Theoretical evidence for random variation of series resistance of elementary diodes in inhomogeneous Schottky contacts

Physica B 373 (2006) 284 290 www.elsevier.com/locate/physb Theoretical evidence for random variation of series resistance of elementary diodes in inhomogeneous Schottky contacts Subhash Chand Department of Applied Sciences, National Institute of Technology, Hamirpur-177 005 (HP), India Received 5 October 2005; received in revised form 24 November 2005; accepted 24 November 2005 Abstract The current voltage (I V) characteristics of Schottky diodes measured at low temperature shows an abnormal behavior, which is attributed to the presence of spatial variation of barrier heights of Gaussian type at metal semiconductor interface. The simulation studies assuming Gaussian distribution of barrier heights also show similar temperature dependence of diode parameters as observed in real Schottky contacts fabricated on various metal semiconductor systems. The simulation studies performed so far on inhomogeneous Schottky contacts assumes constant series resistance (R S ) to all elementary diodes because of the complexity in programming. We have simulated I V characteristics of inhomogeneous Schottky diodes with Gaussian distribution of Barrier heights (BHs) assuming different R S for different elementary diodes in the distribution. It is found that the simulation performed using randomly varying R S having Gaussian distribution yields I V characteristics similar to those obtained using constant R S with its value about which variation in R S is considered for all elementary diodes. The simulation results provide theoretical evidence about the occurrence of random fluctuation of R S of Gaussian type in elementary diodes in inhomogeneous Schottky contact. r 2006 Elsevier B.V. All rights reserved. PACS: 73.30.+y; 73.40.Ns; 73.40.Qv Keywords: Schottky diodes; Gaussian distribution; Numerical simulation; Series resistance Current Voltage characteristics; Inhomogeneous contacts 1. Introduction Corresponding author. Tel.: +91 1972 254136; fax: +91 1972 254005. E-mail addresses: schand@nitham.ac.in, schand@recham.ernet.in. The temperature dependence of Schottky diode parameters derived from experimental current voltage (I V) data exhibit an abnormal behavior, which can only be understood on the basis of the spatial variation of barrier heights (BHs) at contact interface. The variation of barrier heights is described mainly by Gaussian distribution function and is believed to be the only possible model to explain the observed temperature dependence of barrier parameters [1 9]. Simulation performed to see the effect of such inhomogeneities in BHs also favours the existence of Gaussian distribution of BHs in real Schottky diodes and leads to similar temperatures dependence of diode parameters as observed from the experimental I V characteristics [10,11]. The Gaussian distribution of BHs is till date being invoked to explain the temperature dependence of barrier parameters derived from the experimental I V and capacitance voltage (C V) measurements [12 22]. The total current through an inhomogeneous Schottky diode can be expressed as [4,10,11] Z IðVÞ ¼ iðv; fþrðfþ df, (1) where i(v,f) the current at a bias V for a barrier of BH f based on thermionic emission diffusion (TED) theory given as [23] iðv; fþ ¼A d A nn T 2 exp qf exp qv ð ir sþ 1 kt kt (2) 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.11.165

