An accurate approach for analysing an inhomogeneous Schottky diode with a Gaussian distribution of barrier heights

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Semicond. Sci. Technol. 17 (00) L36 L40 SEMICONDUCTOR SCIENCE AND TECHNOLOGY PII: S068-14(0) LETTER TO THE EDITOR An accurate approach for analysing an inhomogeneous Schottky diode with a Gaussian distribution of barrier heights Subhash Chand Department of Applied Sciences, Regional Engineering College, Hamirpur , India schand@recham.ernet.in Received 3 May 00, in final form 3 May 00 Published 17 June 00 Online at stacks.iop.org/sst/17/l36 Abstract An unexpected observation in the current voltage curves of Schottky diodes, containing barrier inhomogeneities generated using the analytical results based on a Gaussian distribution model of barrier heights is reported. Calculations based on these results show that, at very low temperatures, Schottky diodes exhibit higher currents than at higher temperatures. This is an unusual observation, indicating a high current through the Schottky diodes at lower temperatures, which is inconsistent with the thermionic emission diffusion theory. The effects causing this unusual behaviour are explored by analysing a conventional model. A more accurate approach is presented which explains this unusual behaviour and yields results consistent with the theoretical behaviour of the Schottky diodes. In recent years, current transport with respect to the exact nature of contact in real Schottky diodes has been the main thrust of research. The abnormal behaviour of the Schottky diodes has been attributed to the barrier inhomogeneities present in the Schottky contacts. Recently, spatial barrier inhomogeneities have been described mainly with a Gaussian distribution function. This distribution has been widely accepted to correlate experimental data 1 9]. Palm et al 10] have shown direct images of Schottky barrier height (BH) fluctuations in Au/Si contacts using ballistic electron emission microscopy (BEEM) and correlated them with a Gaussian distribution function. Recently, Vanalme et al 11] have also shown BEEM spectra representing a Gaussian distribution of BHs in Au/III V semiconductors. Simulation studies based on the effect of a Gaussian distribution of BHs on current voltage (I V )characteristics have also been reported in the literature 1 14]. This letter reveals the discrepancies observed in the analytical results based on a Gaussian distribution model of barrier inhomogeneities. An accurate approach is presented for an analysis of Schottky diodes containing barrier inhomogeneities, which yields results consistent with the thermionic emission diffusion (TED) theory of current conduction. The total current across a Schottky diode containing barrier inhomogeneities can be written as I(V)= i(v,φ)ρ(φ)dφ (1) where i(v, φ) isthecurrent at a bias V for a barrier of height φ,andρ(φ)isthenormalized distribution function giving the probability of occurrence for BH φ. Theimplicit assumption is that there are a number of parallel diodes of different BHs, each contributing to the current independently. In the case of agaussian distribution of BHs with mean ( φ) andstandard deviation (σ ), the distribution function ρ(φ) is given by ρ(φ) = 1 σ π exp (φ φ) ] () σ where 1/σ π is the normalization constant. The current i(v, φ)through a Schottky barrier at a forward bias V based on TED theory is expressed as 15] i(v,φ) = A d A T exp qφ ] exp { q(v irs ) } ] (3) /0/ $ IOP Publishing Ltd Printed in the UK L36

2 where A d, A, T, q, k and R S are the diode area, the effective Richardson constant, the temperature, the electronic charge, the Boltzmann constant and the diode series resistance, respectively. Substitutingi(V, φ)andρ(φ)inequation (1)and performing integration from to + for values of φ, one obtains 4, 5] ( ) ] q(v IRs ) I(V)= I S exp (4) with I S and φ ap given by I S = A d A T exp qφ ] ap (5) φ ap = φ σ q. (6) It has been the usual practice to find the total current through an inhomogeneous Schottky contact in this way. The experimental data also fit nicely in equation (4), and equation (6) has been used in the past to provide evidence about the existence of a Gaussian distribution of BHs in Schottky contacts 1, 4 8]. Equation (6) hasalso been used to find the mean and standard deviations of the distribution 4 6, 8]. Since equation (4) represents the total current through the inhomogeneous Schottky diode, which has a Gaussian distribution of BHs, it