2.2.7 Volt - Ampere Characteristics 푽Τ푽푻 Equation 2.47 푰풋 = 푰풐 풆 − ퟏ 푬풒. ퟐ. ퟒퟕ indicates the junction current (IJ) related to the biased voltage (v). A positive value of (IJ) means that the current flows from P to N sides and the diode is in forward bias. Also it is known that the constant () in Eq.2.53 푽Τ휼 푽푻 푰 = 푰풐 풆 − ퟏ 푬풒. ퟐ. ퟓퟑ is equal 1 for germanium and equal 2 for silicon and (VT) is the thermal voltage and is given by: 푻 푽푻 = ൗퟏퟏퟔퟎퟎ 푬풒. ퟐ. ퟓퟓ At room temperature VT = 0.026 V. When V is positive and larger than VT, Eq.2.47 will be: 푽푭Τ휼푽푻 푰풋 = 푰풐 풆 푬풒. ퟐ. ퟓퟔ The junction current increased with voltage biased, and when the diode is reverse biased, Eq.2.47 will be: 푰풋 = 푰푶 = 푰푹 = 푰푺 푬풒. ퟐ. ퟓퟕ 1 Understanding the Value of built – in voltage VT 푻 푻 푽푻 = ൗퟏퟏퟔퟎퟎ 푬풒. ퟐ. ퟓퟓ 푽푻 = ൗퟏퟏퟔퟎퟎ 푬풒. ퟐ. ퟓퟓ At room temperature At room temperature ( 27 oC ) = VT = 0.026 V. o 0.026 = 푻Τ 27 + 273 = 300 K ퟏퟏퟔퟎퟎ ퟑퟎퟎ T = 11600 x 0.026 = 301.6 oK 푽푻 = ൗퟏퟏퟔퟎퟎ 301.6 – 273 = 28.6 oC 푽푻 =0.02586 2 I(ma) IF (ma) IF(ma) 5 5 For Ge For Si 80 Devices Devices 4 4 3 3 60 IO 2 2 40 1 1 10 mA VR VF 20 0.2 0.4 0.6 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1.00 V(Volt) VZ V V Fig. (2 – 19) I-V junction characteristics G Si Figure 2.20 difference in V-I characteristics for Si, Ge Form of Eq.2.47; figure (2-19), the reverse current is constant independent of VR , forward and reverse current is shown in figure (2.19b), the dashed portion of curve indicates, at reverse voltage (VZ), at this voltage large reverse current flow and the diode to be in breakdown mode 3 Difference in V - I characteristics V V for Si, Ge is shown in figure (2- 20), determine the cut-in, offset, IJ IJ break point or threshold voltage Rn RP VJ (V) below which forward current Fig. (2 - 21) Equivalent Circuit for a Diode is very small, less than 1% of its Since = 2 for Si, the current maximum value). The reverse increases as eV/2VT for the initial saturation current in Ge is in the bias and then increased as eV/VT. range of microamperes and the This initial dependence of reverse saturation current in Si is current on voltage accounts for the delay in the rise of Si device in the range of nano amperes. characteristics. 4 2.2.8 Diode Resistance Resistance R is defined as the ratio between voltage V and current I. At any point on the I-V characteristics of the diode, R is equal to the reciprocal of the slope of a line joining the operating point to the origin. In the specification sheet maximum VF required to a given IF and also VR for a given IR is indicated, for example, VF=0.8V at IF = 10 ma gives RF = 80 VR=50 V at IR = 0.1 a gives RR = 500 M In the non-ohm devices, the resistance drop depends upon the operating current, and in fact we have two possible resistances to consider. 5 One is the DC resistance; the other is the dynamic or small signal resistance (ac resistance). And useful equivalent circuit for the diode is shown in figure (2-21). The resistor RP and Rn represent the bulk resistance of the structure, the junction voltage is given by: 푽푱 = 푽 = 푰푱 푹풏 + 푹풑 ퟐ. ퟓퟖ The over all resistance of the junction is, 푽 푽푱 RT = = + 푹풏 + 푹풑 ퟐ. ퟓퟗ 푰푱 푰푱 푽푱 RT = + 푹풏 + 푹풑 푰푱 Or RT = 푹풅풄 + 푹풏 + 푹풑 ퟐ. ퟔퟎ The Dc resistance of the combination is 푹풅풄 = RT - (푹풏+ 푹풑) 6 Dc resistance of ideal part of diode, (rdc) determined graphically as shown in fig. (2.22) or it can be calculated from: 푽푱 푽푱 풓 = = ퟐ. ퟔퟏ 풅풄 푰 풒푽 푱 푰 풆풙풑 푱 − ퟏ 푺 푲푻 Dynamic resistance (ra) an important parameter and it is equal, 흏푽 풓 = ퟐ. ퟔퟐ 풂 흏푰 Dynamic resistance ra is not constant depends upon the voltage and we find that the dynamic conductance for PN ideal diode is given by: 푽ൗ ퟏ 흏푰 푰 풆 휼푽푻 푰 + 푰 품 = = = 풐 = 풐 ퟐ. ퟔퟑ 푹 흏푽 휼푽푻 휼푽푻 7 Ideal Diode For reverse bias, where V/VT<<1, conductance g = 1/R will be very small I and dynamic resistance r is very large. Rac For forward bias, where V/ VT >> 1, I >> I1RDC = V1/I1 I and dynamic resistance r is given by: rdc o V V1 휼푽푻 Fig. (2 - 22) Slopes and points which must 풓풂 = ퟐ. ퟔퟒ be selected to obtain the