Temperature, current and doping dependence of non-ideality factor for pnp and npn punch-through structures

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1 Indian Journal of Pure & Alied Physics Vol. 44, December 2006, Temerature, current and doing deendence of non-ideality factor for n and nn unch-through structures Khurshed Ahmad Shah & S S Islam Deartment of Alied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (Central University), New Delhi * Shahkhurshaid@yahoo.co.in Received 16 January 2006; revised 11 Setember 2006; acceted 13 October 2006 The effect of unch-through in devices is of great imortance in electronic industry. In this aer, an efficient, unconditionally stable, simle and accurate method has been resented for solving non-linear transort equations for carriers with aroriate choice of initial values of deendent variables. The methods are so fast that arameter deendence of nonideality for unch-through structures has been incororated easily and their secific effects have been determined recisely. The resent study reveals that the non-ideality factor decreases with temerature and increases with current and doing concentration of the active region of the device. The relative changes are larger at lower temeratures and more effective at higher imurity density of the active region of the device. This study enables to have better understanding of the unchthrough mechanism and design of these devices. Keywords: Punch-through diode, Numerical method, Parametric study, Non-ideality factor IPC Code: H01L29/861 1 Introduction The recent work suorts that a unch-through diode shows good claming characteristics and current handling at a given standoff voltage than zener diode 1, imroved leakage and claming characteristics 2, suresses imact and ionization related effects 3 and its use as a voltage reference element 4. However, it is imortant to have a comlete knowledge of the I-V characteristics and its deendence on device design arameters, so as to exloit fully the device in terms of its use in circuit alications. The exression for voltage at which unch-through currents starts flowing in these devices has been derived in 5, but no current voltage relation has been derived. Lohstroh 6, derived one-dimensional exression for n and nn sandwich structures based on drift and diffusion theory for low current levels. The redicted I-V characteristics were verified exerimentally. Some attemts were also made to comute I-V characteristics of unch-through devices using exact numerical techniques 7,8, and this study brought out reasonable exlanation of I-V characteristics of these devices observed exerimentally. However, the study does not rovide comlete understanding of the mechanism of oeration and design of these structures. Soon after this, finite difference methods were frequently used to model heat flow in semiconductors 9. Sadik and Sing 10, develoed a simle analytical theory to describe the carrier distribution and the current flow in unchthrough structures under the diffusion limited injection condition and negligible generation recombination effects. The current flow rocess is found to be intimately related to the doing concentration and carrier mobility in the base. It has been observed by Ellison et al. 11, that the unchthrough currents and leakage currents are equal and oosite between adjacent + stris on the surface of silicon Microsoft detectors. Kaun and Yuk 12, investigated that unch-through current in sub-micron MOS transistor is essentially injected at the surface near the edge of the source junction. In this aer, an accurate, unconditionally stable, simle and relatively faster method is develoed for the solution of non-linear transort equations for carriers, with aroriate choice of deendent variables involved. A relatively faster and unconditionally stable technique is develoed for the determination of source and drain contact locations. In this technique, solutions from source and drain regions are matched with the solutions in active region for the determination of corresonding

2 954 INDIAN J PURE & APPL PHYS, VOL 44, DECEMBER 2006 junction. The methods are so fast that the arameter deendence has been incororated easily and there secific effects has been determined recisely. The non-ideality factor in unch-through structures is a hysical quantity and is a function of temerature and current. It determines the sloe of the I-V characteristics of unch-through structures, which is necessary to exloit these devices to the best of their use in circuit alication. It has been revealed in this aer that the non-ideality factor decreases with temerature and increases with current and doing. The changes are more ronounced at lower temerature and more effective at higher imurity concentration of the active region of the device. The study enables to have better understanding of unchthrough mechanism and design of unch-through structures, so as to have large use of these devices in circuit alications. 