is possible to calculate the current by numerically solving it using a computer program for iteration, at any mean and standard deviation. The ln(i) V curves thus obtained at various temperatures are shown in figure 1. These curves are calculated using A = Am K, A d = m (for a diode with a 1 mm diameter), R S = 0 and φ = 0.8 V. The interesting observation here is that these curves first shift downwards up to a certain temperature, below which the trend is reversed and they start shifting up thus intersecting the curves at higher temperatures. This is an unusual phenomenon, observed in the curves obtained using equation (4), based on agaussian distribution model of BHs. It is unusual in the sense that it indicates higher currents through the diode at lower temperatures. Below a certain temperature, it leads to a greater increase in the current, even yielding ohmic behaviour at very low temperatures. This effect is more prominent at higher temperatures for large standard deviations (for example, at 40 K for σ = 0.08 V and at 80 Kforσ = 0.10 V, shown in figure 1). The TED theory, however, predicts less current at low temperatures through the diode. This observation is thus inconsistent with the TED theory. To resolve this unusual behaviour depicted in figure 1, we have investigated equation (5), since in this phenomenon it is the saturation current (I S ), i.e. the starting point of the ln(i) V curve, which is shifted up irrespective of applied bias. As far as equation (5) fori S is concerned, it is correct in this form, as it is the standard expression for the saturation current of Schottky diodes 15]. It is the appearance of the apparent BH (φ ap ), given by equation (6), which crucially plays this role in a subtle way by increasing I S below a certain temperature. On critically observing the dependence of I S on T, thediscrepancy can be revealed. The saturation current I S contains φ ap and T.Inturn, φ ap is again temperature-dependent for given mean and standard deviations through equation (6), which implies that φ ap decreases as T decreases. When this φ ap is substituted into equation (5) foranevaluation of I S,the exponential term contains both φ ap and T. With a decrease in 1.E-13 1.E-03 1.E-09 (a) Sigma=0.08V 60K 40K (b) Sigma=0.1V Figure 1. Simulated I V curves of Schottky diodes using equation () for various temperatures with φ = 0.8 V and series resistance R S = 0. Clearly, the curve shows high current at very low temperatures. T, thedecrease of φ ap with a negative sign inthenumerator of the exponential term leads to an increase of I S. On the other hand, a decrease of T in the denominator makes I S decrease. With decreasing temperatures, while evaluating ln(i) V plots atvarious temperatures, initially the effect of T in the denominator is more dominant than the effect of φ ap through equation (6). This occurs up to a certain T (transition temperature) below which the effect of φ ap becomes dominant and I S begins to increase. This is shown graphically in figure where I S is plotted as a function of temperature for various standard deviations using equations (5) and(6). It is clear from this figure that I S first decreases from its value at room temperature, then starts to increase again below the transition temperature. This transition temperature is dependent on σ for given φ. Thegreater the standard deviation σ,thehigher the transition temperature. It is the temperature which corresponds to the onset of the upshifting of the ln(i) V curves shown in figure 1. L37

3 Saturation 1.E+10 1.E+05 1.E+00 1.E-15 1.E-0 1.E-5 1.E-30 1.E-35 sigma Temperature (K) Figure. Variation of I S as a function of T for various values of σ for a diode area of 1 mm diameter φ = 0.8 V. I S first decreases with decreasing T up to a particular T below which it increases with a further decrease of T. The discrepancy appears to be in the derivation of equation (4) based on a Gaussian distribution model. After careful analysis it was found that the problem lies in the limits taken in the integration of equation (1). In real Schottky diodes, the maximum BH cannot exceed the semiconductor energy bandgap and the lowest it can be is close to zero, depending upon the type of semiconductor and the work function of the metal used. For barriers that have a height greater than the mean BH, the current decreases and their probability density function also decreases as one moves away from the mean BH. Thus, the product of the current due to higher barriers