IEE REVIEW
`
`Metal-semiconductor contacts
`
`Prof. E.H. Rhoderick, M.A., M.Sc., Ph.D., C.Eng., F.lnst. P., F.I.E.E.
`
`Indexing terms: Semiconductor devices and materials, Schottky-barrier devices
`
`Abstract: A review is given of our present knowledge of metal-semiconductor contacts. Topics covered
`include the factors that determine the height of the Schottky barrier, its current/voltage characteristics, and
`its capacitance. A short discussion is also given of practical contacts and their application in semiconductor
`technology, and a comparison is made with p-n junctions.
`
`1
`Historical
`The study of metal-semiconductor contacts goes back to 1874,
`when Braun reported the asymmetrical nature of conduction
`between metal points and crystals such as lead sulphide. Their
`application as radio-frequency detectors is almost as old as
`wireless telegraphy itself, and in 1906 Pickard took out a
`patent for a point-contact rectifier using silicon. In 1907
`Pierce published rectification characteristics of diodes made by
`sputtering metals onto a variety of semiconductors, and the
`first copper-oxide plate rectifiers appeared in the early 1920s.
`The point-contact
`rectifier or
`'cat's-whisker' was used
`extensively in the early days of radio, but the first real scien-
`tific study of the device (and indeed the beginning of semi-
`conductor physics) was stimulated by the wartime use of
`silicon and germanium point-contact rectifiers as microwave
`detectors. Point-contact rectifiers were very variable and
`unreliable in their characteristics, and our present understand-
`ing of contact behaviour has come with the realisation that
`metal films evaporated onto single-crystal semiconductor
`surfaces under conditions of high cleanliness can show almost
`ideal rectification characteristics. The intensive study of
`contacts in the 1960s and 1970s was largely stimulated by
`their importance in semiconductor technology, both as rectify-
`ing elements and as low resistance or 'ohmic' contacts. This
`activity continues unabated, with emphasis on the metallur-
`gical and reliability aspects and also on the underlying physics.
`An extensive account of the early history of metal-semi-
`conductor contacts can be found in Henisch's book [ 1 ], and
`a more up-to-date review has recently been given by this
`author [2]. There are also some short reviews that may be
`helpful to the reader [3—5].
`
`2
`Formation of barrier
`2.1 Schottky-Mott theory
`The rectifying properties of a metal-semiconductor contact
`arise from the presence of an electrostatic barrier between the
`metal and the semiconductor. This barrier is due to the
`difference in work functions of the two materials. If the work
`function of the metal 0m exceeds that of the semiconductor
`0S, electrons pass from the semiconductor into the metal to
`equalise the Fermi levels, leaving behind a depletion region in
`the semiconductor in which the bands are bent upwards (Fig.
`\a for the case of an «-type semiconductor). Assuming that
`the region of the semiconductor where the bands are bent
`upwards is completely devoid of conduction electrons (the
`so-called 'depletion approximation'), the space charge is due
`entirely to the uncompensated donor ions. If these are
`uniformly distributed, there will be a uniform space charge in
`the depletion region, and the electric field strength will
`
`Paper 17601, received in final form 1st December 1981. Commissioned
`IEE Review
`Prof. Rhoderick is Professor of Solid-State Electronics, Department of
`Electrical Engineering & Electronics, University of Manchester Institute
`of Science & Technology, PO Box 88, Manchester M60 1QD, England
`
`increase linearly with distance from the edge of the depletion
`region as the metal is approached, in accordance with Gauss's
`theorem. The magnitude of the electrostatic potential will
`increase quadratically, and the resulting potential barrier will
`be parabolic in shape. This is known as a Schottky barrier.
`It can be shown by a straightforward argument (e.g.
`Reference 2) that the amount by which the bands are bent
`upwards (the so-called diffusion potential Vdo) *s given by
`
`0)
`Vdo = 4>m-<t>s
`All energies are measured in electron-volts, and so the energy
`of an electron due to its electrostatic potential is equal to the
`magnitude of the potential expressed in volts. If 0m > 0 S ,
`Vdo is positive and the bands are bent upwards; for the case of
`an «-type semiconductor this produces a barrier which the
`electrons have to surmount in order to pass from the semi-
`conductor into the metal, as in Fig. la, which, we shall see,
`leads to rectifying properties. On the other hand, for a/?-type
`semiconductor
`(Fig.
