For the predeposition heat treatment of a semiconducting device, gallium atoms are to be diffused into silicon at a temperature of \(1150^{\circ} \mathrm{C}\) for \(2.5 \mathrm{~h}\). If the required concentration of \(\mathrm{Ga}\) at a position \(2 \mu \mathrm{m}\) below the surface is \(8 \times 10^{23}\) atoms \(/ \mathrm{m}^{3}\), compute the required surface concentration of \(\mathrm{Ga}\). Assume the following: (i) The surface concentration remains constant (ii) The background concentration is \(2 \times 10^{19} \mathrm{Ga}\) atoms \(/ \mathrm{m}^{3}\) (iii) Preexponential and activation energy values are \(3.74 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) and \(3.39 \mathrm{eV} /\) atom, respectively.

Step by step solution

01

Calculate the diffusion coefficient at the given temperature

The Arrhenius equation relates the diffusion coefficient D to its pre-exponential value, the activation energy, and the temperature: $$ D = D_0 \exp\left(-\frac{E_\mathrm{A}}{k_\mathrm{B}T}\right) $$ where \(D_0\) is the pre-exponential value \(3.74 \times 10^{-5} ~\mathrm{m}^{2} / \mathrm{s}\), \(E_\mathrm{A}\) is the activation energy \(3.39 ~\mathrm{eV}\), \(k_\mathrm{B}\) is Boltzmann's constant, and \(T\) is the temperature in Kelvin. Since the given temperature is \(1150^{\circ} \mathrm{C}\), we need to convert it into Kelvin by adding \(273.15\) to it, giving T = \(1423.15\) K. Now, substitute the given values and constants into the Arrhenius equation and find the diffusion coefficient, D.

02

Calculate the diffusion length

Using the given diffusion time and the diffusion coefficient calculated in step 1, we can calculate the diffusion length, L, using the following formula: $$ L = \sqrt{Dt} $$ where \(D\) is the diffusion coefficient and \(t\) is the diffusion time. Since the given diffusion time is \(2.5 ~\mathrm{h}\), we need to convert it into seconds: \(t = 2.5 \times 3600\) s. Now, compute L with the values calculated.

03

Use Fick's second law and solve for surface concentration

Fick's second law for constant surface concentration is given by: $$ C(x,t) = C_\mathrm{s} \mathrm{erfc}\left(\frac{x}{2\sqrt{Dt}}\right) - C_\mathrm{b} $$ where \(C(x,t)\) is the required concentration at position x below the surface and time t, \(C_\mathrm{s}\) is the surface concentration, \(C_\mathrm{b}\) is the background concentration, and "erfc" is the complementary error function. Now, substitute the values for \(C(x,t)\), \(C_\mathrm{b}\), \(x\), \(D\), and \(t\), and solve for \(C_\mathrm{s}\). Use a table or a calculator with an "erfc" function to compute the complementary error function. Using the obtained value of \(C_\mathrm{s}\), we can determine the required surface concentration of Ga.

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