Solid-State Electronics Vol. 23, pp. 297-303 Pergmon Press Ltd.. 1980 Plinted in Great Britain

ON THE MOBILITY OF POLYCRYSTALLINE SEMICONDUCTORS

J. MARTINEZ and J. PIQUERAS Laboratorio de Semiconductores, Instituto de Fisica de1 Estado Solid0 (C.S.I.C.),

Universidad Autonoma de Madrid, Cantoblanco, Madrid, Spain

(Received IS May 1979; in revised form 9 September 1979)

Abstract-A simnle nhenomenological model based on tunnel and thermionic emission across grain boundary barrier has been developed. -

The .present model has been applied to polycrystalline silicon assuming that the dangling bonds at the grain boundaries behave as electron trans. The calculations have been carried out in two different ways: one, assuming that the interface states density, ki, (cm-* eV-‘), is constant across the energy gap; and second with the boundary states, NT (cm-*), localized around a very narrok energy range at ET.

In the first case no differences in the mobility reduction have been found between n and p type polysilicon, but for the assumption of states localized at an energy Er in the upper ha/f of the gap, the barrier height is larger in n-type than in p-type material and consequently the calculated mobility of n-type polysilicon becomes lower than the p-type mobility.

In general, the mobility increases with the dopant concentration approaching the monocrystalline behaviour for very large dopings, in qualitative agreement with other approaches and with available experiments.

1. INTRODlJCTlON

The technological importance of polycrystalline semi- conductors has grown larger and larger, because of their application in integrated circuits as active and passive layers [ I-31, and for the most part because they represent today a potential solution for low-cost photovoltaic energy conversion.

A large technological effort has been made in order to obtain high-quality polycrystalline films [4-61, but there is a difficulty in comparing the experimental results with theoretical prediction, as there is no theoretical model, at present, which explains satisfactorily all of the aspects of carrier transport in polycrystalline semiconductors. The main problem in this comparison arises from the difficulty and poor repeatability in obtaining polycrystal- line films. The control of grain size and doping concen- tration is not a trivial problem. Furthermore, the different parameters depend on the different substrates used for deposition. Also, in the same deposited film, homo- geneity in doping and grain size is difficult to control.

The first model for an inhomogeneous semiconductor was proposed by Volger[7], who simply treated the material as consisting of two regions differing in resis- tivity and mobility, the boundaries being of a higher resistivity than the grains. The hypothesis is consistent with experimental observations in which the Fermi level at grain boundaries is pinned deep within the forbidden gap because of the interface states arising from the dangling bonds[8]. Furthermore, this model explains well the lower lifetime in polycrystalline semiconductors[9] and the qualitative behaviour of the mobility vs doping concentration[6]. For large dopant concentrations the interface states are filled and the boundary potential decreases towards zero and the polycrystalline semi- conductor approaches the monocrystalline behaviour, as shown by Kamins[6]. However, from the theoretical

point of view, this model is incomplete, in the sense that it does not consider how the transport from grain to grain takes place. For low-angle grain boundary potentials, of the order of a few kT/q’s, the Volger model could be a suitable approach; but for large boundary potentials, the transmission of the carriers across the potential barrier of the grain boundary must be taken into account.

A first approach to current flow across grain boun- daries in terms of thermionic emission and generation- recombination at the boundaries was reported by Mueller[lO]. He also included the effect of non vertical incidence to the barriers, image force and the finite width of the boundary. An excellent review of the earlier results on transport properties of dislocations and grain boundaries can be found in the book of MatarC [ll].

In this paper, we have considered that the transport from grain to grain takes place, either by thermionic emission or well tunneling across the boundary potential for which we have used the WKB approximation.

In order to obtain a simplified scheme for the mobility we disregard the possible finite width of the boundary, non vertical incidence and trapping effects.

