Validity of the Approximation for Absorption/Emission Rates. The derivation of t
ID: 1838246 • Letter: V
Question
Validity of the Approximation for Absorption/Emission Rates. The derivation of the rate of spontaneous emission made use of the approximation g_vo(v) approximatelyequal delta (v - v_0) in the course of evaluating the integral r_sp(v) = integral [1/T_r g_vo(v)] f_e(v_0) e (v_o) dv_o. Demonstrate that this approximation is satisfactory for GaAs by plotting the functions g_vo(v), f_e(v_o), and e (v_o) at T = 300 degree K and comparing their widths. GaAs is collisionally lifetime broadened with T_2 appromimatelyequal 1 ps. Repeat (a) for the rate of absorption in thermal equilibrium. Peak Spontaneous Emission Rate in Thermal Equilibrium.Explanation / Answer
Substituting the given approximation in the given integral we have
(i)
The plot of is the standard form of Dirac delta function under the given approximation and hence the given approximation is sufficient for the standard plot of this function.
The spontaneous emission. rate gives rise to finite width of the spectral line of the order of
which is in the prescribed limit for the given values of T2=1 ps and the given band of .07m0 of GaAs.
The plot of the functions and can also be justified in the similar manner under the given approximation of the function as standard Dirac delta function
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