# CEMTool Tutorial. Semiconductor physics

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1 EMTool Tutorial Semiconductor physics Overview This tutorial is part of the EMWARE series. Each tutorial in this series will teach you a specific topic of common applications by explaining theoretical concepts and providing practical examples. This tutorial is to demonstrate the use of EMTool for solving electronics problems. In this tutorial, a brief description of the basic concepts governing the flow of current in a PN junction is discussed. Both intrinsic and extrinsic semiconductors are discussed. The characteristics of depletion and diffusion capacitance are explored through the use of example problems solved with EMTool. The effect of doping concentration on the breakdown voltage of PN junctions is examined. Table of ontents 1. Intrinsic semiconductors 2. Extrinsic semiconductors 3. PN junction: contact potential, junction current 4. Depletion and diffusion capacitances 5. Breakdown voltages of PN junctions 1. Intrinsic semiconductors 1.1 Energy bands According to the planetary model of an isolated atom, the nucleus that contains protons and neutrons constitutes most of the mass of the atom. Electrons surround the nucleus in specific orbits. The electrons are negatively charged and the nucleus is positively charged. If an electron absorbs energy (in the form of a photon), it moves to orbits further from the nucleus. An electron transition from a higher energy orbit to a lower energy orbit emits a photon for a direct band gap semiconductor. The energy levels of the outer electrons form energy bands. In insulators, the lower energy band (valence band) is completely filled and the next energy band (conduction band) is completely empty. The valence and conduction bands are separated by a forbidden energy gap. 1

2 In conductors, the valence band partially overlaps the conduction band with no forbidden energy gap between the valence and conduction bands. In semiconductors the forbidden gap is less than 1.5eV. Some semiconductor materials are silicon (Si), germanium (Ge), and gallium arsenide (GaAs). Figure 1 shows the energy level diagram of silicon, germanium and insulator (carbon). Figure 1: Energy level diagram of (a) Silicon, (b) Germanium 1.2 Mobile carriers Silicon is the most commonly used semiconductor material in the integrated circuit industry. Silicon has four valence electrons and its atoms are bound together by covalent bonds. At absolute zero temperature the valence band is completely filled with electrons and no current flow can take place. As the temperature of a silicon crystal is raised, there is increased probability of breaking covalent bonds and freeing electrons. The vacancies left by the freed electrons are holes. The process of creating free electron-hole pairs is called ionization. The free electrons move in the conduction band. The average number of carriers (mobile electrons or holes) that exist in an intrinsic semi-conductor material may be found from the mass-action law: é E g ù ê- ú 1.5 ë kt û ni AT e = (1) where: T is the absolute temperature in o K k is Boltzmann s constant ( E g is the width of the forbidden gap in ev. k J ev K K E is 1.21 and 1.1eV for Si at 0 o K and 300 o K, = or ) respectively. It is given as Eg = Ec - Ev (2) where A is a constant dependent on a given material and it is given as 3 * * æ m ( ) m n ö p A = p m0k h ç m0 m è 0 ø h is Planck s constant ( m 0 is the rest mass of an electron h -34 = Js or g evs ) (3) 2