This model has been applied to polycrystalline silicon assuming a constant interface states density and local- ized states at the grain boundaries. In the first case no differences in the mobility lowering have been found between n and p-type materials but for the second one, the calculated values result in strongly reduced mobility for n-type material in comparison with monocrystalline mobility, as a consequence of the larger barrier heights in the n-type than in the p-type polysilicon.

2. -CAL MODEL

We assume that the interface states in the grain boun- daries arising from the dangling bonds, behave as elec- tron traps and give rise to a parabolic boundary potential

SSE Vol. 23, No &A 297

298 J. MARTINEZ and J. PIQUERAS

in the form seen in the abrupt approximation temperatures low enough to prevent impurity migration, the doping concentration at the boundaries will be the same as that of the grain. On this kind of silicon samples

-w<x<w, (1)

where E is the dielectric constant, w is the half-width of the depletion layer, VD is the peak barrier height and ND is the donor density for n-type material. This equation is valid whenever the grain size is larger than the depletion layer width in such a way that the space charge regions of two adjacent boundaries do not overlap.

The barrier height can be calculated equating the depletion layer charge to the amount of charge ac- cumulated at the interface states. For a constant density of interface states the equation writes[9]

qNir(Es - Em - qVD) = 2 (]8qeNbV1~])“’ (2)

where Em is the Fermi level in the grains far enough from the space charge regions of the boundaries and &B is the neutral Fermi level at the boundaries. In the localized states model the charge neutrality condition can be expressed as

+12qNDwl= +(ISqcN,V,/)” =

= 4NT i&p+)

1 (3)

( ,1

L-E ’ l+exp +j+

NT and ET are the density and the energy, referred to the valence band, of the interface states respectively. The + or - sign in both eqns (2) and (3), must be used depending on the material to be n or p-type respectively. Consequently the bands bend upwards in n-type and downwards in p-type crystals.

In the first case, the diffusion potential, VD, has been calculated by Card and Yang[9] for a variety of ND and Ni,, and will not be reproduced here.

Although the energy distribution of the interface states is not well known at present, the localized states model seems to be more suited to explaining the transport properties than a continuum of states[lO]. If the local- ized model is preferred some care must be taken in the choice of the neutral Fermi level at the interface, Em, and the energy of the interface states, ET, because the diffusion potential of the boundaries and thus the material properties depend on this choice. One of the difficulties for determining these two constants arise from the fact that a number of the interface states are additionally filled due to the unavoidable impurity segregation in the neighbourhood of the boundaries if the samples are doped during the growth process or by diffusion. Preferential diffusion along grain boundaries and subsequent lateral diffusion from the boundaries has been reported some time ago[ 131. But if the doping takes place at room temperature, as in the case of neutron transmutation doping, and the samples are annealed at

Seager and Castner[9] have made activation energy measurements and they have found that the Fermi level at the boundaries lies very near to the middle of the gap I& = 0.55 eV above the valence band. The best fit for the dependence of VD and ND was obtained for a level localized at ET = 0.63 eV[9]. This energy is in good agreement with that obtained from capacitance transient technique. An electron trap located at 0.66kO.3 eV above the valence hand was found in polycrystalline silicon solar cells, but no hole traps were found in p-type polysilicon[l4].

The eqn (3) has been used to obtain the barrier heights in polysilicon as a function of impurity concentration and for a variety of interface state densities between 10” and 1Or4 cmF2. Screening and degeneration problems were avoided by maintaining the doping concentration below 10” cm-’ in all the calculations. Figures 1 and 2 show the results for n and p-type materials. As it can be seen the barrier height is always lower in p-type than in n-type material and the diffusion potential becomes negligible, for larger doping concentrations in n-type than in p-type polysilicon. For example, at NT = lOI cme2, the barrier height becomes of the order of kT for an acceptor concentration of 10” cme3, while in n-type a donor concentration around ND = 10”’ cmm3 is needed to obtain the same value of the barrier height. This can be explained, as in n-type material the electrons from the grains must fill all the surface states while, in p-type material, the holes from the grains must neutralize the trapped electrons and the number of charged traps becomes low in this way.

