EXAMPLE 4.1 OBJECTIVE

1. EXAMPLE 4.1 • OBJECTIVE • Calculate the drift current density induced in a semiconductor for a given applied electric field. • Consider a silicon semiconductor at T = 300 K with an impurity doping concentration of Nd = 1016 cm-3 and Na = 0. Assume electron and hole mobilities given in Table 4.1. Calculate the drift current density for an applied electric field  = 35 V/cm. • Solution Since Nd > Na , the semiconductor is n type and, at room temperature, we can assume complete ionization so that n  Nd = 1016 cm-3 Then, the hole concentration is Since n >> p, the drift current density is Jdrf = e (nn + pp)   enn so Jdrf = (1.6  10-19)(1350)(1016)(35) = 75.6 A/cm2 • Comment • Significant drift current densities can be obtained in a semiconductor with relatively small applied electric fields. This result then implies that currents in the mA range can be generated in very small semiconductor devices. (in chip max 10 mA/um2)

2. EXAMPLE 4.2 • OBJECTIVE • Find the electron and hole mobilities in silicon at various temperatures. • (a) Find the electron mobility for (i) Nd = 1017 cm-3, T = 150C and for (ii) Nd = 1016 cm-3, T = 0C. (b) Find the hole mobility for (i) Na = 1016 cm-3, T = 50C and for (ii) Na = 1017 cm-3, T = 150C. • Solution Using Figure 4.2, we find (a) (i) For Nd = 1017 cm-3, T = 150C  n  500 cm2/V-s (a) (ii) For Nd = 1016 cm-3, T = 0C  n  1500 cm2/V-s (b) (i) For Na = 1016 cm-3, T = 50C  p  380 cm2/V-s (b) (ii) For Na = 1017 cm-3, T = 150C  p  200 cm2/V-s • Comment We can see that the mobility decreases as the temperature increases.

3. EXAMPLE 4.3 • OBJECTIVE • Determine the required impurity doping concentration in silicon at T = 300 K to produce a semiconductor resistor with specified current-voltage characteristics. • Consider a bar of silicon uniformly doped with acceptor impurities and having the geometry shown in Figure 4.5. For an applied voltage of 5 V, a current of 2 mA is required. The current density is to be no larger than Jdrf = 100 A/cm2. Find the required cross-sectional area, length, and doping concentration. • Solution The cross-sectional area can be found as The resistance of the device is From Equation (4.22b), the resistance, for the bar of semiconductor, is given by From this relation, we see that there is no unique solution for Na and L. If we choose a very small value of L, then Na might be unreasonably small. On the other hand, if we choose a very large value of L, then Na might be unreasonably large. Hence, we will choose a value of the doping concentration and then determine the required length. Let Na = 1016 cm-3. From Figure 4.3, we find p  400 cm2/V-s. The device length is then found to be L = AR = epNaAR = (1.6  10-19)(400)(1016)(2  10-5)(2.5  103) or L = 3.2  10-2 cm • Comment • We must make sur that we use the mobility that corresponds to the doping concentration in our analysis and design.

4. EXAMPLE 4.4 • OBJECTIVE • Design a p-type semiconductor resistor with a specified resistance to handle a given current density. • A silicon semiconductor at T = 300 K and with the geometry shown in Figure 4.5 is initially doped with donors at a concentration of Nd = 5  1015 cm-3. Acceptors are to added to form a compensated p-type material. The resistor is to have a resistance of R = 10 k, handle a current density of Jdrf = 50 A/cm2 when 5 V is applied, and have an applied electric field no larger than  = 100 V/cm. • Solution For 5 V applied to a 10-k resistor, the total current is For a current density limited to 50 A/cm2, the cross-sectional area must be From the specified voltage and electric field, the device length is found to be The semiconductor conductivity, from Equation (4.22b) is The conductivity of a compensated p-type semiconductor is  = epp = ep(Na - Nd) where the mobility p is a function of the total ionized impurity concentration Na + Nd. Using trial and error, if Na = 1.25  1016 cm-3, then Na + Nd = 1.75  1016 cm-3, and the hole mobility, from Figure 4.3 is p  410 cm2/V-s. The conductivity is then • = ep(Na- Nd) = (1.6  10-19)(410)(1.25  1016  5  1015) = 0.492 (-cm)-1 which is very close to the value we need. • Comment Since the mobility is related to the total ionized impurity concentration, the determination of the impurity concentration to achieve a particular conductivity is not straightforward.

5. EXAMPLE 4.5 • OBJECTIVE • Determine the carrier density gradient to produce a given diffusion current density. • The hole concentration in silicon at T = 300 K varies linearly from x = 0 to x = 0.01 cm. The hole diffusion coefficient is Dp = 10 cm2/s, the hole diffusion current density is Jdif = 20 A/cm2, and the hole concentration at x = 0 is p = 4  1017 cm-3. Determine the hole concentration at x = 0.01 cm. • Solution The diffusion current density is given by or which yields p(0.01) = 2.75  1017 cm-3 • Comment • We can note that, since the hole diffusion current is positive, the hole gradient must be negative, which implies that the value of hole concentration at x = 0.01 must be smaller than the concentration at x = 0.

6. EXAMPLE 4.6 • OBJECTIVE • Determine the induced electric field in a semiconductor in thermal equilibium given a linear variation in doping concentration. • Assume that the donor concentration in an n-type semiconductor at T = 300 K is given by • Nd(x) = 1016  1019x (cm-3) • where x is given in cm and ranges between 0  x  1 m. • Solution Taking the derivation of the donor concentration, we have The induced electric field is given by Equation (4.42), so we have At x = 0, for example, we find  = 25.9 V/cm • Comment • We can recall from our previous discussion of drift current that fairly small electric fields can produce significant drift current densities, so that an induced electric field from nonuniform doping can significantly influence semiconductor device characteristics.

7. EXAMPLE 4.7 • OBJECTIVE • Determine the diffusion coefficient given the carrier mobility. • Assume that the mobility of a particular carrier is  = 1200 cm2/V-s at T = 300 K. • Solution Using the Einstein relation, we have that • Comment • Although this example is simple and straightforward, it is important to keep in mind the relative orders of magnitude of the mobility and diffusion coefficient. The diffusion coefficient is approximately 40 times smaller in magnitude than the mobility at room temperature.

8. EXAMPLE 4.8 • OBJECTIVE • Determine the majority-carrier concentration and mobility, given Hall effect parameters. • Consider the geometry shown in Figure 4.20. Let L = 10-1 cm, W = 10-2 cm, and d = 10-3 cm. Also assume that Ix = 1.0 mA, Vx = 12.5 V, B = 500 gauss = 5  10-2 tesla, and VH = 6.25 mV. • Solution A negative Hall voltage for this geometry implies that we have an n-type semiconductor. Using Equation (4.70), we can calculate the electron concentration as or n = 5  1015 cm-3 The electron mobility is then determined from Equation (4.74) as or n = 1000 cm2/V-s • Comment • It is important to note that the MKS units must be used consistently in the Hall effect equations to yield correct results.