S. Chand / Physica B 373 (2006) 284 290 285 and r(f) being the normalized distribution function giving the probability of barrier of BH, f defined as [1,2,4,10,11] " rðfþ ¼ 1 p s ffiffiffiffiffi exp f 2 # f 2p 2s 2, (3) here A d is diode area, A effective Richardson constant, k the Boltzmann constant, T the diode temperature at which corresponding current voltage data is to be calculated, f and s are the mean and standard deviation of the Gaussian distribution of BHs and R S is the series resistance of the elementary diodes. For the first time, we have performed the numerical integration of (1) by Simpson s 1/3rd rule taking varying values of R S for elementary diodes in the distribution. 2. Results and discussion Simulation of I V data of inhomogeneous Schottky diodes is performed at various temperatures for a diode of area A d ¼ 7:87 10 7 m 2, corresponding to 1 mm diameter metal dot, effective Richardson constant A ¼ 1.12 10 6 Am 2 K 2 (for n-si), f ¼ 0.8 V, s ¼ 0.08 V. In the simulation value of standard deviation s of Gaussian distribution of BHs is assumed to be 10% of the f, the mean BH, which is generally reported in the experimentally fabricated Schottky barrier diodes showing Gaussian distribution of BHs, on various metal semiconductor systems [6,12,15 22]. The R S was considered 10 O in one case while in other case it was varied randomly with 50% deviation on either side of 10 O i.e., in the range 1075 O with values distributed in Gaussian manner centered at 10 O. The total currents through the inhomogeneous contact was generated by performing numerical integration of Eq. (1) using Simpson s one-third rule, over BH range 0 1.6 V (i.e., 0 to 2 f) considering all the barriers symmetrically around mean of the distribution in steps of 0.005 V. This small step increment of 0.005 V in BH yields 320 elementary diodes in the limit 0 1.6 V and enables 320 values of R S to be assigned randomly to these elementary diodes during the numerical integration. Fig. 1 shows ln(i) V plots obtained using constant, R S ¼ 10 O (solid line) and randomly assigned value of R S between 5 and 15 O for each elementary diode (dashed line). Fig. 1 clearly shows that assuming random value of R S to elementary diodes yields integrated current through the diode almost equal to that obtained by taking constant R S 10 O to each elementary diode. Thus, an inhomogeneous Schottky contact with Gaussian distribution of BHs with its elementary diodes having randomly varying series resistance in Gaussian manner behave as an ideal diode with some apparent R S. Fig. 1 also shows ln(i) V plots generated using constant R S of 5 O (m) and 15 O (K) for all elementary diodes. It is clear from Fig. 1 that, assigning 5 or 15 O R S to all elementary diodes makes total current differ over entire bias range from that obtained by assigning 10 O to all. Also for these constant R S values saturation occurs at different currents. Thus taking extreme values of R S for all elementary diodes yields different ln(i) V plot. However, as seen above assigning randomly selected R S values between 5 15 O, with Gaussian distribution centered at 10 O, for the elementary diodes does not make appreciable change in the I V characteristics and yields almost same integrated current as is obtained by taking 10 O resistance for all elementary diodes. Fig. 2(a) shows histogram of R S values assigned randomly to all elementary diodes in the distribution for generating ln(i) V plots, shown in Fig. 1. It implies that inhomogeneous contact with randomly varying R S of its elementary diodes exhibit I V characteristics identical to that of the diode with all its elementary diodes having constant R S equal to the mean of the distribution of R S. Fig. 2(b) shows variation of series resistance of elementary diodes with their BHs. It is clear from Fig. 2(b) that each elementary barrier is assigned random value of R S within the distribution range during numerical simulation of total current through the diode. This insensitiveness of the integrated current to the random variation of R S indicates that real inhomogeneous Schottky contacts with Gaussian distribution of BHs may have its elementary diodes associated with them R S, varying randomly in a Gaussian manner. This variation in R S may occur due to spatial variation of doping concentration, any structural defect/ impurities introduced during surface cleaning prior to metal deposition or any non-stochiometric phase forma- 1x10-1 R S =5 Ω R S =Gaussian about 10Ω R S =10 Ω R S =15 Ω 1x10-6 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 1. Simulated ln(i) V curves generated by numerical integration of Eq. (1) over a BH range for various values of R S of elementary diodes at 300 K.

286 ARTICLE IN PRESS S. Chand / Physica B 373 (2006) 284 290 have thin depletion layer and hence wider neutral region in the semiconductor and vice versa. This different neutral region thickness of semiconductor for different BH s regions will have different R S of that corresponding elementary diode. Thus, the inhomogeneous Schottky contacts with varying R S of its elementary diodes exhibit ln(i) V characteristic similar to that the inhomogeneous Schottky contact will have with constant R S of its all elementary diodes in the distribution. To see the effect of deviation in resistance values of elementary diodes, ln(i) V plots were generated for various resistance ranges about 100 O. Fig. 3 shows these plots for constant series resistance of 100 O and also for varying values of R S of elementary diodes assigned randomly for different deviation from 1 to 200 about 100 O. It is evident from the Fig. 3 that with increasing deviation in R S on either side of the mean R S ¼ 100, integrated current slightly increase in the saturation region (at high bias) only without any change at low bias in the linear region from which the BH and ideality factor of the apparent Schottky diode is derived. It is found that the current at higher bias in the saturation region, is contributed mainly by high BH patches near the mean [24]. This slight increase in current in saturation region is caused by those high BH patches, which acquire low R S in the range 1 100 O during numerical simulation. The low R S of elementary diodes 1 5 Fig. 2. (a) Histogram showing Gaussian distribution of R S of elementary diodes used in simulation of I V data shown in Fig. 1 and (b) random variation of R S of elementary diodes with their BHs. 1 Gaussian (100% deviation) 2 Gaussian (80% deviation) 3 Gaussian (60% deviation) 4 Gaussian (35% deviation) 5 R S =100Ω tion during annealing. Any physical effect/phenomenon at metal semiconductor interface leading to BH distribution may as well give rise to different R S for different elementary diodes. Moreover, the low BH patches will 0.0 0.2 0.4 0.6 0.8 0.0 Fig. 3. Simulated ln(i) V curves obtained by numerical integration over BH range, for various deviations in R S of elementary diodes about mean of 100 O along with that at constant R S ¼ 100 O for all elementary diodes.