`\b),
`the band-bending causes no
`impediment to the motion of holes, and no rectification takes
`place, giving an 'ohmic' contact.
`If 0m < 0S, the bands are bent downwards. This gives an
`ohmic contact for an w-type semiconductor (Fig. lc) and,
`since holes have difficulty in passing underneath a barrier, a
`rectifying contact for a p-type semiconductor (Fig. Id). In
`nearly all cases <pm > 0S for n-type semiconductors, but 0 m
`< 0S for p-type semiconductors, and so most metal-semi-
`conductor combinations form rectifying contacts. Unless the
`contrary is clearly stated, all subsequent discussions will centre
`round the case of n -type semiconductors with 0m > 05, which
`is the most important case in practice.
`What is usually quoted is not the diffusion potential but the
`barrier height 0 b as viewed from the metal (Fig. 1). For an
`«-type semiconductor, this is given by
`
`<Pbn =
`
`Vdo+(EC-EF)
`
`(2)
`
`where xs {= 0S — (Ec - EF)} is the electron affinity of the
`semiconductor, i.e. the difference
`in energy between the
`vacuum
`level and the bottom of the conduction band.
`Although usually attributed to Schottky [6]>, eqn. 2 was first
`stated implicitly by Mott [7], and will be referred to as the
`Schottky-Mott approximation. In obtaining it, the assumption
`is made that the surface dipole contributions to 0 m and x« do
`not change when the metal and semiconductor come into
`contact.
`In most practical metal-semiconductor contacts the ideal
`situation shown in Fig. la is never reached, because there is
`usually a thin oxide layer, about 1—2nm thick, on the surface
`of the semiconductor. Such an oxide film is referred to as an
`interfacial layer. A practical contact is, therefore, more like
`Fig. le; however, the additional barrier presented by the oxide
`layer is usually so thin that electrons can penetrate it quite
`easily by quantum-mechanical tunnelling, and so there is little
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`IEEPROC, Vol. 129, Pt. I, No. 1, FEBRUARY 1982
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`0143-7100/82/010001 +14 $01.50/0
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`1
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`difference between Fig. la and e as far as the electrical
`characteristics are concerned.
`
`theory
`2.2 Modifications to Schottky-Mott
`The Schottky-Mott approximation (eqn. 2) assumes that the
`surface dipole contributions to <pm and Xs do not change when
`the metal and the semiconductor are brought into contact.
`These surface dipole layers arise because at the surface of a
`solid the atoms have neighbours on one side only. This causes
`a distortion of the electron cloud belonging to the surface
`atoms, so that the centres of the positive and negative charge
`distributions do not coincide. It was soon discovered that the
`linear dependence of 0bn on 0 m predicted by eqn. 2 does not
`occur in practice, and so the assumption of constancy of the
`surface dipole layers cannot be true.
`One of the first explanations for the departures of the
`experimental data from eqn. 2 was given by Bardeen [8], who
`pointed out the importance of localised surface states. For our
`present purpose it is sufficiently accurate to regard surface
`states as unsatisfied bonds on the surface of the semiconductor.
