The Poole-Frenkel constant

The Poole-Frenkel constant

Thin Solid Films - Elsevier Sequoia S.A., L a u s a n n e - Printed in Switzerland R21 Short Communication The Poole--Frenkel constant ROBERT M. HIL...

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Thin Solid Films - Elsevier Sequoia S.A., L a u s a n n e - Printed in Switzerland


Short Communication The Poole--Frenkel constant ROBERT M. HILL Chelsea College, Pulton Place, London (Gt. Britain)

(Received June 28, 1971)

The Poole-Frenkel e f f e c t 1'2 is commonly invoked to explain the strong field dependence of current on electric field in semi-insulating, amorphous, materials under high field conditions. The basis o f the effect is the lowering of the effective ionisation energy o f a d o n o r by the applied field. F o r an ionisation energy E1 the field lowered value is E 1 - f l F 1/2, where F is the field and fl the Poole-Frenkel constant e3/2(zCeeo)-1/2, with e the relative dielectric constant for low or high frequencies depending on whether or not the medium surrounding the donor can polarise within the emission time. Experimentally there is difficulty in obtaining realistic values for fl although the exponential dependence o f current on F 1/2 has clearly been observed 3. Two alternative explanations have been proposed depending on the presence o f discrete trapping levels 4 or on compensation within the materials 5. A simpler source for the discrepancy is outlined below, which will apply, particularly, for the case of non-crystalline materials. It has been proposed that the observation of the Poole-Frenkel effect is unlikely in crystalline solids 6. It is generally accepted that in amorphous, or disordered, solids there exists in the normally forbidden gap a large density o f localised states 7 which can be considered as either trap or d o n o r like in nature 8. The Poole-Frenkel effect requires the presence o f donors and we assume that traps co-exist with these donors. F r o m the magnitude o f the ionisation energies El typically observed in the Poole-Frenkel effect (0.5-1.0 eV), from the constancy o f the zero field extrapolated value of El at high fields, and from the form of the field dependence it can be inferred that the current arises from states within the forbidden gap, that these states are probably localised in energy, and that they are donor states. There is no evidence that the ionised density is sufficient to give rise to space-charge effects so the density of ionised donors is small, and we may assume that this only represents a small fraction o f the total density o f ionisable donors. If this is the case then ionisation o f a d o n o r by Poole-Frenkel emission can stimulate further ionisation by processes other than Poole-Frenkel, the initial ionised Thin Solid Films, 8 (1971) R 2 1 - R 2 4



donor acting as a capture center for the electron released from the second donor, giving rise to an effective mobile donor and anomalous values for b. The process is shown diagrammatically in Fig. 1. In (a) non-dissociated donors, i.e.neutral centers, are shown at p, q, Y and s, each lying at an energy Ei below the bottom of the extended states in the conduction band. Immediately below the conduction band lie the localised trapping centres, which are also neutral, being empty. At some time after emission of an electron from the donor r the potential within the system will be as shown in (b), where the emitted electron has been trapped within the coulombic well of the ionised donor. Subsequent de-trapping and movement of this carrier beyond the peak in the well would result in observation of the classic Poole-Frenkel effect with the value of j? given above. However during the time the carrier is trapped it is possible for a second ionisation event to occur. For the electron neutralising the donor at q the potential barrier between q and r is decreased by the overlapping of two coulombic wells, centred at q and r. The effective height of the barrier is2 Ei-


where d is the separation of the centres. Thermal emission can occur over this lowered barrier or alternatively, if d is small, direct tunnelling can occur between q and r with emission of a phonon of energy eFd. Both processes are shown in diagram (c) and act in parallel. The probability of occurrence of whichever of the processes which dominate, is greater than that of the initial Poole-Frenkel emission so that this step is not the rate limiting mechanism for d.c. conduction. Once secondary ionisation has taken place, the potential diagram is as shown in (d) where there is now an increased separation between the “free” carrier and the only remaining “mobile” ionised donor. Defining effective velocities for the “free” carrier, v,, and “mobile” donor uh, by v = Al/At where Al is averaged over a number of transitions and At is dominated by the time lag between individual ionisation events we can obtain the ratio of the distances travelled as

where the total separation between the positive and negative charges Calculation of the effective Poole-Frenkel constant then gives

is xe +x,.

As the ionisation of the second electron must always be dependent on the availability of the first donor it is unlikely that vi, > v, so that the range of the effective Poole-Frenkel constant is from /I to f/3 (=/I Schottky), as observed experimentally3. Obviously in the limiting case of equal effective velocities the Thin Solid Films, 8 (1971) R21-R24









4 0









Fig. 1. Energy diagrams of the Poole-Frenkel induced donors, once r becomes ion&d (b) the potential barrier ionisation for the donor at q (c).


Union&d Electron


+ -

Thin Solid Films, 8 (1971) R21-R24






Ionised Trap



emission. between

In (a) p, q, r and s are union&d q and r is lowered allowing an easy



s y s t e m is e q u i v a l e n t t o S c h o t t k y e m i s s i o n w i t h t h e i m a g i n a r y

electrode plane

p a s s i n g t h r o u g h r. REFERENCES 1 2 3 4 5 6 7 8

J. FRENKEL, Phys. Rev., 54 (1938) 657. R.M. HILL, Phil. Mag., 23 (1971) 59. A.K. JONSCHER, Thin Solid Fihns, 1 (1967) 213. J.G. SIMMONS, Phys. Rev., 155 (1967) 657. J.R. YEARGANAND H. L. TAYLOR. J. Appl. Phys., 39 (I968) 5600. A.K. JONSCHER, J. Phy3. C., (Solid State Phys.), 3 (1970) 2159. N. F. MOTT (ed.), Amorphous and Liquid Semiconductors, North Holland. Amsterdam, 1970 A.K. JONSCHER, Vac. Sci. Technol., 8 (1971) 135.

Thin Solid lqlm~', ~ (1971) R21 R24