S. Chand / Physica B 373 (2006) 284 290 287 Fig. 4. Histograms showing distribution of R S values of elementary diodes for various deviation about mean R S ¼ 100 O, used for simulation of I V data shown in Fig. 3. near mean gives rise to slightly higher current, thereby increasing the integrated current of inhomogeneous Schottky contact in the saturation region. This slight increase in current in saturation region makes inhomogeneous Schottky contact behave with slightly higher apparent R S without any change in its apparent BH and ideality factor. Fig. 3 shows that even with maximum 100 percent deviation in R S values on either side of the mean R S ¼ 100 O, the resultant ln(i) V plot remains almost identical to that obtained for constant R S ¼ 100 O, except slight increase in saturation current. These observations clearly indicate that with random variation in values of R S of elementary diodes in inhomogeneous contact possess resultant I V characteristics having ideal shape without any discontinuity or distortion over entire bias. Fig. 4 shows the histograms of R S distribution

288 ARTICLE IN PRESS S. Chand / Physica B 373 (2006) 284 290 T=300K 1x10-6 0.0 0.2 0.4 0.6 0.8 1.0 For R S pattern of Fig. 6 (a) Fig. 6 (b) Fig. 6 (c) Fig. 6 (d) Fig. 5. Simulated ln(i) V curves obtained by numerical integration, for different distribution pattern of R S of elementary diodes for 100 percent deviation about mean 100 O. for various deviations in values of R S assumed for generating ln(i) V plots shown in Fig. 3. It is evident from Fig. 3 that increasing deviation in R S values lead to increase in integrated current slightly in the saturation region only. Simulation results show that the ln(i) V plots remains same even on changing the distribution pattern of R S values. Fig. 5 shows various ln(i) V plots for different distribution patterns of R S values with 100% deviation on either side of mean of 100 O. Fig. 6 shows histograms of these R S values with 100% deviation. It is evident from Fig. 5 that the ln(i) V plot remains almost identical for these distribution patterns of R S values. It clearly implies that distribution pattern of R S values of elementary diodes did not affect the resultant integrated current of inhomogeneous Schottky contact as long as the distribution is Gaussian in nature with same deviation. In the simulation of these curves the computer program generates infinitely large number of R S values in the range 0 200, obeying normalized Gaussian distribution with mean at 100. From these large numbers, 320 values are arbitrarily selected and assigned randomly to the elementary diodes. The samples of 320 values used in all the calculations are tested statistically using student s t- distribution for existence of normality and significance of mean and found to be consistent at more than 99% level of confidence. Moreover this can be consequence of famous central limit theorem that if population is normal, the sampling distribution is also normal with same mean but standard error equal to standard deviation of population divided by square root of sample size, while for large samples the same result holds even if the distribution of the population is non-normal. Fig. 7 shows these plots at various temperatures. It is clear from Fig. 7 that these plots have similar behavior at low temperatures. The effect of random variation of R S of elementary diodes on inhomogeneous Schottky contacts I V characteristics is much pronounced for higher standard deviation s of Gaussian distribution of BHs. Fig. 8 shows ln(i) V plots for different standard deviation. For higher s, deviation occur even at low bias in these plots. Higher standard deviation s of the BH distribution enables increased number of low BH elementary diodes with higher occurrence probability at contact interface, which contributes more current at low bias [24]. The decrease in the R S of low BH elementary diodes assigned randomly makes ln(i) V plot slightly shift up at low bias in the linear region. On the other hand for low s, low BH elementary diodes disappear and the effect of R S fluctuation become less as the number of elementary diodes contributing effectively to total integrated current also decreases. As shown in Fig. 3, these curves of inhomogeneous Schottky contact with random variation of R S have almost identical features as that of inhomogeneous Schottky contact with constant R S of all elementary diodes. Therefore it is not possible to distinguish whether the elementary diodes in an inhomogeneous Schottky contact exhibits random Gaussian variation of R S or have constant value of R S associated with them all. However, the ln(i) V curves of inhomogeneous contact either with random distribution of series resistance or with constant R S of elementary diodes, can be differentiated from the curves of homogeneous contact by least square fitting into thermionic emission diffusion equation for extracting diode parameters. It is observed that I V data cannot be fitted into thermionic emission diffusion equation with single R S over entire saturation region. It is because these curves continuously bend downward as total current is obtained by summing up the contribution of all elementary diodes [24]. On the other hand the I V data of homogeneous contact can be fitted easily over entire bias range using single R S value. This is how these curves can be discerned from those of the homogeneous contact. 3. Conclusions The I V data of inhomogeneous Schottky diodes with Gaussian distribution of BHs are generated by numerical simulation using TED equation. The R S of the elementary barriers is assumed to be varying with Gaussian distribution and each elementary diode was assigned random value of R S during numerical simulation. It is shown that

S. Chand / Physica B 373 (2006) 284 290 289 Fig. 6. Different distribution patterns of R S values of elementary diodes with 100 percent deviation about mean R S ¼ 100 O. considering random variation in R S of elementary diodes yields ln(i) V characteristic of inhomogeneous Schottky contact having ideal shape without any distortion or discontinuity over entire bias range at all temperatures. It is also shown that distribution pattern of R S values of elementary diodes did not affect the resultant integrated current of inhomogeneous Schottky contact as long as the distribution is Gaussian in nature with same deviation. This insensitiveness of ln(i) V plots to R S variation of elementary diodes provide theoretical evidence about occurrence of Gaussian distribution of R S in real inhomogeneous Schottky contacts.