`At the surface, the atoms have neighbours on one side only,
`and on the vacuum side the valence electrons have no partners
`with which to form covalent bonds. Each surface atom, there-
`fore, has associated with it an unpaired electron in a localised
`
`e e Q e e e e
`
`Fig. 1 Schottky barriers for semiconductors of different types and
`work functions
`b <t>m > 4>s, p - t y pe
`a <t>m> <J>S, M-type
`c <pm< 0S, n-type
`d <t>m < <t>s, p-type
`e <t>m > 0S, rc-type, with interfacial layer
`— acceptor ion
`e electron in conduction band
`+ donor ion
`® hole in valence band
`
`orbital, directed away from the surface. Such an orbital is
`often spoken of as a dangling bond. It can either give up its
`electron, acting as a donor, or accept another, acting as an
`acceptor. The surface states are usually continuously dis-
`tributed in energy within the forbidden gap, and are character-
` such that, if the surface states are
`ised by a 'neutral level' 0O
`occupied up to 0O and empty above 0O, the surface is
`electrically neutral. In general, the Fermi level does not
`coincide with the neutral level, and in this case there will be a
`net charge in the surface states. If, in addition, there is a thin
`oxide layer between the metal and the semiconductor, as will
`happen if the surface of the latter has been prepared by
`chemical polishing, the charge in the surface states together
`with its image charge on the surface of the metal will constitute
`a dipole layer. This dipole layer will alter the potential
`difference between the semiconductor and the metal and will
`upset eqn. 2. It was shown by Cowley and Sze [9] that,
`according to the Bardeen model, the barrier height is given
`approximately by
`
`(3)
`
`where
`
`y =
`
`qbDs
`Eg is the bandgap of the semiconductor, 6 the thickness of the
`oxide layer, and e,- its total permittivity. The surface states are
`assumed to be uniformly distributed in energy within the
`bandgap, with a density Ds per electron-volt per unit area. The
`position of the neutral level 0O is measured from the top of
`the valence band.
`If there are no surface states, Ds = 0 and 7 = 1, and so
`eqn. 3 gives 0bn = 0 m —%s, which is the Schottky-Mott
`approximation. If the density of states is very high, 7 becomes
`very small and 0bn approaches the value Eg—<t>0. This is
`because a very small deviation of the Fermi level from the
`neutral level can produce a large dipole moment, which
`stabilises the barrier height by a sort of negative feedback
`effect. The Fermi level is said to be 'pinned' relative to the
`bandedges by the surface states.
`A similar analysis for the case of a p-type semiconductor
`shows that 0 bp is approximately given by
`
`<t>bP =
`
`Xs) + (1 - 7 ) 0o
`
`(4)
`
`Hence if 0bn and <j>bp refer to the same metal on n- and p-type
`specimens of the same semiconductor, we should have
`
`(5)
`<t>bn
`if the semiconductor surface is prepared in the same way in
`both cases, so that 5, e,-, Ds, and 0O are the same. This
`relationship holds quite well in practice [10]. It is usually true
`that 0 bn >Eg/2, and so 0bn > 0 b p.
`Some experimental measurements of the barrier height
`obtained by depositing films of various metals by evaporation
`onto chemically etched surfaces of n -type silicon are shown in
`Fig. 2. These are plotted against the work function of the
`metal (see Reference 2, Table 2.1). There is considerable
`scatter in the values obtained by different authors with the
`same metal; this is largely because, as we have seen, the barrier
`height depends on the thickness and composition of the thin
`insulating layer, and this is bound to depend on the method of
`preparing
`the surface. The wide scatter in the case of
`aluminium is probably because aluminium oxidises very easily,
`and therefore tends to modify the chemical nature of any
`oxide that may be on the surface of the silicon. It cannot
`really be said that the data conform to eqn. 3, although metals
`with high work functions (e.g. gold and platinum) tend to give
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`IEEPROC, Vol. 129, Pt. I, No. 1, FEBRUARY 1982
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`large barrier heights, whereas metals with low work functions
`(e.g. magnesium and titanium) tend to give small barrier
`heights. Data on barrier heights in other semiconductors are
`given in Reference 2.
`
`o|Au
`
`7 Pt
`
`0.9r
`
`0.8
`
`07h
`;
`>
`i
`*
`£0.6(-
`
`°i
`
`-w
`
`Fe'Cui
`
`.Mo
`
`Mo
`
`sMg
`
`oPb
`
`3 6
`
`3 8
`
`4.0
`
`4 2
`
`4.4
`
`L
`3.4
`0.3L
`Fig. 2 Barrier heights of various metals on chemically etched n-type
`silicon
`Symbols denote results of various authors
`
`4.6
`
`4.8
`
`5.0
`
`5.2
`
`5.4 5 6
`
`In tima te con tac ts
`2.3
`The Bardeen model assumes the existence of an insulating
`layer between the metal and semiconductor, as evidenced by
`the occurrence of 5 and e,- in the expression for 7 (eqn. 3).