2 Exerimental Details The current-voltage characteristics of unchthrough devices 6 is I=I 0 ex (q (V-V t )/ (mkt)), where m is non-ideality factor given by m=[2w / (X j0 +X j )], X j0 is the thickness of the deletion layer into the n material of the non-biased + n junction (for + n + ), X j is the deletion layer width of the forward biased junction, W the distance between + regions (Ref. Fig. 1(a)), V the alied voltage, V t unch-through voltage, k the Boltzman constant and T is the absolute temerature. To determine the deendence of nonideality factor we need to solve the current transort equations of the device numerically, as the zero electric field lane X j0 injection lane (Ref. Fig. 1(b)) moves towards or away from the + n junction, by changing these arameters. The transort equations for unch-through structures are highly non-linear in nature. If one is not careful in selecting roer initial values of variables, the occurrence of solution instability is quite common 13,14. The fastness of solutions of transort equations becomes imortant in order to kee the comutational effort within reasonable limit. The inclusion of arameter deendence is necessary in order to analyze the device oeration more realistically. The transort of injected carriers is described by Poisson s equation, current density and current continuity equations along with the exressions for currents due to mobile carriers. Based on holes as injected carriers in the active region of the + n + unch-through structures, one can write the Fig. 1 (a) + n + unch-through structure. (b) Electric field and otential rofile across the structure. (c) Proosed method of matching of solutions of E and in the active region with those in source and drain. relevant equations given by: q = [ ( + N D ] ε (1) J (=qμ ( (E( qd ( (2) ε x J T ( = J ( + ( J( t = ) t q, (3) (4) where N D, E,, D, J T, ε, μ and q is donar concentration, electric field, hole concentration, hole diffusion coefficient, total current density, dielectric constant, hole mobility and electronic charge, resectively. In Eq. (4), generation and recombination rate has been droed. In order to avoid numerical overflow and underflow during the execution of the comuter rogramme, it is necessary to normalize the above set of equations into dimensionless form using the normalization factors as used by De-Mari 15 and keeing the same symbols one can write the normalized equations as: = [ ( + ] N D (5)

3 SHAH & ISLAM: NON-IDEALITY FACTOR FOR PNP AND NPN PUNCH-THROUGH STRUCTURES 955 J ( = μ ( ( ( E( (6) J T ( = J ( + (7) ( J( t = t q (8) In Eq. (6), the diffusion coefficient is substituted by carrier mobility using the well-known Einstein relation: D kt ( = μ ( (9) q Under steady state, the normalized onedimensional Poisson s equation and transort equation for one tye of carrier is given by: = [ ( + N D ] (10) d( J ( = ( E( dt μ ( (11) Eqs 10 and 11 are for holes in + n + structure, as the current transort mechanism is mainly dominated by the flow of minority carriers in active region of the device. The deendence of carrier velocity on electric field and temerature is incororated in the solution through using an emirical relation 16. Taking ohmic contacts as the hysical boundaries, Eqs 10 and 11 can be solved as boundary-value roblem in a finite difference aroximation 17. In this case, these equations are converted into two ordinary differential equations using otential function. A large number of grid oints are needed to be considered for the accuracy of the derivatives. The actual number of iterations required for the convergence deends on the closeness of starting guess of the otential function to exact one besides the method used for solving the resultant simultaneous equations. However, it can be converted into an initial value roblem 18 by taking aroriate values of the deendent variables and in the Eqs 10 and 11 to start with, inside active region of the device. Referring to Fig. 2, it can be noted that either zero or the highest field oint E max in the active region can be chosen as the starting oint for integration. In case of zero electric field, the carrier diffusion being resonsible for current flow, some guess for the hole concentration is to be made and the solutions for E and are advanced. If the guess for is not roer, the solution for E diverges. Several iterations are required for finding out the correct value of for convergence. On the other hand, a backward integration towards the source from the electric field maximum at the drain junction is not exected to face any such instability as the absence of diffusion term simlifies the differential equations at the start. This has been found unconditionally stable. However, in the resence of the initial value solution, the interfaces with the source and drain regions are not yet established. This can be