Solar Cell Design Rules

Course subject(s) 3. Solar Cell Operation, Performance and Design Rules

In this video Prof. Arno Smets talks about the different ways solar cells make use of the mechanisms that generate electricity. He talks about the direct and indirect band gap, and the requirements for certain band gap materials to generate current.

You will find the chapter associated to the lecture below the videos.

In the case of an intrinsic material:

\( n=N_C*exp\Big( \frac{E_F-E_C}{k_B*T} \Big) \), for \( E_C-E_F\ge 3K_B*T \tag{1}\)
\( p=N_V*exp\Big( \frac{E_V-E_F}{k_B*T} \Big) \), for \( E_F-E_V\ge 3K_B*T \tag{2} \)

The product of \( (1) \) and \( (2) \) yields:

\[ n^2_i=n*p=N_C*N_V*exp\Big( \frac{E_F-E_C}{k_B*T} \Big)*exp\Big( \frac{E_V-E_F}{k_B*T} \Big) \] \[ =N_C*N_V*exp\Big( \frac{E_F-E_C+E_V-E_F} {k_B*T} \Big)\] \[ =N_C*N_V*exp\Big( \frac{-(E_C-E_V)} {k_B*T} \Big) \] \[ =N_C*N_V*exp\Big( \frac{-E_{gap}}{k_B*T} \Big) \tag{3} \]

For excess carriers:

\( n_p=N_C*exp\Big( \frac{E_{Fn}-E_C}{k_B*T} \Big) \), for \( E_C-E_{Fn}\ge 3K_B*T \tag{4}\)
\( p_n=N_V*exp\Big( \frac{E_V-E_{Fp}}{k_B*T} \Big) \), for \( E_{Fp}-E_V\ge 3K_B*T \tag{5} \)

The product of \( (4) \) and \( (5) \) yields:

\[ n_p*p_n=N_C*N_V*exp\Big( \frac{E_{Fn}-E_C}{k_B*T} \Big)*exp\Big( \frac{E_V-E_{Fp}}{k_B*T} \Big) \] \[ =N_C*N_V*exp\Big( \frac{E_{Fn}-E_C+E_V-E_{Fp}} {k_B*T} \Big)\] \[ =N_C*N_V*exp\Big( \frac{-(E_C-E_V)+E_{Fn}E_{Fp}} {k_B*T} \Big) \] \[ =N_C*N_V*exp\Big( \frac{-E_{gap}+q*V_{oc}}{k_B*T} \Big) \] \[ =N_C*N_V*exp\Big( \frac{-E_{gap}}{k_B*T}\Big) *exp\Big( \frac{q*V_{oc}}{k_B*T} \Big) \]

which from \( (3) \):

\[=n^2_i*exp\Big( \frac{q*V_{oc}}{k_B*T} \Big) \tag{6}\] But \( \tau_0 \propto N^{-1}_{t} \)

Therefore:

\[ n_p=p_n\approx G_L*\tau_0 \tag{7} \]

If we solve \( n_p*p_n=n^2_i*exp\Big( \frac{q*V_{oc}}{k_B*T} \Big)\) for \( V_{oc} \)

\[ V_{oc}= \frac{k_B*T}{q}*ln \Big(\frac{n_p*p_n}{n^2_i} \Big) \]

which from \( (7) \) can be written:

\[ V_{oc}\approx \frac{k_B*T}{q}*ln \Big( \frac{G_L*\tau_0}{n_i} \Big)^2 \] \[ V_{oc}\approx \frac{2*k_B*T}{q}*ln \Big( \frac{G_L*\tau_0}{n_i} \Big) \tag{8}\]