and their probability distribution decrease appreciably and their contribution towards the total current becomes negligible. Also, a BH that is less than zero, or a negative BH, has no relevance to the actual device behaviour. Therefore, in the analytical integration of equation (1) limits to + have no physical relevance. In the past, these limits might have been chosen for ease of integration. Thus, to obtain the total current across the inhomogeneous Schottky contact, equation (1)mustbeintegrated from 0 to φ to include all the barriers symmetrically around the mean, i.e. I(V)= φ 0 i(v,φ)ρ(φ)dφ. (7) On performing the above integration, the total current through the diode becomes ( ) ] ( ) q(v IRs ) erf(f1 ) erf(f ) I(V) = I S exp (8) with the same I S (given by equation (5)) as obtained previously in terms of the same apparent BH φ ap (given by equation (6)). Equation (8) can further be written as I(V)= I S exp ( q(v IRs ) ) ] (9) Saturation 1.E+00 1.E-15 1.E-0 1.E-5 1.E-30 1.E-35 Sigma Temperature (K) Figure 3. Modified saturation current I S calculated using equation (10) with same parameters as those used to calculate I S shown in figure. Clearly, I S now continuously decreases with the decrease of T. where the modified saturation current I S now becomes I S = I erf(f 1 ) erf(f )] S (10) in terms of the error functions of two factors, f 1 and f.these factors are functions of φ, σ and T as ( σ ) q 1 f 1 = + φ σ (11) ( σ ) q 1 f = φ σ. (1) Thus, changing the limits of integration as suggested above leads to a similar expression for the total current with the multiplication of an additional term m ={erf(f 1 ) erf(f )}/ in terms of the error function. This multiplier is a function of temperature and the mean and standard deviations. This additional factor in terms of the error function attains low values at low temperatures but is close to unity (m = 0.99) at higher temperatures. As an example, for φ = 0.8 V and σ = 0.08 V, m attains a value of more than 0.99 for temperatures in the range K. Below this temperature, it acquires a value of less than 0.99 which, when multiplied by I S,makes it less. For the same value of mean and standard deviations, m attains values of 1 10 at 75 K, at 65 Kand at 55 K. The modified value of the saturation current (I S )isshownin figure 3 as a function of temperature for various standard deviations. It is clear from figure 3 that the saturation current continuously decreases and shows no such increase at low temperatures, as depicted in figure. However, at very low temperatures, m has such a small value that it is recorded as zero and I S becomes equal to zero. This is why the curves, which show the variation of I S in figure 3, are terminated at a particular temperature. This temperature ishigher for large standard deviations. The appearance of this factor in the total current equation is very effective in scaling down the ln(i) V curves at low temperatures giving rise to acontinuous downshifting of the curves. Its effect is clearly L38

4 (a) Sigma=0.08V (a) Sigma =0.08V 1.E-1 1.E-14 60K 1.E-18, n=1, n=1, n=1 1.E-1, n=1 1.E-14, n=1.16, n= K, n= E-18 (b) Sigma =0.1V (b) Sigma = 0.1V Figure 4. Simulated ln(i) V curves using equation (8) at various temperatures with φ = 0.8 V and R S = 0. Thecurves continuously shift down with decreasing T. AtverylowT the curves are steeper and thus intersect the curves at higher T. evident from the curves shown in figure 4 using equation (8) for the same parameters as those used for obtaining the curves infigure 1. Clearlythese curves shift continuously downwards and show no suchupshifting at low temperatures. However, these curves are steeper and thus intersect other curves at higher temperatures, which exhibit lesser slope. This is probably due to the high ideality factor exhibited by the total current through a diode with a Gaussian distribution of BHs. In generating these curves shown in figure 4,anideality factor of unity is taken. However, these curves show no such crossing if one generates them using an ideality factor greater than unity in equation (8). This is depicted in figure 5 where no crossing is observed in the curves with a higher ideality factor at low temperatures. The high ideality factor arises due to the bias dependence of the BHs. Needless to say, the BH is known to depend on the applied bias 15]. This bias dependence of BHs in thedistribution through mean and standard deviations leads to the temperature-dependent ideality factor in inhomogeneous Schottky diodes 4, 5, 14]. 