done by advancing the solutions of transort equations from either side of the contacts, once again treating them as initial-value roblem and matching them with the solutions in the active region. To comlete the secification of the mathematical model, a set of initial and boundary conditions must be defined. The details of determining the initial values of E and in three regions are given below. The initial values of the deendent variables are decided by the device structure and the articular region of the device under consideration. At reverse biased junction on drain side, the electric field is largest and above the value required for carrier drift velocity saturation 19. This ensures ractically the absence of diffusion of carriers and current consists of drift comonent alone. Due to this, the solutions of E and from Eqs 10 and 11 are relatively simler. With the integration advancing towards the source end, the electric field decreases initially with constant rate and mobile carrier density remains constant, as the electric field goes below the value required for the drift velocity saturation the mobile carrier density rises and causes the growth of the diffusion current. The diffusion current aroaches the total current where electric field is zero, known as injection lane and it coincides with the otential maximum V max as shown in Fig. 1(b). Further integration lowers the electric field to negative values. The value of E is decided by the geometry of the structure. In the resent work, the value of E is chosen as V/cm. Deending uon current density, the mobile carrier density can be determined by the relationshi given by: = J/ (q V s ) (12) where V s is the saturated carrier velocity. In general, a heavy uniform doing is considered in the source and drain regions of the + n + (n + n + ) structures 14. The initial values of E and for + n +

4 956 INDIAN J PURE & APPL PHYS, VOL 44, DECEMBER 2006 structure in source region are taken from the relation s, given by: = N A (1 ) (13) E = J/(q µ ) (14) where is a small erturbation, which is chosen as 10-6 in the resent work. This choice is arbitrary and can be taken even lower. However, the resent value of is reasonably good from the oint of view of comutational efficiency. Starting with these initial values from Eqs 13 and 14, the solution of and E icks u consistent values over a short distance. The concentration of holes decreases while electric field grows in ositive or negative direction deending uon the direction of carrier flow [Ref. Fig. 1(c)]. Starting with these initial values for E and the solution for E and icks u consistent constant values over a short distance. For the determination of source and drain contact locations, the solutions from source and drain regions are matched with solutions in active region for determination of corresonding junction. With advancing the solutions of E and, an adequate number of these values are stored as function of osition. On other side of the junction, while advancing the solution in active region, the following rocedure is adoted while the electric field becomes negative. For the current value of electric field at the end of the integration ste, one equivalent oint is searched and corresonding value of carrier concentration is calculated by interolation from the stored values of E and in the source region. In case the two values of from two regions are not equal, integration is advanced further from active region side otherwise interface is established as exlained in Fig. 1(c). A situation arises, where the value of from active region side is more than the one from the source side, and this ensures that the osition is somewhere in the last ste of the distance. Further, the rocess of interval having and reeated search leads to the final location within the limit of accuracy defined. Moreover, similar rocedure can be followed for drain contact location, however it deends uon other conditions too. 3 Results and Discussion The integration of Eqs 10 and 11 can be started from drain side as described, the choice is based on the advantage of having unconditional stability of solution. The electric field sign changes in the vicinity of injection lane, and this coincides with otential maximum in the resent case [Fig. 1(b)]. In the resent study, a uniform doing of /cm 3 has been taken in the source and drain regions of the + n + unch-through diode. The temerature and current are varied in the range K and A/cm 2 resectively. Doing in the active region is varied in the range /cm 3.The width of the active region of the device is taken as cm in the resent study. These chosen values cover most of the ractical structures. The lots of non-ideality factor for + n + unch-through structure as a function of temerature with bias current density as arameter is shown in Fig. 2(a). The variations of non-ideality factor with current and doing densities are shown in Figs 