Semiconductors — Question 10

Question 10

A moving hole represents a sequence of electron hops

A one-dimensional chain of lattice spacing a has one vacant electron site. Each time a neighboring electron hops right into the vacancy, the vacancy moves one site left. Such steps occur at mean rate f. Treat the vacancy as a hole of charge +e.

1. Find hole drift velocity and the current contribution of a uniform hole density p through area A.

2. For a periodic chain with N sites and exactly one hole, find the mean conventional current through one fixed bond and verify it using circulation time.

Question 10 - Solution

A moving hole represents a sequence of electron hops

(a) Method and calculation

$$v_h=-af,\quad J_h=epv_h=-epaf.$$$$I_h=J_hA=-epAaf.$$

(b) Interpretation and further calculation

$$T_{\mathrm{circuit}}=N/f,\quad I_{\mathrm{bond}}=\frac{-e}{T_{\mathrm{circuit}}}=-ef/N.$$$$pA=1/(Na)\Rightarrow I_h=-e(1/Na)af=-ef/N.$$

Physical check

An electron moving right and the corresponding hole moving left carry the same conventional-current sign.