3 m n * is the effective mass of an electron in a material m p * is effective mass of a hole in a material The mobile carrier concentrations are dependent on the width of the energy gap, Eg, measured with respect to the thermal energy kt. For small values of T (kt<<e g ), n i is small implying, there are less mobile carriers. For silicon, the equilibrium intrinsic concentration at room temperature is ni 10 3 = electrons / cm (4) Of the two carriers that we find in semiconductors, the electrons have a higher mobility than holes. For example, intrinsic silicon at 300 o K has electron mobility of 1350 cm 2 /volt-sec and hole mobility of 480 cm 2 /volt-sec. The conductivity of an intrinsic semiconductor is given by ( p ) s = q n m + m i i n i p where q is the electronic charge (1.6 x ) n i is the electron concentration p i is the hole concentration. p i = n i for the intrinsic semiconductor µ n electron mobility in the semiconductor material µ p hole mobility in the semiconductor material. Since electron mobility is about three times that of hole mobility in silicon, the electron current is considerably more than the hole current. The following example illustrates the dependence of electron concentration on temperature. (5) Example 1: Given that at T= 300 o K, the electron concentration in silicon is 1.52 x 1010 electrons /cm 3 and E g = 1.1 ev at 300 o K. (a) Find the constant A of Equation (1). (b) Use EMTool to plot the electron concentration versus temperature. Solution 10 From Equation (1), we have = A( 300) 1.5 e é /300*8.62*10 ù ë û We use EMTool to solve for A. The width of energy gap with temperature is given as. E g ( T ) 2-4 æ T ö = ç è T ø Using Equations (1) and (6), we can calculate the electron concentration at various temperatures. EMTool script % alculation of the constant A k = 8.62e-5; na = 1.52e10; ta = 300; ega = 1.1; ka = -ega/(k*ta); t32a = ta.^1.5; A = na/(t32a*exp(ka)); fprintf("constant A is"), A (6) 3

4 % Electron oncentration vs. temperature for( i = 1;i<=10;i++) { t(i) = *(i-1); eg(i) = e-4*(t(i)*t(i))/(t(i) + 636); t32(i) = t(i).^1.5; ni(i) = A*t32(i)*exp(-eg(i)/(k*t(i))); } semilogy(t,ni) title("electron oncentration vs. Temperature") xlabel("temperature, K") ylabel("electron oncentration, cm-3") Result for part (a) constant A is A = e+024 Figure 2 shows the plot of the electron concentration versus temperature. Figure 2: Electron concentration versus temperature 4

5 2. Extrinsic semiconductors Electron and hole concentrations Extrinsic semiconductors are formed by adding specific amounts of impurity atoms to the silicon crystal. An n-type semiconductor is formed by doping the silicon crystal with elements of group V of the periodic table (antimony, arsenic, and phosphorus). The impurity atom is called a donor. The majority carriers are electrons and the minority carriers are holes. A p-type semiconductor is formed by doping the silicon crystal with elements of group III of the periodic table (aluminum, boron, gallium, and indium). The impurity atoms are called acceptor atoms. The majority carriers are holes and minority carriers are electrons. In a semiconductor material (intrinsic or extrinsic), the law of mass action states that where pn = constant (7) p n is the hole concentration For intrinsic semiconductors, and Equation (5) becomes is the electron concentration. and n i is given by Equation (1). p = n = ni (8) 2 pn = n i (9) The law of mass action enables us to calculate the majority and minority carrier density in an extrinsic semiconductor material. The charge neutrality condition of a semiconductor implies that where p + N = n + N (10) N D N A p n D A is the donor concentration is the acceptor concentration is the hole concentration is the electron concentration. In an n-type semiconductor, the donor concentration is greater than the intrinsic electron concentration, i.e., N D is typically cm -3 and n i = 1.5 x10 10 cm -3 in Si at room temperature. Thus, the majority and minority concentrations are given by n p n N (11) n N D 2 i (12) D In a p-type semiconductor, the acceptor concentration N A is greater than the intrinsic hole concentration p i =n i. Thus, the majority and minority concentrations are given by p n P N (13) n N 2 i A A (14) 5

6 The following example gives the minority carrier as a function of doping concentration. Example 2 For an n-type semiconductor at 300 o K, if the doping concentration is varied from to atoms/cm3, determine the minority carriers in the doped semiconductors. Solution From Equation (11) and (12), Electron concentration = N D and Hole concentration = where ni = electrons / cm 10 3 The EMTool program is as follows: % hole concentration in a n-type semiconductor nd = logspace(13,18); n = nd; ni = 1.52e10; ni_sq = ni*ni; p = ni_sq./nd; semilogx(nd,p,"b") title("hole concentration") xlabel("doping concentration, cm-3") ylabel("hole concentration, cm-3") Figure 3 shows the hole concentration versus doping. n N 2 i D 6

