Point Defects — Donor Impurities, Vacancy Concentration, and Diffusion: 17-Phys-A6 Solid State Physics, May 2013, Q7

Question 7: Point Defects — Donor Impurities, Vacancy Concentration, and Diffusion (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Na density $\rho$	$0.971\ \text{g/cm}^3$
Na atomic weight	23 amu
Na vacancy formation energy $E_v$	1.05 eV
Temperature (part b)	300 K
Si–Al diffusion constants (from table)	$D_0=8.0\ \text{cm}^2/\text{s}$, $E=3.47\ \text{eV}$
Temperature (part c)	1200 K

Find. (a) why As-doping is useful; (b) equilibrium vacancy concentration in Na at 300 K; (c) diffusion coefficient of Al in Si at 1200 K.

Approach. (a) reason from donor doping and carrier control. (b) use the paper's own vacancy-equilibrium relation, eq. (18), with $n\ll N$. (c) use the Arrhenius diffusion law, eq. (19), reading $D_0,E$ for the Si–Al pair straight off the given table.

Part (a) — usefulness of As impurities in Si. Arsenic is a Group-V element (5 valence electrons) substituting for a Group-IV Si atom (4 valence electrons); 4 of As's electrons bond into the lattice and the 5th is only weakly bound, occupying a shallow donor level just below the conduction-band edge. At room temperature this electron is thermally ionized into the conduction band, controllably increasing the free-electron concentration (and hence conductivity) by many orders of magnitude relative to intrinsic Si, while leaving a fixed, immobile positive donor ion behind. This controlled, reproducible n-type doping — paired with p-type (acceptor) doped regions — is what creates the p-n junctions underlying diodes, transistors, and essentially all planar silicon devices.
Part (b) — vacancy concentration in Na at 300 K. Number density of Na sites: $$N=\frac{\rho N_A}{M}=\frac{(0.971\ \text{g/cm}^3)(6.02217\times10^{23}\ \text{mol}^{-1})}{23\ \text{g/mol}}=2.542\times10^{22}\ \text{cm}^{-3}$$ For $n\ll N$, eq. (18), $\dfrac{n}{N-n}=\exp\!\left(\dfrac{-E_v}{k_BT}\right)$, reduces to $n\approx N\exp(-E_v/k_BT)$. With $E_v=1.05\ \text{eV}=1.682\times10^{-12}\ \text{erg}$, $k_BT=(1.38062\times10^{-16})(300)=4.142\times10^{-14}\ \text{erg}$ (so $E_v/k_BT=40.62$): $$n=(2.542\times10^{22}\ \text{cm}^{-3})\,e^{-40.62}$$ $\boxed{n_{vac}\approx5.83\times10^{4}\ \text{cm}^{-3}}$ — an extremely dilute defect population, since $E_v=1.05\ \text{eV}$ is large compared with $k_BT\approx0.026\ \text{eV}$ at room temperature; the equilibrium vacancy fraction $n/N\sim2\times10^{-18}$ confirms $n\ll N$, justifying the approximation.
Part (c) — Al diffusion coefficient in Si at 1200 K. Reading $D_0=8.0\ \text{cm}^2/\text{s}$ and $E=3.47\ \text{eV}=5.560\times10^{-12}\ \text{erg}$ for the Si–Al pair straight off the given defect table, and applying the Arrhenius law eq. (19), $D=D_0\exp(-E/k_BT)$, with $k_BT=(1.38062\times10^{-16})(1200)=1.657\times10^{-13}\ \text{erg}$ (so $E/k_BT=33.56$): $$D=(8.0\ \text{cm}^2/\text{s})\,e^{-33.56}$$ $\boxed{D\approx2.13\times10^{-14}\ \text{cm}^2/\text{s}}$ — this is the rate (diffusion coefficient) at which Al atoms migrate through the Si lattice at 1200 K; it lands within an order of magnitude of measured Al-in-Si diffusivities near this temperature, another useful internal check.

Quantity	Result
(a) Role of As in Si	shallow donor → controlled n-type conductivity (basis of p–n devices)
(b) Na vacancy concentration, 300 K	$5.83\times10^4\ \text{cm}^{-3}$
(c) $D$(Al in Si), 1200 K	$2.13\times10^{-14}\ \text{cm}^2/\text{s}$

Topic: Point defects — doping, thermal vacancy equilibrium, and Arrhenius diffusion.

Key relations: donor/acceptor doping model; vacancy equilibrium $n/(N-n)=\exp(-E_v/k_BT)$ (eq. 18); Arrhenius diffusion law $D=D_0\exp(-E/k_BT)$ (eq. 19).

Why this works: both vacancy formation and diffusive hopping are thermally-activated processes over an energy barrier, so both obey Boltzmann/Arrhenius statistics; the same mathematical form ($k_BT$ in an exponential) recurs throughout solid-state physics (carrier statistics, vacancy populations, diffusion, ionization) because all are governed by the same Boltzmann factor for a rare activated event.

Common pitfall: forgetting that eq. (18)'s exact form has $N-n$ in the denominator (only negligible when $n\ll N$, which must be checked, not assumed); reading the wrong row of the diffusion-constants table (the paper lists several Si-host dopants — Al, Ga, In, As, Sb, Li, Au — each with very different $D_0,E$).

Source: Kittel, ISSP 8th ed., Ch. 8 (semiconductor doping) & Ch. 18 (point defects, diffusion).

Question 7 of 7: Point Defects — Donor Impurities, Vacancy Concentration, and Diffusion

Question 7: Point Defects — Donor Impurities, Vacancy Concentration, and Diffusion (20 marks)