2(b and c), resectively. The results lotted in Fig. 2(a-c) reveal that the non-ideality factor decreases with temerature and increases with current and doing. The decrease is much faster at lower current density and is not more ronounced at higher current densities. The increase is faster at lower doing and current densities. The relative changes in non-ideality factor over a current density range A/cm 2 are larger at lower temeratures, and these changes are more effective at higher imurity density of the active region of the device. An increase in temerature causes the moment of injection lane away from the source region 8. This results in negative temerature coefficient of unch through voltage of the device. The deletion layer width of the forward biased junction increases with temerature and therefore, results in decrease in nonideality factor. The decrease is exected, as this accounts for negative temerature coefficient of unch-through voltage of these devices observed exerimentally 20. The non-ideality factor decreases faster with temerature at lower current density, as the injection region changes faster with current density below 50 A/ cm 2 and varies relatively slow above this value of current. The effect of mobile sace charge increases with current in the vicinity of injection lane. The voltage across the device increases with current density according to the sace charge resistance. The location of injection lane shifts towards the source side at higher current densities 19. This results in reduction in deletion layer width at forward biased junction and therefore, the non-ideality factor increases. The deletion layer width of the forward biased junction

5 SHAH & ISLAM: NON-IDEALITY FACTOR FOR PNP AND NPN PUNCH-THROUGH STRUCTURES 957 Fig. 2 (a) Temerature deendence of non-ideality factor for + n + unch-through structure with bias current density as arameter. (b) Current deendence of non-ideality factor for + n + unch-through structure with imurity density as arameter. (c) Doing deendence of non-ideality factor for + n + unch-through structure with bias current density as arameter. reduces with current density, as an aroriate barrier lowering is needed to suly the extra current flowing across the device. In case of higher imurity density of the active region of the device, the injection region is narrower, and therefore areciable rise in the mobile concentration over that required for causing the current through drift alone takes lace, at relatively smaller distance from the injection junction. The deletion layer width of forward biased junction reduces with the increase in doing of the active region and therefore, the non-ideality factor increases. Moreover, this increase is more ronounced at lower temeratures. 4 Conclusions An exact comuter aided numerical method has been described for solving the non-linear transort equations for unch-through structures. The results reveal that the non-ideality factor decreases with temerature and increases with current and doing concentration of the active region of the device. The relative changes are larger at lower temeratures and more effective at higher imurity density of the active

6 958 INDIAN J PURE & APPL PHYS, VOL 44, DECEMBER 2006 region of the device. At lower current densities, the effect of change in temerature on non-ideality factor is dominant. The resent results enable to have better understanding of the unch-through mechanism and to exloit these devices to the best of their use in circuit alications. References 1 Dalan Van R, Koos J E G & Pflennigsterf O, USA Patent, (2004) Bin U Y, Chenming Hu, King, Chin Ya, Pohlman, T Jerffrey, Trivedi Rita, USA Patent, (2000) Dalan Van R & Koos J E G, Electrostatic discharge symosium roceedings, EOS-23 Portland OR USA (2001) Set Beasom D James, USA Patent, (1999) Hien N Ba, Ph.D Dissertation, University of Michigan (1977). 6 Lohstroh J & Kooman M J J, Solid State Electronics, 24 (1981) Mustafa M & Ahmad S, Symosium on semiconductor materials and devices, (1980) Set Mustafa M & Ahmad S, J Al Phys, 53 (1982) Seviron M & L Lee, Det of electrical and electronic Engg, University of Nottingham Notingha UK, J Phys D Al Phys, 15 (1982) 10 Esener Sadik & Sing Lee H, J Al Phys, 53 (1985) Ellison Hall J, Roe G, Wheadon S, Avset R, L Evensen, IEEE Transaction, 36 (1989) Kaun Yn Fu & Yukl Tsang, Motorola Inc Semiconductor roducts sector 350 Ed Bluesteein Blvd Austin, Th USA (1996). 13 Ahmad S, Freyer J & Claasen M, Solid State & Electron Devices, 4 (1977) Ahmad S, IEE Proc, 127 (1980) Mari De A, Solid State Electronics, 11 (1968) Canali C, Majni G, Minder R & Ohaviani G, IEEE Transaction on Electron Devices, 12 (1975) Karasek M, Solid state Electron, 19 (1976) Wright W & Sultan B N, Solid State Electron, 16 (1973) Vanoverschelde A & Salmer G, IEE J Solid State & Electron Devices, 2 (1978) Ahmad S & Freyer J, IEEE Electron Devices, 26 (1979) 1370.