`This corresponds quite closely to the majority of practical
`contacts, which are fabricated in such a way that there is a
`thin oxide layer on
`the surface of the semiconductor.
`Occasionally, for research purposes, contacts are made by
`cleaving a crystal of the semiconductor in an ultra-high-
`vacuum system and then evaporating a metal film before there
`is time for an oxide layer to form on the freshly created
`surface. Such a contact is known as an 'intimate' contact, and
`is devoid of any interfacial layer. We must now ask whether
`Bardeen's model of the effect of surface states can be applied
`to such an intimate contact.
`In Bardeen's analysis the interface states are regarded as
`point charges, and the term q8 in the expression for 7 is
`simply the dipole moment of a charged interface state together
`with its balancing charge on the surface of the metal. But the
`interface states actually extend into the semiconductor to a
`distance of about 1 nm, and even if there is no interfacial layer
`there is still a dipole between a charged interface state and the
`surface of the metal. This point of view has been developed
`by Heine [13], who regards the interface states as the
`.exponentially decaying tails of the wave functions of the
`conduction electrons in the metal, which can penetrate into
`the bandgap of the semiconductor by tunnelling. The situation
`is much more difficult to analyse theoretically than when
`there is an interfacial layer present, because the density of the
`interface states and the position of the neutral level 0O both
`depend on the metal as well as on the semiconductor [14].
`Our present ideas of barrier formation at intimate metal-
`semiconductor contacts are influenced very largely by four
`recent pieces of experimental evidence.
`(a) It is now well established (e.g. from the work of
`Thanailakis and co-workers [11, 12]) that 0 bn is not a mon-
`otonically increasing function of 0m . Indeed, there are some
`groups of metals for which 0bn actually decreases as 0m
`increases.
`(b) It has recently been discovered that the nature of the
`barrier produced when a particular metal makes contact with
`a semiconductor can be correlated with the chemical properties
`of the metal, i.e. with whether or not a chemical reaction takes
`place at the interface [15].
`(c) It is now recognised that ideally sharp boundaries at
`metal-semiconductor junctions hardly ever occur in practice,
`
`IEEPROC, Vol. 129, Pt. I, No. 1, FEBRUARY 1982
`
`even if the metal is deposited at room temperature on a
`cleaved semiconductor in an ultra-high-vacuum. There may be
`an alloy phase at the interface, or there may be a gradual
`transition from metal to semiconductor over a distance of
`10 A or more due to interdiffusion effects [16].
`(d) It has been found that certain crystal faces of some
`semiconductors (e.g. the cleaved (110) surface of gallium
`arsenide) do not possess any intrinsic surface states in the
`bandgap, yet the Fermi level may be 'pinned' by as little as
`one-tenth of a monolayer of an evaporated metal [17]. (By
`'intrinsic' surface states are meant states which exist at an ideal
`semiconductor surface exposed to vacuum.)
`
`All these findings can be explained in terms of the following
`picture:
`(i) The short-range dipole at an intimate metal-semi-
`conductor interface is not equal to the difference between the
`surface dipoles at the metal-vacuum and semiconductor-
`vacuum surfaces.
`(ii) The interface dipole depends on the precise spatial
`arrangement of the constituent atoms at the interface, and on
`whether or not they form chemical bonds with each other.
`This stage depends very much on the chemical reactivity of the
`metal, and there seems to be an empirical correlation between
`0b and the 'heat of reaction' between the metal and semi-
`conductor.
`(iii) Barrier heights can be significantly affected by inter-
`diffusion effects, even in the absence of heat-treatment.
`(iv) In the case of the (110) surfaces of III—V compounds
`(e.g. GaAs, InP) which show no intrinsic surface states in the
`bandgap, the 'pinning' of the Fermi level by a thin overlayer
`of metal is probably due to the creation of extrinsic surface
`states. These are thought by Spicer et al. [17] to be crystal
`defects created by the heat of condensation of the metal. Such
`defects may be missing group III or group V atoms, or 'anti-
`sites' in which a pair of atoms have exchanged places.