7 Figure 3: Hole concentration in N-type semiconductor (Si) 3. PN junction: contact potential, junction current 3.1 ontact potential An ideal pn junction is obtained when a uniformly doped p-type material abruptly changes to n- type material. This is shown in Figure 4. Practical pn junctions are formed by diffusing into an n-type semiconductor a p-type impurity atom, or vice versa. Because the p-type semiconductor has many free holes and the n-type semiconductor has many free electrons, there is a strong tendency for the holes to diffuse from the p-type to the n-type semiconductors. Similarly, electrons diffuse from the n-type to the p-type material. When holes cross the junction into the n-type material, they recombine with the free electrons in the n-type. Similarly, when electrons cross the junction into the p-type region, they recombine with free holes. In the junction a transition region or depletion region is created. 7

8 Figure 4: Ideal pn Junction (a) structure, (b) concentration of Donors (Nd) and acceptor (Na) impurities In the depletion region, the free holes and electrons are many magnitudes lower than holes in p- type material and electrons in the n-type material. As electrons and holes recombine in the transition region, the region near the junction within the n-type semiconductor is left with a net positive charge. The region near the junction within the p-type material will be left with a net negative charge. This is illustrated in Figure 5. Because of the positive and negative fixed ions at the transition region, an electric field is established across the junction. The electric field creates a potential difference across the junction, the potential barrier. The latter is also called diffusion potential or contact potential, V. The potential barrier prevents the flow of majority carriers across the junction under equilibrium conditions. The contact potential, V, may be obtained from the relations n n ( -V ) éq VG S ù ê ú n kt p ë û p = e = (15) p p n The following example illustrates the effect of source voltage on the junction potential. 8

9 Figure 5: pn junction (a) depletion region with positive and negative lons (b) energy band diagram near a pn junction Example 3 For a Silicon pn junction with n = cm - at T = 300 K i N cm D = 10 and N cm A = 10 and with (a) alculate the contact potential. (b) Plot the junction potential when the source voltage VS of Figure 6 increases from -1.0 to 0.7 V. Solution: EMTool script % Junction potential versus source voltage % using equation(15) contact potential is t = 300; na = 1.0e17; nd = 1.0e14; nisq = 1.04e20; q = 1.602e-19; k = 1.38e-23; % calculate contact potential vc = (k*t/q)*(log(na*nd/nisq)) 9

10 vs = -1.0:0.7:0.1; jct_pot = vc - vs; % plot curve plot(vs,jct_pot) title("junction potential vs. source voltage") xlabel("source voltage, V") ylabel("junction potential, V") Figure 6: PN junction (a) with depletion layer and source connection (b) contact potential with no source voltage (c) junction potential for forward-biased pn junction and (d) junction potential for reversed-biased pn junction (a) The contact potential is vc = (b) Figure 7 shows the graph of the junction potential versus the source voltage. 10

11 Figure 7: Junction potential versus source voltage 3.2 Junction current æ qv s ö ç è kt ø The pn junction current is given as I = I e -1 where V S is the voltage across the pn junction [see Figure 6 (a)] q is the electronic charge T is the absolute temperature k is Boltzmann s constant I S is reverse saturation current. It is given as where A is the diode cross-sectional area Lp, Ln are the hole and electron diffusion lengths p n n p, are the equilibrium minority carrier concentrations s é ê êë ù ú úû I S æ Dp pn Dn p ö p = qa + ç Lp L è n ø (16) (17) 11

12 Dp, Dn, are the hole and electron diffusion coefficients, respectively. The following example shows how I S is affected by temperature. Example 4 A silicon diode has I S = A at 25 o and assuming I S increases by 15% per o rise in temperature, find and plot the value of I S from 25 o to 125 o. Solution From the information given above, the reverse saturation current can be expressed as I S = ( ) ( - 25) T EMTool is used to find I S at various temperatures. % Saturation current % t = 25:125:5; is = 1.0e-15*(1.15).^(t-25); plot(t,is) title("reverse Saturation urrent vs. Temperature") xlabel("temperature, ") ylabel("urrent, A") Figure 8 shows the effect of temperature on the reverse saturation current. 12

