The Transfer Length Method or the "Transmission Line Model" (both abbreviated as TLM) is a technique used in semiconductor physics and engineering to determine the specific contact resistivity between a metal and a semiconductor. TLM has been developed because with the ongoing device shrinkage in microelectronics the relative contribution of the contact resistance at metal-semiconductor interfaces in a device could not be neglected any more and an accurate measurement method for determining the specific contact resistivity was required.
General description
The goal of the transfer length method (TLM) is the determination of the specific contact resistivity <math>\rho_C</math> of a metal-semiconductor junction. To create a metal-semiconductor junction a metal film is deposited on the surface of a semiconductor substrate. The TLM is usually used to determine the specific contact resistivity when the metal-semiconductor junction shows ohmic behaviour. In this case the contact resistivity <math>\rho_C</math> can be defined as the voltage difference <math>\Delta V</math> across the interfacial layer between the deposited metal and the semiconductor substrate divided by the current density <math>J</math> which is defined as the current <math>I</math> divided by the interfacial area <math>A</math> through which the current is passing:
:<math>\rho_C = \frac{\Delta V}{J} = \frac{(V_{Semiconductor} - V_{Metal})A}{I}</math>
In this definition of the specific contact resistivity <math>V_{Semiconductor}</math> refers to the voltage value just below the metal-semiconductor interfacial layer while <math>V_{Metal}</math> represents the voltage value just above the metal-semiconductor interfacial layer. There are two different methods of performing TLM measurements which are both introduced in the remainder of this section. One is called just transfer length method while the other is named circular transfer length method (c-TLM).
In the image to the right the distance between the pads <math>d_i</math> increases from the bottom to the top. Therefore, when the resistance between adjacent pads is measured the total resistance <math>R_{Tot}</math> increases accordingly as it is indicated in the graph beneath the depiction of the metal pads. In this graph the abscissa represents the distance <math>d</math> between two adjacent metal pads while the circles represent measured resistance values.
The total resistivity <math>R_{Tot}</math> can be separated into a component due to the uncovered semiconductor substrate and a component that corresponds to the voltage drop in two metal-covered areas. The former component can be described with the formula <math>\frac{R_S}{Z}d_i</math>, whereas <math>R_S</math> represents the sheet resistance of the semiconductor substrate and <math>Z</math> the width of the metal pads. The other component that contributes to the total resistance is denoted by <math>2R_C</math> because when two adjacent pads are characterized two identical metallized areas have to be considered. This means that the total resistance can be written in the following functional form, with the pad distance <math>d</math> as independent variable:
:<math>R_{Tot} = \frac{R_S}{Z}d+ 2R_C</math>
If the contribution of the metal layer itself is neglected then <math>R_C</math> arises because of the voltage drop at the metal-semiconductor interface as well as in the semiconductor substrate underneath. This means that during a total resistance measurement, the voltage drops exponentially (and hence also the current density) in the metallic regions (see also theory section for further explanation). Physically speaking this means that the main part of the area underneath a metallic contact through which current enters the metal via the metal-semiconductor interface is given by the transfer length multiplied with the width of the pad <math>Z</math>. This situation is also depicted in the figure in this section where the current density distribution underneath two adjacent metal pads during a resistance measurement is depicted with a green colouring.