`
`2.4 Summary
`Our present understanding of the mechanism of barrier
`formation is still highly imperfect, especially where intimate
`contacts are concerned, although progress is being made at
`quite a rapid rate. 'Real' contacts, i.e. those in which there is
`an oxide layer between metal and semiconductor, are rather
`better understood than intimate contacts because the oxide
`tends to reduce the importance of interdiffusion, and also
`because, to a certain extent, it 'decouples' the electron states
`in the semiconductor from the influence of the metal, and so
`they can be analysed more simply. Nevertheless, it should be
`emphasised that the Bardeen theory, as extended by Cowley
`and Sze, uses a very idealised model and should not be
`expected to explain the finer points of barrier formation,
`although for many purposes it provides a useful working
`picture.
`Anyone wishing to know what barrier height is likely to
`occur in a particular metal-semiconductor combination made
`by a particular technique has little alternative to searching the
`literature to find out whether that particular combination has
`been reported using the same recipe. A summary up to 1977
`of results for the more common semiconductors is given in
`Reference 2, together with a description of methods of
`measuring barrier heights.
`
`3
`
`Current/voltage relationship
`
`The various ways in which current can be transported through
`a metal-semiconductor contact under forward bias are shown
`in Fig. 3 and include the following:
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`(a) emission of electrons from the semiconductor over the
`top of the barrier into the metal
`(b) quantum mechanical tunnelling through the barrier
`(c) recombination in the space-charge region
`(d) recombination in the neutral region ('hole injection').
`
`The inverse processes occur under reverse bias.
`It is possible to make practical Schottky barrier diodes in
`which process (a) is far and away the most important, and
`such diodes will be referred to as 'nearly ideal'. Processes (b),
`(c), and (d) cause departures from this ideal behaviour.
`
`Fig. 3 Transport processes in forward-biased Schottky barrier on
`n-type semiconductor
`Fermi level according to thermionic-emission theory
`
`3.1 Emission over the barrier
`3.1.1 Theory: Before an electron can be emitted over the
`barrier into the metal, it must first be transported through the
`depletion region of the semiconductor. In the latter process its
`motion is determined by the usual mechanisms of diffusion
`and drift, while the emission process is controlled by the
`number of electrons that impinge on unit area of the metal per
`second. These two processes are essentially in series, and the
`current is determined predominantly by whichever causes the
`larger impediment to the flow of electrons.
`Historically, the first theory of conduction in Schottky
`theory' of Wagner [18] and
`diodes was the 'diffusion
`Schottky and Spenke [19]. According to this theory, the
`current is limited by diffusion and drift in the depletion
`region, and the assumption is made that the conduction
`electrons in the semiconductor immediately adjacent to the
`metal are in thermal equilibrium with those in the metal. In
`contrast, the 'thermionic-emission theory' proposed by Bethe
`[20] assumes that the current is limited by the emission
`process, and that the quasi-Fermi level for electrons remains
`horizontal
`throughout the depletion region, as in a p-n
`junction (see Fig. 3). One can avoid the necessity of postulat-
`ing a discontinuity in the quasi-Fermi level at the metal-semi-
`conductor interface by regarding the electrons emitted into
`the metal as 'hot' electrons with their own quasi-Fermi level
`[21].
`relationship
`the J/V
`semiconductor,
`For an «-type
`predicted by the diffusion theory can be shown [22] to be
`=
`qNefin^maxexp(-q0bn/kT){exp(iiV/kT)-l}
`
`(6)
`
`where J is the current density per unit area, Nc the effective
`density of states in the conduction band of the semiconductor,
`Hn the electron mobility, %?max the maximum electric field,
`and 4>bn the barrier height. This is not quite of the form of the
`
`rectifier equation J — Jo {exp(qV/kT) — 1} because
`ideal
`^max is voltage dependent.