290 S. Chand / Physica B 373 (2006) 284 290 Acknowledgment 300 K 200K 100K The author is very thankful to Dr. J. N Sharma, Professor of Mathematics, Department of Applied Sciences, National Institute of Technology, Hamirpur, for valuable suggestions and statistical interpretation of the findings reported in the manuscript. References 1x10-6 1x10-7 1x10-8 1x10-9 1x10-10 100 Ω Gaussian 0.0 0.2 0.4 0.6 0.8 1.0 Forward Bias(V) Fig. 7. Simulated ln(i) V curves generated by numerical integration of Eq. (1) over a BH range, using constant R S for all elementary diodes (solid lines) and randomly assigned value of R S for elementary diodes (dashed lines) at various temperatures. T=300 K [1] Y.P. Song, R.L. Van Meirhaeghe, W.F. Laflere, F. Cardon, Solid State Electron. 29 (1986) 633. [2] V.W.L. Chin, M.A. Green, J.W.V. Storey, Solid State Electron. 33 (1990) 299. [3] A. Singh, K.C. Reinhardt, W.A. Anderson, J. Appl. Phys. 68 (1990) 3475. [4] J.H. Werner, H.H. Guttler, J. Appl. Phys. 69 (1991) 1522. [5] S. Chand, J. Kumar, J. Appl. Phys. 80 (1996) 288. [6] S. Chand, J. Kumar, Semicond. Sci. Technol. 11 (1996) 1203. [7] P.G. McCafferty, A. Sellai, P. Dawson, H. Elabd, Solid State Electron. 39 (1996) 583. [8] S. Zhu, R.L. Van Meirhaeghe, C. Detavernier, G.P. Ru, B. Z Li, F. Cardon, Solid State Commun. 112 (1999) 611. [9] S. Zhu, R.L. Van Meirhaeghe, C. Detavernier, F. Cardon, G.P. Ru, X.P. Qu, B.Z. Li, Solid State Electron. 44 (2000) 663. [10] E. Dobrocka, J. Osvald, Appl. Phys. Lett. 65 (1994) 575. [11] S. Chand, J. Kumar, Semicond. Sci. Technol. 12 (1997) 899. [12] A. Gumu, A. Turut, N. Yalcin, J. Appl. Phys. 91 (2002) 245. [13] K. Maeda, Appl. Surf. Sci. 190 (2002) 445. [14] N.L. Dmitruk, O.Y. Borkovskaya, I.N. Dmitruk, S.V. Mamykin, Zs.J. Horvath, B. Mamontova, Appl. Surf. Sci. 190 (2002) 455. [15] M. Biber, Physica B 325 (2003) 138. [16] C. Coskun, M. Biber, H. Efeoglu, Appl. Surf. Sci. 211 (2003) 360. [17] S. Karatas, S. Altindal, A. Turut, A. Ozmen, Appl. Surf. Sci. 217 (2003) 250. [18] K. Akkilic, A. Turut, G. Cankaya, T. Kilicoglu, Solid State Commun. 125 (2003) 551. [19] M. Saglam, F.E. Cimilli, A. Turut, Physica B 348 (2004) 397. [20] S. Acar, S. Karadeniz, N. Tugluoglu, A.B. Selc-uk, M. Kasap, Appl. Surf. Sci. 233 (2004) 373. [21] S. Karatas, S. Altindal, Mater. Sci. Eng., B 122 (2005) 133. [22] H. Cetin, B. Sahin, E. Ayyildiz, A. Turut, Physica B 364 (2005) 133. [23] E.H. Rhoderick, Metal-Semiconductor Contacts, Second ed, Clarendon Press, Oxford, 1978. [24] S. Chand, S. Bala, Appl. Surf. Sci. 252 (2005) 358. 1 2 1 σ = 0.10 V 2 σ = 0.06 V R S =100 Ω R S =Gaussian 1x10-6 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 8. Simulated ln(i) V curves for different standard deviation of Gaussian distribution of BHs for constant R S of all elementary diodes (solid lines) and randomly varying R S of elementary diodes (dashed lines).