When the impurity concentration ND is small or when the interface states density NT is high

0.5 ,,

0.4 -

ND (cme3 1

Fig. 1. Grain boundary barrier heights vs donor concentration for several interface states densities in the localized level model. ET = I& t 0.63 eV and Em = Ev t 055 eV. The interface state

densities are given beside each curve.

On the mobility of polycrystalline semiconductors

where we have defined

299

N*(cm-)) Fii. 2. Grain boundary barrier heights vs acceptor concentration for several interface state densities in the localized level model. ET = Ev t 0.63 eV and &+ = Ev t 0.55 eV. The interface state

densities are given beside each curve.

(8q&,VD)“2/qNT -PO and from the eqn (3) the diffusion potential will be approximately

qVo=Em-EFB. (4)

In this case qVD increases initially proportional to ND, because Em approaches the conduction band as ND increases, up to some point at which ND becomes large enough to till nearly all the interface states. After a maximum is reached, qVD begins to decrease. See Fig. 1 for example, for an interface states density of 10” cm-*.

3. ‘lRANMON ACROSS THE GRAIN

BOUNDARY BARRIERS

The transmission probability across a potential barrier for an electron of mass m* and an energy E associated with momentum perpendicular to the boundary plane. can be written, in the WKB approximation as

D(E) = exp (-$ r [2m*(qV(x)- E)l’*] dx],

(9

where x, and x2 are the classical turning points. Introducing the eqn (1) in (5) and solving the integral,

the transmission coefficient can be expressed as

---&cosh-I d[$+])}; for E< qVm (6)

and

D@‘)=l; for E*qVD

& = 2 (A$“*.

If a potential difference AV occurs at the barrier, the symmetry of the potential is lost. But, if AV+ VD, the change in the potential shape can be represented by a reduction AV/2, of the barrier height seen from one of the sides and an increase of AV/2 of the barrier seen from the other side.

Hence the net current can be calculated as the difference between the current from left to right, J,,, and in the reverse direction, J,+,

where A* is the Richardson constant and f, and f, are the occupancies of the left and right sides respectively. Introducing the eqn (6) in the eqn (8), after some manipulation, the current density can be written as

where the function G is given by

G(VD,&)=exp(-$$)

1 +iT 0

jqvD D(E) exp (-6) dE. (10)

When the integral of the eqn (10) is negligible com- pared to the exponential term, the current is entirely due to thermionic effect and it can be expressed ap- proximately as

J thrrm=2A*Fexp(-w)

exp (-e) sinh (g). (11)

In order to have an idea of the ranges of validity for which the thermionic emission is a good approximation, we have compared’ Itherm given by eqn (11) and the general expression for the current density, eqn (9), for different doping concentrations and interface state den- sities. We have assumed, as mentioned before, EFB = 0.55 eV and a localized energy level at the interface ET = 0.63 eV. The calculated ratio, Jther,,,/J, at room temperature is shown in Fig. 3. As it can be seen for low values of ND and NT the thermionic emission could be a good approximation, but for large ND and NT the contribution due to tunneling becomes more important and has to be considered. This contribution is enhanced as the temperature is decreased. In oxygen doped poly-

J. MARTINEZ a md J. PIQUERAS

0.4 1 I I I

lo’& lo15 UP lo” lo’*

No (cmS3)

Fig. 3. Jlhermionicl-Ll ratio dependence on doping concentration for several interface state densities in the localized level model. ET = Ev + 0.63 eV and EFB = Ev t 0.55 eV. The interface state

densities are parameter.

crystalline silicon, where the existence of a boundary layer could lower considerably the tunnel contribution, it is also necessary to consider this contribution to explain the experimental I-V characteristics [ 151.