`The I/V characteristic for the case of the thermionic-
`emission theory can easily be derived if one realises that, under
`the application of a forward bias V, a flat quasi-Fermi level
`implies that the electron concentration in the semiconductor
`just inside the interface is given by
`
`n = Ncexp{-q(<j>bn-V)/kT}
`
`(see Fig. 3). The flux of these electrons across the interface
`into the metal can be shown by elementary kinetic theory to
`be niT/4, where ins the average thermal velocity of electrons in
`the semiconductor. The flux in the reverse direction, which is
`independent of V, must exactly balance the flux from semi-
`conductor to metal when no bias is applied, and so the net
`current density is given by
`
`qNcv
`
`/ =
`
`exp (-q<j)bn/kT){exp
`
`(qV/kT) - 1}
`
`(7)
`
`= A*T2
`exp(-q<}>bJkT){exp(qV/kT)-l}
`(8)
`where A* = 4irm* qk2 {h3
`constant
`is
`the Richardson
`corresponding to the effective mass in the semiconductor.
`Crowell [23] has shown that, for a semiconductor with
`ellipsoidal constant energy surface, the appropriate effective
`mass for a single valley is (l2mymz +m2mzmx
`+n2mxmy)in,
`where I, m, n are the direction cosines of the normal to the
`interface relative to the axes of the ellipsoid, and mx,my and
`mz are the components of the effective mass tensor. For the
`case of silicon with the junction parallel to a {111} plane
`m* = 6{(m2
`t +2mtml)/3}1/2
`where mt and ml are the transverse and longitudinal effective
`masses. This can be easily understood because Nc is pro-
`portional to the density-of-states effective mass {m]mi)113
`raised to the three-halves power, and v is inversely pro-
`portional to the square root of the conductivity effective mass
`\(2/mt + l/m i)/3}~ 1. The factor 6 comes from the number of
`valleys.
`It is often assumed that the condition for the validity of the
`thermionic emission theory is that the mean free path of the
`electrons should exceed the width of the depletion region.
`This is an unnecessarily stringent requirement, since the theory
`merely requires that the electron density at the top of the
`barrier is in equilibrium with the bulk of the semiconductor,
`and this can be the case even if the electrons make many
`collisions in negotiating the depletion region. Bethe argued
`that the condition for the validity of the thermionic-emission
`theory is merely that the mean free path should exceed the
`distance within which the barrier falls by kT/q from its
`maximum value. This criterion has also been derived by
`Gossick [21] and by Crowell and Sze [24]. Crowell and Sze
`also take into account the effects of optical phonon scattering
`in the region between the top of the barrier and the metalt
`and of the quantum mechanical reflection of electrons which
`have sufficient energy to surmount the barrier. For the case of
`the thermionic-emission theory, their combined effect is to
`replace the Richardson constant A * with
`
`A** =
`
`fpfQA*
`where fp is the probability of an electron reaching the metal
`without being scattered by an optical phonon after having
`passed the top of the barrier, and/ g is the average transmission
`
`tin general the top of the barrier occurs within the semiconductor
`owing to the effect of the image force (see Section 3.1.2)
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`coefficient. fp and fq depend on the maximum electric field in
`the barrier, on the temperature, and on the effective mass.
`They have been calculated by Crowell and Sze for silicon,
`germanium, and gallium arsenide. Generally speaking, the
`is of the order of 0.5. The values of A** given by
`product fpfq
`Crowell and Sze for {111}-oriented silicon and gallium arsenide
`104 Am~2K~2 and 4.4 x 104 Am"2K"
`are 96 x
`respect-
`ively.
`
`3.1.2 Effect of bias-dependence of barrier height: There are
`several reasons why the barrier height 0 b may depend on the
`applied bias. If there is an interfacial layer, the voltage drop in
`this layer reduces the barrier height by an amount pro-
`portional to the maximum field in the barrier, which in turn
`depends on the applied bias. Furthermore, even if there is no
`interfacial layer, the barrier height depends on the bias because
`of the effect of the image force. The image force arises because
`an electron near to the surface of the metal is attracted to it
`by the positive image charge. This force has the effect of
`reducing the barrier height by an amount that depends on the
`electric field in the semiconductor, and hence on the applied
`bias (see e.g. Reference 2, p. 37).