13 Figure 8: Reverse saturation current versus temperature 4. Deplention and diffusion capacitances 4.1 Depletion capacitance As mentioned previously, a pn junction is formed when a p-type material is joined to an n-type region. During device fabrication, a p-n junction can be formed using process such as ionimplantation diffusion or epitaxy. The doping profile at the junction can take several shapes. Two popular doping profiles are abrupt (step) junction and linearly graded junction. In the abrupt junction, the doping of the depletion region on either side of the metallurgical junction is a constant. This gives rise to constant charge densities on either side of the junction. This is shown in Figure 9. 13

14 Figure 9: PN junction with abrupt junction (a) depletion region (b) charge density (c) electric field and (d) potential distribution In general, we may express the depletion capacitance of a pn junction by j j0 1 1 = m m é V ù 3 2 S ê1 - V ú ë û where j0 = zero-biased junction capacitance. V is contact potential VS is source voltage m=1/3 for linearly graded junction and m =1/2 for step junction Equations 18 are, strictly speaking, valid under the conditions of reversed-biased V S <0. The equations can, however, be used when V S <0.2V. The positive voltage, V, is the contact potential of the pn junction. As the pn junction becomes more reversed biased (V S < 0), the depletion capacitance decreases. However, when the pn junction becomes slightly forward biased, the capacitance increases rapidly. This is illustrated by the following example. (18) 14

15 Example 5 For a certain pn junction, with contact potential 0.065V, the junction capacitance is 4.5 pf for V S = -10 and j is 6.5 pf for V S = -2V. (a) Find m and j0 of Equation (18). (b) Use EMTool to plot the depletion capacitance from -30V to 0.4V. Solution From Equation 18 therefore j0 j1 = m é VS1 ù ê1 - ë V j1 év -V ù S 2 = ê ú j2 ëv -VS1 û m ú û à and j0 j2 = m é VS 2 ù ê1 - V ú ë û é ù j1 log10 ê ú ê j2 ú é V ù S1 m = ë û and j0 = j1 ê1 - ú év V -V ù S 2 log ë û 10 ê V V ú ë - S1 û EMTool is used to find m and j0. It also used to plot the depletion capacitance. m (19,20) EMTool script % depletion capacitance % cj1 = 4.5e-12; vs1 = -10; cj2 = 6.5e-12; vs2 = -2; vc = 0.65; num = cj1/cj2; den = (vc-vs2)/(vc-vs1); m = log10(num)/log10(den); cj0 = cj1*(1 - (vs1/vc))^m; vs = -30:0.4:0.2; k = length(vs); for (i = 1;i<=k;i++) cj(i) = cj0/(1-(vs(i)/vc))^m; plot(vs,cj) xlabel("voltage,v") ylabel("apacitance,f") title("depletion apacitance vs. Voltage") axis([-30,2,1e-12,14e-12]) 15

16 (a) The values of m, j0 are m = cj0 = e-012 (b) Figure 10 shows the depletion capacitance versus the voltage across the junction. Figure 10: Depletion capacitance of a pn junction 4.2 Diffusion capacitance When a pn junction is forward biased, holes are injected from the p-side of the metallurgical junction into the n-type material. The holes are momentarily stored in the n-type material before they recombine with the majority carriers (electrons) in the n-type material. Similarly, electrons are injected into and temporarily stored in the p-type material. The electrons then recombine with the 16

17 majority carriers (holes) in the p-type material. The diffusion capacitance, d, is due to the buildup of minority carriers charge around the metallurgical junction as the result of forward biasing the pn junction. hanging the forward current or forward voltage, V, will result in the change in the value of the stored charge Q, the diffusion capacitance, d, can be found from the general DQ expression d = (21) D V It turns out that the diffusion capacitance is proportional to the forward-biased current. That is = K I (22) d d DF where Kd is constant at a given temperature I DF is forward-biased diode current. The diffusion capacitance is usually larger than the depletion capacitance. Typical values of d ranges from 80 to 1000 pf. A small signal model of the diode is shown in Figure 11. Figure 11: Small-signal model of forward-biased pn junction In Figure 11, d and j are the diffusion and depletion capacitance, respectively. R S is the semiconductor bulk and contact resistance. The dynamic resistance, rd, of the diode is given as nkt rd = (23) qi DF where n is constant k is Boltzmann s constant T is temperature in degree Kelvin q is electronic charge. When a pn junction is reversed biased, d = 0. The model of the diode is shown in Figure

