Here Is denotes the reverse saturation current. In practical junctions, the p region is usually much more heavily inEq.(2233 Is=qA DpPn/W=qA Dp n//WnND (2234) The reverse saturation current in short diodes is mainly determined by the diffusion constant D, and the width Wh of the n region, by intrinsic concentration n, by the doping concentration Np in the n region, and by the diode area A;. (In reality, Is is also slightly dependent on the reverse voltage[Phillips, 1962J-) If Vp is made positive, the exponential term in Eq (22.32)rapidly becomes larger than one; thus Ip=Is exp Vp/VI (22.35) where I, is the diode forward current and Is is the reverse saturation current Another mechanism predominates the reverse current Is in silicon. Due to the recombination centers in the depletion region, a generation-recombination hole-electron current IG is generated in the depletion region Phillips, 1962; Sze, 1985] IG=KgA Here e is the generation rate unit volume, A is the junction area, q is the elementary charge, Xd is the depletion layer thickness, and K is a dimensional constant. IG is proportional to the thickness X, of the depletion layer and to the junction area A; Since Xa increases with the square root of the reverse voltage, IG increases accordingly, yielding a slight slope in the reverse I-V characteristic. The forward I-V characteristic of the practical diode is only slightly affected (slope m=2)at very small forward currents(In=I nA to 1 HA). In practical diodes n I at small to medium currents(Ip=1 HA to 10 mA). At large currents(Ip>10 mA), m=1 to 2 due to the high current effects [Phillips, 1962] and due to the series bulk resistance of the diode The reverse current Ig in silicon is voltage dependent. The predominant effect is the voltage dependence of the generation-recombination current IG and to a smaller extent the voltage dependence of Is. The total reverse current of the diode is thus equal to (2236b) Forward-Biased Diode For most practical applications Ip=Is exp vp/mVr where Is is the reverse saturation current(about 10-4 A for a small-signal diode);V=kT/q is the thermal voltage equal to :6 mv at room temperature;k= Boltzmanns constant 1.38. J/K; T is the absolute temperature in kelvin; g is the elementary charge 1.602. 9C; m is the ideality factor, m=1 for medium currents, m=2 for very small and very large currents; I s is part of the total reverse current Ig of the diode IR=Is+ I c and Is is the reverse saturation current and IG is the generation recombination current, also called diode leakage current because IG is not a part of the carrier diffusion process in the diode Ip is exponentially related to Vp in Fig. 22.10 FIGURE 22.10 Ip versus Vp of a diode. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Here IS denotes the reverse saturation current. In practical junctions, the p region is usually much more heavily doped than the n region; thus np << pn. Also, since Wn << Lnp in Eq. (22.33), we obtain IS = qAj Dp pn /Wn = qAjDp ni 2/Wn ND (22.34) The reverse saturation current in short diodes is mainly determined by the diffusion constant Dp and the width Wn of the n region, by intrinsic concentration ni , by the doping concentration ND in the n region, and by the diode area Aj . (In reality, IS is also slightly dependent on the reverse voltage [Phillips, 1962].) If VD is made positive, the exponential term in Eq. (22.32) rapidly becomes larger than one; thus ID = IS expVD /VT (22.35) where ID is the diode forward current and IS is the reverse saturation current. Another mechanism predominates the reverse current IS in silicon. Due to the recombination centers in the depletion region, a generation-recombination hole–electron current IG is generated in the depletion region [Phillips, 1962; Sze, 1985]. IG = KqAj eXd (22.36a) Here e is the generation rate unit volume, Aj is the junction area, q is the elementary charge, Xd is the depletion layer thickness, and K is a dimensional constant. IG is proportional to the thickness Xd of the depletion layer and to the junction area Aj . Since Xd increases with the square root of the reverse voltage,IG increases accordingly, yielding a slight slope in the reverse I-V characteristic. The forward I-V characteristic of the practical diode is only slightly affected (slope m = 