The Debye length is a characteristic length for semiconductor. Arrange the doping density given below in such a way that Debye length changes from maximum to minimum. (A) N = 1015 cm–3 Choose the most appropriate answer from the options given below :
(B) N = 1017 cm–3
(C) N = 3 × 1015 cm–3
(D) N = 1016 cm–3
(E) N = 5 × 1016 cm–3
(A), (C), (D), (E), (B)
Debye length falls as the doping rises, so ordering it from maximum to minimum means ordering the doping from minimum to maximum — option 3.
\(L_{D}=\sqrt{\dfrac{\varepsilon kT}{q^{2}N}}\quad\Rightarrow\quad L_{D}\propto\dfrac{1}{\sqrt{N}}\)
Step 1 — sort the doping densities. Writing them all with the same exponent makes the comparison immediate:
| Label | N (cm–3) | As × 1015 | Rank |
|---|---|---|---|
| (A) | 1015 | 1 | Smallest |
| (C) | 3 × 1015 | 3 | 2nd |
| (D) | 1016 | 10 | 3rd |
| (E) | 5 × 1016 | 50 | 4th |
| (B) | 1017 | 100 | Largest |
Step 2 — invert. Since \(L_{D}\propto N^{-1/2}\), the largest Debye length belongs to the smallest N. Ordering the Debye length from maximum to minimum therefore gives
\(A\rightarrow C\rightarrow D\rightarrow E\rightarrow B\)
— exactly option 3.
What the Debye length means. It is the distance over which mobile carriers screen out an electric field, or equivalently the width of the transition region at the edge of a depletion layer, where the charge density falls off gradually rather than abruptly. Heavier doping supplies more carriers to do the screening, so the field is neutralised in a shorter distance — which is the physical reason for the inverse square root.
The numbers are worth a feel. At room temperature in silicon, \(N=10^{15}\) cm–3 gives about 130 nm, and \(10^{17}\) gives about 13 nm — a factor of ten for a hundredfold change in doping, exactly as the square root demands.
Why it sets a limit in device scaling. The abrupt-depletion approximation is only valid while the depletion width greatly exceeds \(L_{D}\). In a heavily scaled device the two become comparable, so the depletion edge is no longer sharp and the simple textbook analysis breaks down — one of the reasons short-channel modelling is difficult.
Hence, the order is (A), (C), (D), (E), (B).
Similar Questions
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