It is now in order to briefly discuss the validity of the WKB method in the present case. At room temperature, in the low doping ranges below 10’6cm-3, the maximum of the tunnel contributibution occurs at energies very near to the top of the barrier height, at 0.9982qVD for ND = lOI crnm3 for example[l6], but in this range the tunnel current is low, around 10% of the total current (see Fig. 3), and it is not meaningful as calculated. For larger doping concentrations, also at room temperature, the maximum position lowers to 0.843qVD for ND = lO’*~!rn-~, and the WKB method begins to be a suitable approach. At low temperatures the limit of validity is displaced towards lower doping concentrations. For ND = 10” cme3 at 77°K the maximum occurs at 0.774qVD Therefore when the WKB method is not valid the tunnel contribution is low but the WKB method becomes valid as the tunnel current increases. The lar- gest deviations using this approach are expected for barrier heights of the order of kT for which erroneous transmission factors may occur at the highest contribu- tion to the transport. Moreover, the abrupt ap- proximation assumed in eqn (1) should be invalid.

4. MOBILlTY OF A POLYCRYSFALLINR MATERIAL.

APPLICATION TO SILICON

In the simplest form a polycrystalline material can be considered in a one-dimensional way, as consisting of N grains of a uniform size, and with the boundaries per-

pendicularly oriented with respect to the current direc- tion.

If a voltage, V, is applied to a polycrystalline sample, some part of the potential drops in the grains, VG, and another in the boundaries, Ve, in such a way that

N(VGt V,)= I’. (12)

Assuming that the boundary potential drop is much less than kT/q, the current across a grain boundary, eqn (9), can be approximated by

J, =A*T’exp c-w)gG. (13)

This condition Ve & kflq corresponds to the case of low applied field one in monocrystalline materials.

On the other hand, each individual crystallite can be considered as monocrystalline, and thus the current density, out of the space charge region, can be written as

where n is the carrier concentration in the neutral regions of the grains.

Because the current density must be maintained at any point of the sample, we can equate (13) to (14) and taking into account the relation (12), Vs and VC can be elli- minated, and the current density results

1 V J=qnL-2w &‘N’ (15)

~mono + A* 7-G

where NC is the conduction band density if we put E,,, = V/LN in the above equation this is similar to the monocrystalline case, and an effective mobility for the polycrystal can be defined

1 1 L-2w kNc _=-. peff pmono L + A*TGL’ (16)

This is just the form of the mobility with two different scattering mechanisms. The second term of eqn (16) is associated to the grain boundary effect while the first one is the reciprocal of the monocrystalline mobility except for the geometrical factor (L-2w)lL. But if, 2w < L, this geometrical factor tends to the unity and the expression for peff becomes entirely analogous to that of a monocrystalline semiconductor with two different scattering mechanisms.

For the monocrystalline mobility we have used the phenomelogical formula obtained by Scharfetter and Gummel[l7] for zero electric field, fitted to the known experimental results in monocrystalline silicon[ll].

where the numerical values for the different parameters

On the mobility of polycrystalline semiconductors 301

I I I 103, I I I I

p (mod

No (cm-3) NA (cmm3)

Fig. 4. Mobility dependence on doping concentration for several interface state densities in the continuum of states model. EFB = Ev +O.UeV. Grain size: L= lOOfiLm. The interface states den-

sities are parameter.

Fig. 6. Mobility as a function of acceptor concentration for several interface state densities in the localized states model. ET =Ev t0.63eV and II& =Ev t0.55eV. Grain size; L=

IOO~m. The interface state densities are parameter.

The mobilities for p-type material were calculated under the same conditions as for n-type. In the case of a continuum of states no differences in the reduction of mobility were found with those of n-type but for the localized states model, see Fig. 6, the calculated mobil- ities were larger for p-type than for n-type. In some cases this difference was as large as orders of magnitude, see for example the curves for NT = 10’3cm-2 in the Figs. 5 and 6 in the range of doping concentrations between lOI and 10” cmm3.