2) at very small forward currents (ID = 1 nA to 1 mA). In practical diodes n ª 1 at small to medium currents (ID = 1 mA to 10 mA). At large currents (ID > 10 mA), m = 1 to 2 due to the high current effects [Phillips, 1962] and due to the series bulk resistance of the diode. The reverse current IR in silicon is voltage dependent. The predominant effect is the voltage dependence of the generation-recombination current IG and to a smaller extent the voltage dependence of IS. The total reverse current of the diode is thus equal to IR = IG +IS (22.36b) Forward-Biased Diode For most practical applications ID = IS expVD /mVT (22.37) where IS is the reverse saturation current (about 10–14 A for a small-signal diode); VT = kT/q is the thermal voltage equal to 26 mV at room temperature; k = Boltzmann’s constant, 1.38 · 10–23 J/K; T is the absolute temperature in kelvin; q is the elementary charge 1.602·10–19 C; m is the ideality factor, m = 1 for medium currents, m = 2 for very small and very large currents; IS is part of the total reverse current IR of the diode IR = IS + IG; and IS is the reverse saturation current and IG is the generationrecombination current, also called diode leakage current because IG is not a part of the carrier diffusion process in the diode. ID is exponentially related to VD in Fig. 22.10. FIGURE 22.10 ID versus VD of a diode
62>81 FIGURE 22.11 (a) Ip versus Vp of a diode at three different temperatures 8, >82>8,(b)Vo =f(Temp), Ipc >IDB >Ipa- Temperature Dependence of Vp Equation(22.37)solved for Vp yields Vp=mVr In(Ip/Is) (2238) at constant current Ip, the diode voltage Vp is temperature dependent because Vr and Is are temperature dependent Assume m= 1. The reverse saturation current Is from Eq(22.34)is Is=qAn D/WnND=B1n:D,=B2n;Hp where D,= VrL. with u,= B,T-and for n2 n?=B,TY exp(-VG/Vr) (2239) where y=4-n, and VGo is the extrapolated bandgap energy [Gray and Meyer, 1993]. With Eq (22.39)into Eq(22.38), the derivative dvp /dT for Ip= const yields dalat -VGO)/T-yk1 q (2240) At room temperature(T= 300 K), and Vp=0.65V, VGo=1. 2 V, Y=3, Vr=26 mV, and k/q=86 uv/degree one gets dvp/dT=-2.1 mv/degree. The temperature coefficient TC of Vp is thus TC=dVD/Vp dT=1/T-VGo/VpT-yk/g vp 22.41) For the above case TC=-032%/degree. In practical applications it is more convenient to use the expression V(62)=V(81)-TC(62-81) where 8, and 8, are temperatures in degrees Celsius. For TC =-0.32%/degree and V,=0.65 V at 8,=270 Vp=0.618 V at 82=37C. Both dvp/dT and TC are Ip dependent. At higher Ip, both dv/dT and TC are smaller than at a lower Ip, as shown in Fig. 22.11 Ip- Vp Characteristic From the I-V characteristic of the diode one can find for m=1 IDI Is exp(Vp/vr)and Is exp(Vp /Vr) 22.43 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Temperature Dependence of VD Equation (22.37) solved for VD yields VD = mVT ln(ID /IS) (22.38) at constant current ID , the diode voltage VD is temperature dependent because VT and IS are temperature dependent. Assume m = 1. The reverse saturation current IS from Eq. (22.34) is IS = qAj ni 2Dp /Wn ND = B1ni 2Dp = B2ni 2mp where Dp = VT mp. With mp = B3T –n and for ni 2 ni 2 = B4T g exp(–VG0 /VT) (22.39) where g = 4 – n, and VG0 is the extrapolated bandgap energy [Gray and Meyer, 1993]. With Eq. (22.39) into Eq. (22.38), the derivative dVD /dT for ID = const yields dVD /dT = (VD – VG0)/T – gk/q (22.40) At room temperature (T = 300 K), and VD = 0.65 V, VG0 = 1.2 V, g = 3, VT = 26 mV, and k/q = 86 mV/degree, one gets dVD /dT ª –2.1 mV/degree.The temperature coefficient TC of VD is thus TC =dVD /VD dT = 1/T – VG0 /VD T – gk/qVD (22.41) For the above case TC ª –0.32%/degree. In practical applications it is more convenient to use the expression VD (d2) = VD (d1) – TC(d2 – d1) (22.42) where d1 and d2 are temperatures in degrees Celsius. For TC = –0.32%/degree and VD = 0.65 V at d1 = 27°C, VD = 0.618 V at d2 = 37°C. Both dVD /dT and TC are ID dependent. At higher ID , both dVD /dT and TC are smaller than at a lower ID , as shown in Fig. 22.11. ID-VD Characteristic From the ID-VD characteristic of the diode one can find for m = 1 ID1 = IS exp(VD1 /VT) and ID2 = IS exp(VD2/VT) (22.43) FIGURE 22.11 (a) ID versus VD of a diode at three different temperatures d3 > d2 > d1. (b) VD = f(Temp), IDC > IDB > IDA