through a comparatively narrow band of conducting states with a mobility Pb, which may not be very different from that in the trigonal form. However, the presence of a large density of localised states close to the band edges, arising out of the disordered structure, is the main factor controlling the drift mobility. Carriers are trapped and thermally released during transit and experimentally one determines the sum of the drift time and the total time of localisation. If the latter dominates, it can easily be seen that = ~b rf/Tt-

(6)

Neither the free life time rf, nor the trapping time ~t are likely to be affected to any extent by the pressures used here; the same would apply to /~b, as indicated by the results for trigonal Se. The activated temperature dependence arises through rt. A similar model may well be applicable to the electron transport. Finally, let us briefly consider the implications of our drift mobility results on the pressure dependence of the conductivity in Se. Fig. 6 shows the steady dark current id as a function of pressure in a vitreous Se specimen with thin gold electrodes. The applied field was kept constant and subsidiary experiments showed that for fields of this magnitude id ~: V 2. Results similar to those in fig. 6 have been obtained by Kozyrev20) and by Krishchunas and Daukantaite21) for trigonal Se crystals. On the basis of our drift mobility results, we conclude that the marked rise in electrical conductivity with pressure found in both vitreous and trigonal Se must be associated with an increase in carrier density. If we illuminate the specimen with highly absorbed radiation, additional carriers will be injected from a narrow region near the top electrode. It can be seen from fig. 6 that the pressure dependence of the photocurrent /ph is

104

F . K . D O L E Z A L E K A N D W . E. S P E A R

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>

\

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o

15

,

,:

\ amorph. Se .d.97t~, 0

O.g kbar

•

4.2 kbc~r

I ~1

I

3

Fig. 5.

3.5

4

4.5

,O~'T('K'~

Temperature dependence of the electron and hole drift mobilities in vitreous selenium at two pressures.

v#reous 5e,~-STpm illurnine~ctrode ~ . illtcn~ e~ctrodent~ •

C3 5

20"C

,I

~

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o~ o

,

a:

8

I

.~o,

.i.

05

o

el

I

2

3

4

5 P R E S S U I~E (kbor)

Fig. 6.

Pressure dependence of the steady state dark current, id, and the photocurrent, iph, in vitreous selenium. The applied field is 8 kV/cm.

exactly the same as for ia. This suggests that we are not dealing with a volume controlled process, but that the application of pressure modifies surface conditions in such a way as to enhance the injecting properties of the contact. A similar conclusion was drawn by Rice and Jortner 13) in their discussion of the very pronounced increase of conductivity with pressure found in a

ELECTRONIC TRANSPORT PROPERTIES

105

number of highly resistive organic molecular solids. This effect is associated with a decrease in the thermal activation energy of the conductivity under compression, which has also been observed in trigonal Se zo). In explanation Rice and Jortner suggest that the enhanced injection under pressure is caused by an increase in the electronic polarisation energy [PI of the molecular lattice by the excess charge; IPI is proportional to the inverse fourth power of the intermolecular separation. This should lead to a decrease in the energy difference between the Fermi level of the metal and the conducting states near the crystal surface and could cause an appreciable increase in the injection of excess carriers. A similar explanation may apply in the case of trigonal and vitreous Se. The above results, however, underline the difficulty of obtaining meaningful information on the pressure dependence of transport from conductivity measurements alone. In summary we can say that the study of the drift mobility in low mobility solids over a modest range of applied pressures, provides us with information on the overlap energy between conducting states. As J is of central importance in the theoretical treatment, the experimental approach used here should be of value in identifying predicted transport mechanisms and in providing a more extensive experimental background for meaningful tests of existing theories.

Acknowledgments The authors would like to thank the Xerox Corporation for their support of this project and the British Council for a Research Studentship. We are indebted to Dr. J. Lees for help and advice with the high pressure system and to Dr. J. Mort for a number of trigonal Se crystals.

References 1) W. Paul, in:High Pressure Physics and Chemistry, Vol. 1, Ed. R. S. Bradley (Academic Press, London, 1963) p. 299. 2) See for instance: H. Fritzsche, in: Physics of Solids at High Pressures, Eds. C.T. Tomizuka and R. M. Emrick (Academic Press, New York, 1965) p. 184. 3) T. Holstein, Ann. Phys. (N.Y.) 8 (1959) 343. 4) J. Appel, Solid State Phys. 21 (1968) 193. 5) I. G. Austin and N. F. Mott, Advan. Phys. 18 (1969) 41. 6) W. E. Spear, J. Non-Crystalline Solids 1 (1969) 197. 7) D. J. Gibbons and W. E. Spear, J. Phys. Chem. Solids 27 (1966) 1917. 8) P. K. Ghosh and W. E. Spear, J. Phys. C 1 (1968) 1347. 9) B. E. Cook and W. E. Spear, J. Phys. Chem. Solids 30 (1969) 1125. 10) G. Landwehr, Z. Naturforsch. lla (1956) 257; 14a (1959) 520; see also A. C. Smith, Techn. Report HP 2 (Gordon McKay Laboratory, Harvard University, 1959), unpublished. 11) See for instance: H. Fr6hlich and G. L. Sewell, Proc. Phys. Soc. (London) 74 (1959) 643;

106

12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

F . K . DOLEZALEK AND W . E. SPEAR

L. Friedman, Phys. Rev. 133 (1964) A1668; I. G. Lang and Yu. A. Firsov, Soviet Phys.-Solid State 5 (1964) 2049. R. G. Kepler, in: Organic Crystals Conference, Eds. J. J. Brophy and J. W. Buttery (McMillan, New York, 1962) p. 1. S. A. Rice and J. Jortner, in: Physics of Solids at High Pressure, Eds. C. T. Tomizuka and R. M. Emrick (Academic Press, New York, 1965) p. 63. J. Mort, Phys. Rev. Letters 18 (1967) 540. W. E. Spear, Proc. Phys. Soc. (London) B 70 (1957) 1139. W. E. Spear, Proc. Phys. Soc. (London) 76 (1960) 826. J. L. Hartke, Phys. Rev. 125 (1962) 1177. H. P. Grunwald and R. M. Blakney, Phys. Rev. 165 (1968) I006. See for instance: N. F. Mott, Advan. Phys. 16 (1967) 49; Festk6rperprobleme 9 (1969) 22. P. T. Kozyrev, Soviet Phys.-Solid State 1 (1959) 94. V. Yu. Krischunas and O. K. Daukantaite, Soviet Phys.-Solid State 8 (1966) 471.