As a general result, we note that for large interface state densities the mobility decreases initially, as the doping concentration is increased, reaching an abrupt minimum. This is because of the initial increase of qV, as mentioned above. A similar result was experimentally found by Seto[l9], the mobility seems to colllapse for a particular doping concentration.

N, (cm-3)

Fig. 5. Mobility dependence on donor concentration for several interface states densities in the localized states model. Er = Ev t 0.63 eV and EFB = Ev t 0.55 eV. Grain size: L = IOO~m.

The interface state densities parameter.

As it would be expected, this simple model predicts the asymptotic behaviour of the mobility with the grain size. In Fig. 7 we show the variation of the mobility with the grain size for a localized interface states density of 10”cm-* and different donor concentrations. As it can be seem in Fig. 7 the mobility approaches asymp- totically the corresponding monocrystalline mobility as the grain size increases.

are, for n-type silicon po= l.400cm2 V-’ SC’, Nrer = 3 X 1016 cme3, S= 350 and for p-type p0=

480 cm*V-' s-‘, N,, = 4 x lOI cmm3, S = 81. In Figs. 4 and 5 we show the calculated mobilities in

n-type silicon for two different approaches: a continuum of states and the localized model respectively. The grain size considered for these calculations was 1OOp.m. As it can be seen, for given donor and interface state den- sities, the mobility is lower in the second case, (Fig. S), than in the first one, (Fig. 4). In the two cases the mobility approaches the value of monocrystalline silicon as the doping concentration becomes large, but this occurs more abruptly for the localized model than for the con- tinuum of states model.

Finally the drift mobility dependence with the tem- perature has been calculated assuming that the donors inside the space charge layer are fully ionized in all the temperature range. In Fig. 8 this dependence is shown for a donor concentration of ND = 10’scm-3 and different interface states densities. Although the mobility of a monocrystal increases as the temperature decreases (because lattice scattering becomes negligible), the mobility of polycrystalline materials decreases quickly because of the growth of the barrier height with lower temperature. As can be seen in Fig. 8 this decrease is very strong except for very low interface states den- sities. The monocrystalline mobility used for the com- parison has been taken from the experimental data[20].

0.1 I dL dS

302 J. MARTINEZ and J. PIQUERAS

10'

ld

7 Id P

_.

i:

“E 10 z a.

1

0.1

I I I I I

XT4 lo-’ lr lo” 1 lo

L (cm) Fig. 7. Mobility as a function of grain size for a localized interface state density of Nr = 10’2cm-2 and several

donor concentrations Er = Ev t 0.63 eV and &a = Ev t 0.55 eV. Donor concentrations are parameter.

106 I I I

I I I

0 5 10 15 20

103/w +

Fig. 8. Mobility as a function of the temperature for a donor concentration of No = lOI cme3 and several interface state den- sities in the localized states model. Er = Ev + 0.63 eV and J&s = Ev + 0.55 eV. Grain size: L = IOOpm. Interface state densities

are parameter.

5. CONCLUSIONS

The above calculations show that the transport in polycrystalline semiconductors can be explained from the point of view of thermionic emission and carrier tunneling across the grain boundary potentials. In the low doping ranges the transport can be seen as essen- tially thermionic but the tunnel contribution is enhanced as the doping concentration increases and has to be considered for large values of the concentration. This model agrees qualitatively well with that of Volger[7]

and with the reported experimental results[6, IS] and can be seen as a further refinement of such a model.

The most significant aspect is that the present model can predict the asymptotic approach towards the mono- crystalline behaviour, as it has been experimentally observed. Furthermore a different behaviour of p and n-type polysilicon is predicted if it is assumed that the interface states are localized in a very narrow energy range, at ET in the upper half of the gap.

A strong decrease of the mobility with lower tem- perature is also predicted except for very low interface state densities. This point requires experimental confirmation, but at present there are no available data in the literature because of the difficulty in the inter- pretation of Hall mobility measurements in polycrystal- line semiconductors.

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