`Let us consider the general case of either «-type or p-type
`semiconductors, and suppose that 0b depends linearly on the
`applied bias V, which is true for small values of V, so that we
`can write 0b = 0 bo + 0F. The coefficient |3 is positive because
`0b always increases with increasing forward bias. We can now
`rewrite eqn. 8 as
`= A**T2
`J = A**T
`
`+PV)/kT}{exp(qV/kT)-
`= Jo exp (- q(3V/kT){exp (qV/kT) - 1}
`
`(9)
`
`where
`
`Jo =
`
`A**T2exp(-q<l)b0lkT)
`Eqn. 9 can be written in the form
`
`J = Jo exp (qV/nkT){l
`
`- e xp
`
`(~qV/kT)}
`
`(10)
`
`where
`
`where %=EC —EF for n -type material, oris^ — Ev for p -type,
`and es(= ere0) is the total permittivity of the semiconductor.
`Hence n is not constant but depends on V. If V is restricted to
`values less than about 0b/4, n is roughly constant. Taking
`es — 10"10 Fm" 1, appropriate to silicon or gallium arsenide,
`and 0 b - £ ^ 0.5 eV, n has the value 1.02 for Nd ^ 1023 m " 3 .
`The effect of image-force lowering is therefore negligible for
`Schottky barriers with Nd < 1023 m" 3, although, as we shall
`see, it may be more important under reverse bias.
`
`3.1.3 Comparison with experiment: There are good theoretical
`reasons for believing
`that Schottky diodes made
`from
`reasonably high mobility semiconductors should conform to
`the thermionic-emission theory, for moderate forward voltages
`at least [25], and this conclusion has been confirmed by an
`analysis of experimental data on silicon and gallium arsenide
`diodes made by Rhoderick [26]. The only data known to the
`author which appear to conform to the diffusion theory are
`those which relate to copper oxide [1] and some recent results
`on amorphous silicon [27].
`
`3.2 Tunnelling through the barrier
`It is possible for electrons with energies below the top of the
`barrier to penetrate the barrier by quantum mechanical
`interest because it was
`tunnelling. This is of historical
`postulated by Wilson [28] in 1932 that tunnelling is the
`dominant process in Schottky barriers. It was soon realised,
`however, that this process alone gives the wrong direction
`for rectification. It may, nevertheless, modify the ordinary
`thermionic emission process in one of two ways which may be
`understood by reference to Fig. 4. In the case of a degenerate
`
`TF emission
`
`(11)
`
`Fig. 4 Field and thermionic-field emission under forward bias [29]
`
`n is often called the 'ideality factor'. If 30 b/8Fis constant,n
`is also constant. For values of V greater than 3kT/q, eqn. 10
`can be written as
`
`exp(qV/nkT)
`
`J = Jo
`Eqn. 10 is often written in the literature in the form
`
`(10a)
`
`/ = J0{exp(qV/nkT)-l}
`This form is incorrect, because the barrier lowering must affect
`the flow of electrons from metal to semiconductor as well as
`the flow from semiconductor to metal, and so the second term
`on the right of eqn. 10 must contain n. The correct form, eqn.
`10, has the advantage that a plot of In [J/{\ — exp (— q V/kT)}]
`against V should be a straight line, even for values of V less
`than 3kT/q. The intercept of the straight line on the vertical
`axis gives the value of Jo, and a knowledge of A** allows the
`zero-bias barrier height 0bo to be deduced.
`If there is no interfacial layer, the bias dependence arises
`solely from the effect of the image force, and it can be shown
`that
`
`1/4
`
`87r2e3
`
`(12)
`
`JEEPROC, Vol. 129, Pt. I, No. 1, FEBRUARY 1982
`
`semiconductor at low temperature, where the donor density
`is so high, and the potential barrier so thin, that tunnelling can
`easily occur, the current in the forward direction arises from
`electrons with energies close to the Fermi energy in the semi-
`conductor. This is known as field (F) emission. If the
`temperature is raised, electrons are excited to higher energies,
`and the tunnelling probability increases very rapidly because
`the electrons 'see' a thinner and lower barrier. On the other
`hand, the number of electrons having a particular energy
`decreases very rapidly with increasing energy, and there will
`be a maximum contribution to the current from electrons
`which have an energy Em above the bottom of the conduction
`band. This is known as 'thermionic-field' (TF) emission. If the
`temperature is raised still further, a point is eventually reached
`at which virtually all of the electrons have enough energy to go
`over the top of the barrier; the effect of tunnelling is negligible,
`and so we have pure thermionic emission.
