Advanced Condensed State Physics - Problem Sheet 9(Problems 3, 4 and 5 are some of the assessed problems be handed in to the School Officebefore 14:00 Thursday 9th April)

1 Interpretation of experimental data

The following figure shows schematically the result of a typical experiment studying a semi-conductor doped with both donors and acceptors. The density of carriers is plotted in loga-rithmic scale as a function of 1/T .

ni

N

1/T

S

(i) Describe qualitatively why there are three different regions, and what is the meaning ofintrinsic, saturation and freeze-out in the graph.

(ii) What is the value of NS if the donor and acceptor densities are ND and NA, respectively?[Hint: you have to distinguish two different cases, depending on whether electrons orholes are the majority carrier].

2 Capacitance of a p-n junction

Consider a p-n junction at room temperature. The p-side is doped with a density NA of accep-tors, and the n-side is doped with a density ND of donors. The potential drop in the absenceof bias is 0. In principle, there is some accumulation of charge even in the absence of anyapplied voltage bias.

Let us assume that a voltageV is applied to the p-n junction. We will also assume the sameapproximations that we used to study the depletion layer, namely, that charge accumulates ina region with sharp boundaries.

(i) Calculate the widths, wn and wp, of the depletion layer on the two sides of the junctionin the presence of the voltage bias.

(ii) Calculate the change in the magnitude of the charge per unit area associated with the p-njunction as a result of the applied voltage.

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(iii) Calculate the capacitance of the p-n junction as a function of the voltage, C(V ).

3 p-n junction at low temperaturesTo be assessed in Week 12 (% refers to the contribution to the coursework mark in week 12)

In the lectures, we have calculated the potential drop across a p-n junction, 0, at roomtemperature. Here we will study the low-temperature case.

Consider a p-n junction. The p-side is doped with a majority of acceptor impurities anda residual small density of donors. The n-side is doped with a majority of donor impurities,and a residual small density of acceptors. Assume that the temperature is very low, suchthat kBT ED, and kBT EA, with EA and ED the acceptor and donor ionization energies,respectively.

(i) (15 %) Deduce the value of the chemical potential at the two sides of the p-n junction.

(ii) (15 %) Calculate the potential drop across the junction.

4 Graded p-n junctionTo be assessed in Week 12 (% refers to the contribution to the coursework mark in week 12)

Consider a p-n junction where the densities of donors and acceptors depend on the the positionin the following way:

ND(x)NA(x) = N0 tanh(ax).Note that the p-side corresponds to x< 0 (NA(x)>ND(x)), and the n-side corresponds to x> 0(ND(x)> NA(x)). Assume that the junction is at room temperature.

(i) (15 %) Calculate the chemical potential relative to the valence band edge in the p-sidep(x), and the chemical potential relative to the valence band edge in the n-side, n(x).

(ii) (15 %) Calculate the potential drop along the p-n junction, 0(x), for values of x outsidethe depletion layer.

5 Current through a p-n junction with an applied biasTo be assessed in Week 12 (% refers to the contribution to the coursework mark in week 12)

(20 %) A current of 5 A flows through a p-n junction diode at room temperature (T = 300K) when it is reverse biased with 0.15 V. Calculate the current flow when it is forward biasedwith the same voltage.

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