four distinct 푰 resistances for ideal and real diode Ac resistance rac of ideal diode determined from figure (2-22) as slope of C.C at the operating points or determined analytically as follows. It is convenient to calculate ac conductance (gac), 풒푽푱 푰 = 푰 풆풙풑 − ퟏ 푱 푺 푲푻 8 풅푰푱 풒 풒푽푱 Slope at 품 = = 푰 풆풙풑 ퟐ. ퟔퟓ Operating 풂풄 푺 I 풅푽푱 푲푻 푲푻 F point 1/RF 풒 ퟏ 품풂풄 = 푰푱 + 푰풔 = ퟐ. ퟔퟔ 푲푻 풓풂풄 Ac resistance of real diode is then, 푹풂풄 = 풓풂풄 + 푹풏 + 푹풑 ퟐ. ퟔퟕ 푲푻 ퟏ V VF 푹풂풄 = + 푹풏 + 푹풑 ퟐ. ퟔퟖ 풒 푰풔 + 푰푱 Fig. (2 - 24) Diode I-V At room temperature, value of Rac, Linear Approximation ퟐퟓ 풓풂풄 = 풐풉풎 ퟐ. ퟔퟗ 푰풔 + 푰푱 9 Note that in some cases the ohmic series resistance will be important and in other is not. At very high current the forward ideal diode resistance becomes negligible compared with the series resistance. Since the series resistance in the real diode C.C becomes ohmic at high currents. 2.2.9 Calculation of hole and electron currents current in the junction depend upon physical properties of the junction, properties of the crystal, density of impurities, impurity distribution, and many other factors. When junction reverse biased, current of electrons from n- region to p-region is essentially zero. Also at the p-side of the transition region, the electron density must be almost zero, since electrons arriving at the point are swept down the potential hill. 10 p- side There is a density of electrons n-side PP n in p-side of junction going n from equilibrium value far ni pi from junction, to zero, at edge Forward bias of transition region. A similar pn n situation is for holes in the n- p side of the junction. This is Reverse bias shown in figure (2 - 25). The Xn Xp forward direction is shown as Transition region well. It is observed that large Fig. (2 - 25) Carriers densities on the two sides of the transition region, the p-side number of electrons from the is more doped than n-side n-side enriches the majority carrier density on the p-side. 11 Density gradient is now in the opposite direction, and electron flow to the right. Density gradient within crystal near the junction are very important because there is no electric field in the crystal body itself and current is almost by diffusion due to density gradient. Let us consider the minority electron current in the p-side. Differential equation describes the density gradient of electrons in the p-side in the steady state is (i.e., the continuity equation). 풅풏풑 풏풑 − 풏풑풐 ퟏ 풅푰풏풑 = − + ퟐ. ퟕퟎ 풅풕 풏 풒 풅풙 Where, (np) is electron density as function of (x) as shown in figure (2-26). 12 Since field strength ε is negligible, we have only to consider diffusion current, of electrons Density 풅풏풑 Direction of forward current 푰풏풑 = 풒푫풏 ퟐ. ퟕퟏ 풅풙 n (0) = n exp qV /KT In steady state flow of current, we have p po J np(x) dn / dt = 0, then from equation (2.70) and p 1/e (2.71), we find, n 풏 −풏 풅ퟐ풏 po − 풑 풑풐 + 푫 풑 = ퟎ ퟐ. ퟕퟐ 풏 풏 풅풙ퟐ L From eq. (2.72) n 풅ퟐ풏 풅ퟐ풏 풏 풙 = 풏 + 푫 흉 풑 = 풏 푳ퟐ 풑 Fig. (2 - 26) Electron concentration 풑 풑풐 풏 풏 풅풙ퟐ 풑풐 풏 풅풙ퟐ in the p-side decreases with Where, 푳풏 = 푫풏흉풏 and since npo is increasing distance constant, then, 풅ퟐ 풏 − 풏 ퟐ 풑 풑풐 풏풑 풙 − 풏풑풐 = 푳풏 ퟐ 풅풙 13 General solution of equation is, 풙 풙 풏풑 풙 − 풏풑풐 = 푨풆풙풑 – + 푩풆풙풑 – ퟐ. ퟕퟑ 푳풏 푳풏 Where, A, and B are two constants which determined from the boundary conditions, np(x) = npo at x = infinity, consequently B = 0 because np would go to infinity for x → infinity np(0) = npo exp qVJ/KT at x = 0 , which gives 풒푽풋 풒푽풋 푨 = 풏 풆풙풑 − 풏 = 풏 풆풙풑 − ퟏ 풑풐 푲푻 풑풐 풑풐 푲푻 Then the solution to this equation is, 풒푽풋 −풙 풏풑 풙 = 풏풑풐 풆풙풑 − ퟏ 풆풙풑 + 풏풑풐 ퟐ. ퟕퟒ 푲푻 푳풏 The carriers diffuse down, decreasing density gradient, and on the average they move a distance (Ln) before recombining. 14 From Eq. (2.73) we see that the density of electrons in the p-type material depend upon the number being injected across the junction. This number is controlled by the bias voltage (VJ). As we move away from the junction the density of electrons becomes that of the steady state equilibrium density in the p-material (npo).