All in all this means that (if the metal pad length <math>w</math> is much larger than the transfer length) that a relation between <math>R_C</math> and <math>\rho_C</math> can be stated:
:<math>\frac{\mathrm{d}V}{\mathrm{d}x} = -\frac{R_S}{Z} I(x)</math>
:<math>\frac{\mathrm{d}I}{\mathrm{d}x} = -\frac{Z}{\rho_C} V(x)</math>
These two coupled differential equations can be separated by differentiating one with respect to <math>x</math> such that the other can plugged in. By doing so finally, two differential equations are obtained which do not depend on each other:
:<math>\frac{\mathrm{d}^2V}{\mathrm{d}x^2} = \frac{R_S}{\rho_C} V(x)</math>
:<math>\frac{\mathrm{d}^2I}{\mathrm{d}x^2} = \frac{R_S}{\rho_C} I(x)</math>
Both differential equations have solutions of the form <math>f(x)=Acosh(\alpha x) + Bsinh(\alpha x)</math> whereat <math>A</math> and <math>B</math> are constants which need to be determined by using appropriate boundary conditions and <math>\alpha</math> is given by <math>\sqrt{R_S/\rho_C}</math> which is the inverse of the previously defined transfer length <math>L_T</math>. Two boundary conditions can be obtained by defining the voltage as well as the current at the beginning of a metallic pad area as <math>V_0</math> and <math>I_0</math> respectively. In a formal manner this means that <math>V(x=0)=V_0</math> and <math>I(x=0)=I_0</math> when using the settings in the figure in this section. By using the pair of coupled differential equations above two more boundary conditions are obtained, namely <math>\frac{\mathrm{d}V(x=0)}{\mathrm{d}x} = -\frac{R_S}{Z} I_0</math> and <math>\frac{\mathrm{d}I(x=0)}{\mathrm{d}x} = -\frac{R_S}{Z} V_0</math>. Eventually two equations, describing the voltage and the current as a function of distance <math>x</math> are obtained by using the four stated boundary conditions:
:<math>\frac{\mathrm{d}^2V}{\mathrm{d}r^2} + \frac{1}{r} \frac{\mathrm{d}V}{\mathrm{d}r} -\frac{R_S}{\rho_C}V = 0</math>
A general solution to this type of differential equations is given as follows, whereat <math>A</math> and <math>B</math> are unspecified constants and <math>\alpha</math> is <math>\sqrt{R_S/\rho_C}</math>. The functions <math>I_0</math> and <math>K_0</math> are zero-order modified Bessel functions of the first and second kind respectively.
:<math>V(r) = AI_0(\alpha r) + BK_0(\alpha r)</math>
By utilizing the coupled differential equations above and the differentiation rules for modified Bessel functions (<math>I_0'(x) = I_1(x)</math>, <math>K_0'(x) = -K_1(x)</math>) This means, that multiplication of the matrix-vector equation with the transposed matrix of <math>X</math> yields:
:<math>\begin{bmatrix}
A \\
B
\end{bmatrix} = {(X^TX)}^{-1}X^T\mathbf{R}</math>.
Since all components of <math>X</math> can be calculated and the components of <math>\mathbf{R}</math> are provided by the resistance measurements, the coefficients <math>A</math> and <math>B</math> can be calculated. Finally from the two coefficients, the values for <math>L_T</math>, <math>R_S</math> and the specific contact resistivity <math>\rho_C</math> can be calculated as well. A plot to the left shows the measured resistance values in dependence of the gap length together with the fitting function corresponding to the determined coefficients <math>A</math> and <math>B</math>.
The following GNU Octave script corresponds to the performed measurement series and also includes the obtained resistance values. A plot of the measurement points together with the fitting function is created and the values for <math>L_T</math>, <math>R_S</math> and the specific contact resistivity <math>\rho_C</math> are calculated as well.
<syntaxhighlight Lang="octave">
% vectors that contain the obtained measurement data
d = 20:20:200; # this vector contains the gap lengths
R_row = [112.258772, 125.071437, 130.619235, 138.959548, 139.110758, 148.420932, 148.474871, 160.83128, 166.670412, 167.614947];
R = transpose(R_row);
% Here the column vectors of X are defined
r_i = 200;
x1 = transpose(log((r_i.+d)/r_i));
x2 = transpose(1./(r_i.+d) + 1/r_i);
% Define the matrix X
X = [x1, x2];
% Obtain the values A and B
beta = inv(transpose(X)*X)*transpose(X)*R;
A = beta(1);
B = beta(2);
% Define the fitting function
d_fit = 0:1:200;
R_fit = A*log((r_i.+d_fit)/r_i) + B*(1./(r_i.+d_fit) + 1/r_i);
% Plot the fitting function and the measurement values
scatter(d,R, "r");
hold on;
plot(d_fit, R_fit);
set(gca, 'fontsize', 14);
xlabel('Gap length [μm]');
ylabel('Total Resistance [Ohms]');
% Calculate the physical properties
R_S = A*2*pi; # given in ohms
L_T = 2*pi*B/R_S; # given in μm
rho_c = (R_S*(L_T)^2)*10^(-8); # given in Ohm*cm^2
</syntaxhighlight>