Measuring Resistance of Diodes

Transfer Length Model (TLM)

Linear Transfer Length Model (LTLM)

$$R_{Total}=\frac{R_s}{W}L+2R_C$$

• ${\rho }_c=\text{specific contact resistivity}$ $[{\rho }_c]=\Omega \cdot c{m}^2$

• $L_T=\sqrt{\frac{{\rho }_c}{{\rho }_{sheet}}}$ $L_T$ is the transfer length — the distance over which most of the current transfers from the semiconductor into the contact. It’s derived from the ratio of contact resistivity to sheet resistance.

• ${\rho }_c=L_T^2\cdot {\rho }_{sheet}$ This is just the rearrangement of the above — once you extract $L_T$ from the x-intercept of the LTLM plot (resistance vs. contact spacing) and know ${\rho }_{sheet}$ from the slope, you get ${\rho }_c$.

Link to original

Circular Transfer Length Model (CTLM)

No need for mesa etch, easier to manufacture

$$R_{Total}\approx \frac{R_S}{2\pi }\left[\ln \left(\frac{r_o}{r_o-d}\right)+L_T\left(\frac{1}{r_o-d}+\frac{1}{r_o}\right)\right]$$Link to original

C-V Profiling

The depletion region of a p-n junction acts as a parallel plate capacitor:

$C=\frac{dQ}{dV}=\frac{\epsilon A}{W}(1)$

where $\epsilon$ is the permittivity, $A$ is the junction area, and $W$ is the depletion width.

When $W$ increases by $dW$, the charge exposed is $dQ=qn(x)dW$, so:

$C=qn(x)\frac{dW}{dV}(2)$

From (1), $W=\epsilon A/C$, so $dW=-\frac{\epsilon A}{C^2}dC$ (3).

Substituting (3) into (2) and rearranging gives a separable ODE:

$\frac{dC}{dV}=-\frac{C^3}{\epsilon Aqn(x)}$

Integrating:

$\frac{1}{C^2}=\frac{2}{\epsilon Aqn(x)}V+\text{const}$

The boundary condition is that when $1/C^2=0$, all the depletion region has collapsed, which occurs at $V=V_{bi}$ (the built-in voltage). This gives the key result:

$\frac{1}{C^2}=\frac{2}{\epsilon Aqn(x)}(V+V_{bi})$

How you use it: plot $1/C^2$ vs reverse bias $V$. For uniform doping it’s a straight line where:

• The slope gives the doping concentration $n(x)=\frac{2}{\epsilon Aq}\cdot \frac{1}{\text{slope}}$
• The x-intercept gives the built-in voltage $V_{bi}$

For non-uniform doping, the slope varies with $V$ and you extract the local doping profile $n(x)$ at each bias point from the local gradient.