18 Figure 12: Model of a reverse-biased pn junction In Figure 12, j is the depletion capacitance. The diffusion capacitance is zero. The resistance Rd is reverse resistance of the pn junction (normally in the mega-ohms range). Example 6 A certain diode has contact potential; V = 0.55V, j0 = diffusion capacitance at zero biased is 8 pf; the diffusion capacitance at 1mA is 100 pf. Use EMTool to plot the diffusion and depletion capacitance for forward-biased voltages from 0.0 to 0.7 V. Assume that I S =10-14 A, n = 2.0 and stepjunction profile. Solution Using Equations (16) and (22), we write the EMTool program to obtain the diffusion and depletion capacitance. % Diffusion and depletion apacitance cd1 = 100e-12; id1 = 1.0e-3; cj0 = 8e-12; vc =0.55; m = 0.5; is = 1.0e-14; nd = 2.0; k = 1.38e-23; q = 1.6e-19; T = 300; kd = cd1/id1; vt = k*t/q; v = 0.0:0.5:0.05; nv = length(v); for (i = 1;i<=nv;i++) { id(i) = is*exp(v(i)/(nd*vt)); cd(i) = kd*id(i); ra(i) = v(i)/vc; cj(i) = cj0/((1 - ra(i)).^m); 18

19 } subplot(1,2,1) plot(v,cd) title("diffusion ap.") xlabel("voltage, V"), ylabel("apacitance, F") subplot(1,2,2) plot(v,cj) title("depletion ap.") xlabel("voltage, V"), ylabel("apacitance, F") Figure 13 shows the depletion and diffusion capacitance of a forward-biased pn junction. Figure 13: (a) Depletion and (b) diffusion capacitance 19

20 5. Breakdown voltages of PN junctions The electric field E is related to the charge density through the Poisson s equation de x ( ) r ( x) dx = (24) e e S 0 where ε S is the semiconductor dielectric constant ε 0 is the permittivity of free space, 8.86 * F/cm ρ(x) is the charge density. The maximum electric field can be shown to be E = aq 2 W (25) where a is slope of charge density and W is width of depletion layer. For an abrupt junction, where one side is heavily doped, the electrical properties of the junction are determined by the lightly doped side. Experimentally, the breakdown voltage of semiconductor step junction ( n+p or p+n ) as the function of doping concentration in the lightly doped side is given as V é NB ù BR = k ê 16 ú ë10 û where k=25v for Ge and 60V for Si. N B is the doping concentration of lightly doped side. The following example shows the effect of doping concentration on breakdown voltage. max 8 S e e 0 (26) Example 7 Use EMTool to plot the breakdown voltage versus doping concentration for a one-sided step junction for silicon and germanium, and using doping concentration from to cm -3 Solution Using Equation 26, we calculate the breakdown voltage for various doping concentration. % % Breakdown voltage % k1 = 25; k2 = 60; nb = logspace(14,19); n = length(nb); for( i = 1;i<=n ; i++) { vbr1(i) = k1*(nb(i)/1.0e16)^(-0.75); % Ge breakdown voltage vbr2(i) = k2*(nb(i)/1.0e16)^(-0.75); % Si breakdown voltage 20

21 } semilogx(nb,vbr1, nb,vbr2) xlabel("impurity oncentration, cm-3") ylabel("breakdown Voltage,V") title("breakdown Voltage vs. Impurity oncentration") legend("ge","si") Figure 14 shows the plot of breakdown voltage of one-sided abrupt junction. Figure 14: Breakdown voltage versus impurity concentration References 1. EMTool 6.0 User s Guide 21

22 2. John O. Attia, Electronics and ircuit analysis using MATLAB, R Press, first edition

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