Copper (Cu) has D0=0.16 cm^2/sec, ΔH=2.07 eV, and Aluminum (Al) has D0=0.047 cm2/sec, ΔH=1.28 eV.
(a) Calculate the diffusion coefficient: D_Cu and D_Al at 600K.
(b) Calculate D_Cu and D_Al at their melting point. (T_M for Cu is 1083 °C, and for Al is 660 °C).
(c) Which temperature should you consider for the diffusion issue when you select a metal contact for electronic circuit? Why? Which one is better for interconnects, Al or Cu? (Hint: D = Do exp( -ΔH / kT) )

a) At T = 600 K

D_Cu = (6.48 × 10⁻¹⁹) cm²/sec

D_Al = (8.21 × 10⁻¹³) cm²/sec

b) At their respective melting points

D_Cu at 1083°C = (2.00 × 10⁻⁹) cm²/sec

D_Al at 660°C = (5.70 × 10⁻⁹) cm²/sec

c) Check Explanation.

Explanation:

D = D₀ e^(-ΔH/kT)

k = Boltzmann's constant = (1.38 × 10⁻²³) J/K

And T is in Kelvin

For Cu

D₀ = 0.16 cm²/sec

ΔH = 2.07 eV = (2.07 × 1.602 × 10⁻¹⁹) J

= (3.316 × 10⁻¹⁹) J

For Al

D₀ = 0.0407 cm²/sec

ΔH = 1.28 eV = (1.28 × 1.602 × 10⁻¹⁹) J

= (2.051 × 10⁻¹⁹) J

a) The diffusion coefficient: D_Cu and D_Al at 600K.

D_Cu = D₀ e^(-ΔH/kT)

D₀ = 0.16 cm²/sec

ΔH = (3.312 × 10⁻¹⁹) J

k = (1.38 × 10⁻²³) J/K

T = 600 K

D = 0.16 × e^[(-3.316 × 10⁻¹⁹)/(1.38 × 10⁻²³ × 600)]

D = 0.16 × e^(-40.048)

D = (6.48 × 10⁻¹⁹) cm²/sec

D_Al

D₀ = 0.0407 cm²/sec

ΔH = (2.051 × 10⁻¹⁹) J

k = (1.38 × 10⁻²³) J/K

T = 600 K

D = 0.047 × e^[(-2.051 × 10⁻¹⁹)/(1.38 × 10⁻²³ × 600)]

D = 0.047 × e^(-24.77)

D = (8.21 × 10⁻¹³) cm²/sec

b) D_Cu and D_Al at their melting point. (T_M for Cu is 1083°C, and for Al is 660°C).

D_Cu = D₀ e^(-ΔH/kT)

D₀ = 0.16 cm²/sec

ΔH = (3.312 × 10⁻¹⁹) J

k = (1.38 × 10⁻²³) J/K

T = 1083°C = 1,356.15 K

D = 0.16 × e^[(-3.316 × 10⁻¹⁹)/(1.38 × 10⁻²³ × 1,356.15)]

D = 0.16 × e^(-40.048)

D = (2.00 × 10⁻⁹) cm²/sec

D_Al

D₀ = 0.0407 cm²/sec

ΔH = (2.051 × 10⁻¹⁹) J

k = (1.38 × 10⁻²³) J/K

T = 660°C = 933.15 K

D = 0.047 × e^[(-2.051 × 10⁻¹⁹)/(1.38 × 10⁻²³ × 933.15)]

D = 0.047 × e^(-24.77)

D = (5.70 × 10⁻⁹) cm²/sec

c) The diffusion coefficient for substances increase as we move from solid state to liquid state to gaseous state.

And from the calculations, it is evident that these metals have a higher diffusion coefficient at/close to their melting points.

- So, to obtain high diffusion coefficients for metal contact in electronic circuits, temperatures as high and close to their melting points should be used.

- Of the two metals, Al and Cu, at the two points where the calculations were carried out, Aluminium has a higher diffusion coefficient at both points, hence, it's a better choice for interconnects.

Hope this Helps!!!

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