CHAPTER 4: P-N JUNCTION Part 2

C – V CHARACTERISTICS (18) (19) Eq(18) for depletion capacitance/area is the same as for a parallel-plate capacitor, where spacing between two plates represents the depletion width. It’s valid for any arbitrary impurity distribution. For one-sided abrupt junction: (18) (19)

EVALUATION OF IMPURITY DISTRIBUTION For the p+ - n junction with an arbitrary impurity distribution, the corresponding charge in applied voltage (shaded area) in Fig. 4.14(c) is The expression for the impurity concentration at the edge of the depletion region: For linearly graded junction, the depletion layer capacitance is (20) (21) (22)

Figure 4.14. (a) p+-n junction with an arbitrary impurity distribution. (b) Change in space charge distribution in the lightly doped side due to a change in applied bias. (c) Corresponding change in the electric-field distribution.

VARACTOR Varactor (variable reactor) – the p-n junction that employ the voltage-variable properties of reverse-biased. Reverse-biased depletion capacitance is given by where for linearly graded junction, n = 1/3, and abrupt junction, n = ½. Fig. 4.15 – shows 3 p+ - n doping profiles with the donor distribution ND(x) given by B(x/x0)m. B and x0 – constants. m = 1 (for linearly graded junction) and m = 0 (for abrupt junction), and m = -3/2 (for hyperabrupt junction). The hyperabrupt profile – from epitaxial growth techniques (23)

Figure 4-15. Impurity profiles for hyperabrupt, one-sided abrupt, and one-sided linearly graded junctions.

VARACTOR (cont.) (24) for n = 2 (25) By solving Poisson’s equation with the appropriate boundary conditions, therefore From eq. Cj above, thus n = 1/(m+2), and for hyperabrupt junction, n > ½, and m = negative number. When this varactor is connected to an inductor L in a resonant circuit, the resonant frequency varies linearly with the voltage applied to the varactor, where (24) for n = 2 (25)

I – V CHARACTERISTICS By applying voltage to a p-n junction – it may disturb the precise balance between diffusion current & drift current of electrons and holes. From Fig. 4.16: - Forward Bias: the applied voltage reduces the electrostatic potential across the depletion region (middle of Fig. 4.16(a)). Drift current is reduced in comparison to the diffusion current - enhanced hole diffusion from p-side to n-side and electron diffusion from n-side to p-side. Reverse Bias: the applied voltage increases the electrostatic potential across the depletion region (middle of Fig. 4.16(b)). Greatly reduced the diffusion currents – resulting small reverse current. Note: minority carrier densities at the boundaries (-xp and xn) increase substantially above their equilibrium values under forward bias and decrease for reverse bias.

Figure 4-16. Depletion region, energy band diagram and carrier distribution. (a) Forward bias. (b) Reverse bias.

IDEAL CHARACTERISTICS Ideal I-V characteristics are derived based on: (a) The depletion region has abrupt boundaries/outside the boundaries – the s/c is assumed to be neutral. (b) Carriers densities at the boundaries are related by the electrostatic potential difference across the junction. (c) The majority carrier densities are changed negligibly at the boundaries of neutral regions by the applied bias. (d) Neither generation nor recombination current exists in the depletion region, and the electron and hole currents are constant throughout the depletion region. In ideal case, the expression for the built-in potential may be rewrite as (26) Where nno and ppo – equilibrium electron densities in the n and p sides respectively

IDEAL CHARACTERISTICS (cont.) The mass action law nnoppo = ni2, and by rearrange eq (26), thus Electron & hole density at two boundaries of the depletion region are related through the electrostatic potential difference Vbi at thermal equilibrium. When forward bias : Vbi – VF. While for reverse bias: Vbi + VR, thus for both cases: where nn and np – are the nonequilibrium densities at the boundaries of the depletion region in the n and p sides respectively (+V – forward bias, and –V for reverse bias). and (27) (28)

IDEAL CHARACTERISTICS (cont.) For low injection condition, nn ~ nno, thus at x = -xp (29) and at x = xn (30) THESE EQUATIONS ARE THE MOST IMPORTANT BOUNDARY CONDITIONS FOR THE IDEAL I-V CHARACTERISTICS.

IDEAL CHARACTERISTICS (cont.) Fig. 4.17 – illustrated the injected minority carriers recombine with the majority carriers as minority carriers move away from the boundaries. Electron and hole currents shown in the bottom of Fig. 4.17. Hole diffusion current will decay exponentially in the n-region with diffusion length Lp, and electron diffusion behave the same as hole diffusion but at p-region with Ln. The total current is constant throughout the devices and represents ideal diode equation (Generally for Ge p-n junction): (31) Where Js – saturation current density, and defined as (32) The ideal I-V characteristic is shown in Fig. 4.18(a) and (b) for Cartesian and semilog plots.