1.E-03 1.E-09, n=1, n=1, n=1, n=1.05, n=1.40, n=1.70 Figure 5. Simulated ln(i) V curves with the same parameters as those used for generating curves in figure 4 but with a higher ideality factor at low T. Clearly, the curves at low T now do not intersect the curves at high temperatures. Alternatively, equation (8) can now be used in place of equation (4) for the case of barrier inhomogeneities at all temperatures. Equation (8) isidentical to equation (4) except that it contains a multiplier m inthesaturation current term. This multiplier is always very close to unity (0.999) at higher temperatures generally above 100 K, the temperature regime where most of the experimental studies are reported in the literature. This may be the reason why it has not yet been noticed previously. In the higher temperature regime, this factor is close to unity and the results are identical to those obtained using a conventional approach, i.e. using equation (4). As long as one uses the higher temperature regime, it makes no difference whether one uses equation (8)orequation (4)to reveal the inhomogeneities in BHs, and the ln(i) V curves obtained will be identical. However, when the working temperature is lower than a particular temperature, it is more appropriate to use equation (8) ratherthanequation (4) either for fitting experimental data into a Gaussian distribution model L39

5 or in theoretical calculations of current voltage data based on this model. About the limits of integration, it is sufficient to take the higher limit a few σ above φ. As the current due to high barriers (BH > φ)and the probability of their occurrence both decrease, these have a negligible contribution towards the total current. In equation(8), obtained by integration of equation (1) from 0 to φ, thevalueoff 1 is so high that erf(f 1 ) is always unity over the entire temperature range. On the other hand, in the derivation of equation (8) taking the upper limit to + in place of φ, one obtains m = 1 erf(f )]/, which is identical to that with an upper limit φ. Thus, there is no difference in taking the upper limit to anywhere above φ. For the extraction of φ ap from the saturation current obtained by fitting the experimental data, equation (8) can only be used as long as m 0.1 for which φ ap is a simple function of I S (equations (4) and(8) areidentical). Thus, as an approximation for m 0.1, the lowest temperature up to which φ ap can be extracted from the experimental data is T = σ q/k(0.9σ + φ). Below this temperature, the saturation current obtained from the intercept of the experimental ln(i) V plot will appear to be more than 10% overestimated for the φ ap evaluation using equation (5). Hence, below this temperature, equation (10) should be used for the determination of φ ap but, as it is not an explicit function of φ and σ,thedetermination of distribution parameters is not possible and only data in the higher temperature regime can be used. This can be treated as the limitation of the Gaussian distribution model for revealing BH inhomogeneities at very low temperatures. In conclusion, the conventional Gaussian distribution model can be applied more effectively using the proposed approach. It is shown that the calculated curves using the suggested approach yield results that are consistent with the thermionic emission diffusion theory. Also, at very low temperatures, the distribution model cannot be applied to extract the distribution parameters, i.e. the mean and standard deviations from the experimental data. References 1] Song Y P, Van Meirhaeghe R L, Laflere W F and Cardon F 1986 Solid-State Electron ] Chin V W L, Green M A and Storey J W V 1990 Solid-State Electron ] Singh A, Reinhardt K C and Anderson W A 1990 J. Appl. Phys ] Werner J H and Guttler H H 1991 J. Appl. Phys ] Chand S and Kumar J 1996 J. Appl. Phys ] Chand S and Kumar J 1996 Semicond. Sci. Technol ] McCafferty P G, Sellai A, Dawson P and Elabd H 1996 Solid-State Electron ] Gumu A, Turut A and Yalcin N 00 J. Appl. Phys ] Zhu S Y, Van Meirhaeghe R L, Detavernier C, Cardon F, Ru GP,QuXPand Li B Z 000 Solid-State Electron ] Palm H, Arbes M and Schulz M 1993 Phys. Rev.Lett ] Vanalme G M, Goubert L, Van Meirhaeghe R L, Cardon F and Daele P V 1999 Semicond. Sci. Technol ] Dobrocka E and Osvald J 1994 Appl. Phys. Lett ] Chand S and Kumar J 1997 J. Appl. Phys ] Chand S and Kumar J 1997 Semicond. Sci. Technol ] Rhoderick E H 1978 Metal-Semiconductor Contacts nd edn (Oxford: Clarendon) L40