`The theory of F and TF emission has been developed by
`Padovani and Stratton [29] and by Crowell and Rideout [30].
`Both these analyses are very mathematical, but the essential
`features are as follows:
`(i) Field emission occurs only in degenerate semiconduc-
`tors, and because of the very small effective mass, it shows up
`at lower concentrations in gallium arsenide than in most other
`semiconductors. The ranges of temperatures and concentrations
`
`GF Exhibit 1023 - 5/14
`
`

`

`over which Au-GaAs Schottky barriers exhibit F and TF
`emission are shown in Fig. 5 [31].
`(ii) Except for very low values of V, the forward current-
`voltage relationship is of the form
`
`J = Js exp (V/Eo)
`where, in the notation of Padovani and Stratton.
`
`(13)
`
`Eo = £00
`
`and
`
`1/2
`
`m*el
`
`emission. For this reason field emission is of considerable
`importance in connection with ohmic contacts to semi-
`conductors, which often consist of Schottky barriers on very
`highly doped material. What matters in this case is the specific
`resistance around zero bias [i.e. (dV/dJ)v=Q\,
`for which the
`current/voltage relationship referred to above is not valid. Yu
`[32] has compared the zero-bias calculations of Padovani [31]
`with experimental results on silicon with7Vd in the range 1024
`rn~3 to 1Q26 m~3, and Vilms and Wandinger [33] have done
`similar work using calculations of their own based on a
`simplified barrier model. Their results for Si Schottky barriers
`are shown in Fig. 7.
`
`= 18.5 x 10~15
`
`mrer
`
`1/2
`
`electron-volts
`
`Here m* (= mrm0) is the effective mass of electrons in the n-
`type semiconductor, es (=ere0) its permittivity, and the donor
`concentration ~Nd is expressed in m" 3. The pre-exponential
`term Js is a complicated function of the temperature, barrier
`height, and semiconductor parameters, and is given graphically
`by Crowell and Rideout as a function of kT/qEoo.
`
`121-
`
`1.3
`
`1.2
`
`1.1
`
`1.0
`
`,-
`
`11.0
`
`0.8
`
`0.6
`
`0.2
`
`KT/qE 00
`
`1.0
`
`2.0
`
`5.0 10.0
`
`-3
`
`N d ( S i ).
`
`1025
`
`102*
`
`1026
`
`102^
`
`23
`10
`
`1025
`
`1
`10,23
`1025
`102
`Fig. 6
`Ideality factor n and position of maximum of energy distri-
`bution Em of emitted electrons as function ofkT/qEou
`
`1023
`
`i
`1O22
`
`102A
`
`1022
`
`1021
`
`1023
`
`300K
`
`77K
`
`300K
`
`1022
`
`77K
`
`o Al
`Q Chromel
`+ Co
`A Mo
`? Ni
`x V
`
`L A
`
`\°
`
`\ 0
`x\
`\T
`
`TF
`
`i
`101
`
`V
`
`1
`
`1020
`
`=
`
`- - - -^
`
`x
`0
`
`+
`
`A
`X
`
`x t
`
`T emission
`
`A 3 2 1 0
`
`-1
`-2
`-3
`-A
`-5
`-6
`-7
`
`101'
`
`i
`10"
`
`i
`101
`
`i
`101
`
`Fig. 7
`Specific contact resistance as function of donor density for
`contacts to n-type silicon [33]
`
`3.3 Recombination in depletion region
`The importance of recombination in the depletion region has
`been convincingly demonstrated in a classic paper by Yu and
`Snow [34]. The current density due to recombination via a
`Shockley-Read centre near the middle of the gap is given
`approximately by
`
`= Jrexp(qV/2kT)
`
`(14)
`where Jr = qntw/T. Here nt is the intrinsic concentration,
`which is proportional to exp (- qEg/2 kT), w is the thickness
`of the depletion region, and T the lifetime within the depletion
`region.
`The relative importance of th