Figure 4-17. Injected minority carrier distribution and electron and hole currents. (a) Forward bias. (b) Reverse bias. The figure illustrates idealized currents. For practical devices, the currents are not constant across the space charge layer.

Figure 4-18. Ideal current-voltage characteristics. (a) Cartesian plot Figure 4-18. Ideal current-voltage characteristics. (a) Cartesian plot. (b) Semilog plot.

GENERATION-RECOMBINATION & HIGH- INJECTION EFFECTS For Si and GaAs p-n junction the ideal equation (eq. 31) can give qualitative agreement because of generation or recombination of carrier in the depletion region. For reverse bias case with large values of ni, i.e Ge, the diffusion current dominates at T = 300K, and the reverse current follows the ideal diode equation, but if ni <<<, i.e Si and GaAs the generation current in the depletion region may dominate, and the total reverse current for p+ - n junction (for NA >> ND and for VR > 3kT/q): (33) and g – generation lifetime, and for simple case n = p = o, the rate of electron hole-pair generation, G is (34)

GENERATION-RECOMBINATION & HIGH- INJECTION EFFECTS (cont.) For forward bias, concentration of both electrons and holes exceed their equilibrium values. The carriers will attempt to return to their equilibrium values by recombination – the dominant generation-recombination processes in the depletion region are the capture processes. The total forward current (for pno >> npo, and V > 3kT/q) is (35) Generally, the experimental results can be represented empirically by (36) - ideality factor. Ideal diffusion current dominant with  = 1, and for recombination current dominant,  = 2.

GENERATION-RECOMBINATION & HIGH- INJECTION EFFECTS (cont.) Fig. 4.19 – measurement of forward characteristics of a Si and GaAs p-n junction at T = 300K. At low current levels, recombination current dominates and  = 2, while at higher current levels, diffusion current dominates with  = 1. For higher current levels, it increases more gradually with forward current and caused by two effects: series resistance & high injection. For series resistance effect: at low- and medium current levels, IR drop across the neutral regions and it usually compared with kT/q (I – forward current, R – series resistance). This IR drop reduces the bias across the depletion region, the current becomes (37)

Figure 4-19. Comparison of the forward current-voltage characteristics of Si and GaAs diodes at 300 K. Dashed lines indicate slopes of different ideality factors η.

EXERCISE An ideal Si p-n junction has ND = 1018cm-3, NA = 1016cm-3, ni=9.65x109, Dp=10, Dn=21, p=n=10-6s, and the device area of 1.2 x 10-5cm2. Calculate the theoretical saturation current at T = 300K.

The saturation current calculation and from the cross-sectional area A = 1.210-5 cm2, we obtain

TEMPERATURE EFFECT (38) (39) Temperature has a profound effect on device performance. Both forward and reverse case, the magnitude of the diffusion and the recombination-generation currents depend strongly on temperature. Forward bias: The ratio of hole diffusion current to the recombination is given by The ratio depends on both temperature and the s/c band gap. From Fig. 4.20(a), the ideal diode equation will be followed over a wide range of forward biases as the temperature increases. The temperature dependence of saturation current density Js, where (38) (39)

TEMPERATURE EFFECT (cont.) Reverse bias: The ratio of diffusion current to the generation is given by The ratio is proportional to the ni, as the temperature increases, the diffusion current dominates. Fig. 4.20(b) – at low temperature, the generation current dominates and reverse current varies as (VR)1/2 for an abrupt junction (i.e W ~ (VR)1/2). As T > 175oC, the current demonstrates a saturation tendency for VR  3kT/q, at which point the diffusion current becomes dominates. (40)

Figure 4-20. Temperature dependence of the current-voltage characteristics of a Si diode. (a) Forward bias. (b) Reverse bias.

CONCLUSION The p-n junction is the basic building block for other s/c devices. Understanding of junction theory serve as the foundation to understanding other s/c devices. Modern p-n junctions are fabricated using “planar technology”. When p-n junction is formed – the uncompensated –ve ions (NA-) on the p-side and uncompensated +ve ion (ND+) on n-side. Thus, depletion region formed at the junction. At thermal equilibrium, the drift current (due to the electric field) balanced by diffusion current (due to concentration gradients of the mobile carriers). When +V applied to p-side – large current will flow through the junction, while when –V applied virtually no current flows. Practical devices depart from ideal characteristics because of carrier generation & recombination in the depletion layer, high injection under forward bias and series-resistance effect.

~ Ahmed Zewail ~ 1999 Chemistry Nobel Laureate “There is a beauty and IMPORTANCE in understanding fundamental science & not everything we do has to have immediate applications” ~ Ahmed Zewail ~ 1999 Chemistry Nobel Laureate

“An education isn't how much you have committed to memory, or even how much you know. It's being able to differentiate between what you do know and what you don't” ~ Anatole France (1844 - 1924) ~

Tonight Wednesday 13/8/2008 Test 1 (Chapters 1-4) in K. Perlis (DKP1) at 8.30pm-9.30pm