Physics 3313, Homework #3 (due 2/3)

P1 Calculate the upper boundary of the lowest allowed energy band in the Kronig-Penneymodel if the potential width a=5 A and V0b=10 eVA.

P2,3 (a) The E(k) dependence for the conduction band in some material can be describedas E(k) = E0−E1 cos ka, where a=5 A, E0=20 eV, and E1=5 eV (Fig. 1). Whatis the effective mass m∗ of the electron at k=0?

(b) What is m∗ if E(k) = E0 − E1(cos ka)5/2 (Fig. 2)?

1/A-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

eV

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E(k)

1/A-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

eV

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E(k)

Fig. 1: E(k) = E0 − E1 cos ka Fig. 2: E(k) = E0 − E1(cos ka)5/2

P4 The forbidden energy band of GaAs is 1.42 eV. What is the minimum frequency ofan incident photon that can elevate a valence electron to the conduction band?

P5 The energy bandgap in Si decreases with increasing temperature according to theformula

Eg(T ) = Eg(0) −αT 2

T + β

where Eg(0)=1.170 eV, α = 4.73× 10−4 eV/K, β=636 K. A specially designed laptopis able to operate at temperatures from −30◦C to +60◦C. What is the relative changeof the Si bandgap energy ∆Eg/Eg in this temperature range?