Transfer length method

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

Graphical description of the transfer length method (TLM)

The goal of the transfer length method (TLM) is the determination of the specific contact resistivity $\rho _{C}$ 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 $\rho _{C}$ can be defined as the voltage difference $\Delta V$ across the interfacial layer between the deposited metal and the semiconductor substrate divided by the current density $J$ which is defined as the current $I$ divided by the interfacial area $A$ through which the current is passing:

$\rho _{C}={\frac {\Delta V}{J}}={\frac {(V_{Semiconductor}-V_{Metal})A}{I}}$

In this definition of the specific contact resistivity $V_{Seminductor}$ refers to the voltage value just below the metal-semiconductor interfacial layer while $V_{Metal}$ represents the voltage value just above the metal-semiconductor interfacial layer. To determine the specific contact resistivity $\rho _{C}$ an array of rectangular metal pads is deposited on the surface of a semiconductor substrate as it is depicted in the image to the right. The definition of the rectangular pads can be done by utilizing photolithography while the metal deposition can be done with sputter deposition, thermal evaporation or electroless deposition.

In the image to the right the distance between the pads $d_{i}$ increases from the bottom to the top. Therefore, when the resistance between adjacent pads is measured the total resistance $R_{Tot}$ increases accordingly as it is indicated in the graph beneath the depiction of the metal pads. In this graph the abscissa represents the distance $d$ between two adjacent metal pads while the circles represent measured resistance values. The total resistivity $R_{Tot}$ 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 ${\frac {R_{S}}{Z}}d_{i}$, whereat $R_{S}$ represents the sheet resistance of the semiconductor substrate and $Z$ the width of the metal pads. The other component that contributes to the total resistance is denoted by $2R_{C}$ because when two adjacent pads are characterized two identical metallized areas have to be considered. This means that the rotal resistance can be writtten in the following functional form, with the pad distance $d$ as independent variable:

$R_{Tot}={\frac {R_{S}}{Z}}d+2R_{C}$

If the contribution of the metal layer itself is neglected then $R_{C}$ 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). As it is derived in the next section of this article the majority of the voltage drop underneath a metallic pad takes place within in the length ${\sqrt {\frac {\rho _{C}}{R_{S}}}}$ which is defined as the transfer length $L_{T}$. Metaphorically 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 $Z$. 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 $w$ is much larger than the transfer length) that a relation between $R_{C}$ and $\rho _{C}$ can be stated:

$R_{C}={\frac {\rho _{C}}{ZL_{T}}}={\frac {\sqrt {R_{S}\rho _{C}}}{Z}}$

Since $R_{S}$ can be extracted from a linear fit through the data points and $R_{C}$ can be obtained from the y-intercept of the linear fit an estimation of $\rho _{C}$ is possible.

Theory

Resistor network for derivation of the TLM differential equations and a plot of the voltage drop across two adjacent measurement pads

In the last section the basic principle of TLM was introduced and now more details about the theoretical background are given. The main purpose here is to find an expression that relates the measurable quantity $R_{C}$ with the specific contact resistivity $\rho _{C}$ which is intended to be determined with TLM. Therefore in the image to the right a resistor network is illustrated that describes the situation when a voltage is applied between two adjacent metallic pads. The resistor (${\frac {R_{S}d_{i}}{Z}}$) in the middle takes account for the part that is not covered with metal while the rest describes the situation for the metallic pads. The horizontal resistor elements (${\frac {R_{S}\Delta x}{Z}}$) represent the resistance due to the semiconductor substrate and the vertical resistor elements (${\frac {\rho _{C}}{Z\Delta x}}$) take account for the resistance due to the metal-semiconductor interfacial layer. In this description pairs of horizontal and vertical resistor elements describe the situation within a volume element of length $\Delta x$ in a metallic pad area. This methodology is also used for the derivation of the telegrapher's equations which are used to describe the behaviour of transmission lines. Because of this analogy, the described measurement technique in this article is often called the transmission line method.

By using Kirchhoff's circuit laws the folllowing expressions for the voltage as well as for the current within the above considered length element (read square in the figure in this section) are obtained for a steady state situation where both voltage and current are not a function of time:

$V(x)={\frac {R_{S}\Delta xI(x)}{Z}}+V(x+\Delta x)$

$I(x)={\frac {Z\Delta xV(x)}{\rho _{C}}}+I(x+\Delta x)$

By taking the limit $\Delta x\to 0$ the following two differential equations are obtained:

${\frac {\mathrm {d} V}{\mathrm {d} x}}=-{\frac {R_{S}}{Z}}I(x)$

${\frac {\mathrm {d} I}{\mathrm {d} x}}=-{\frac {Z}{\rho _{C}}}V(x)$

These two coupled differential equations can be separated by differentiating one with respect to $x$ such that the other can plugged in. By doing so finally, two differential equations are obtained which do not depend on each other:

${\frac {\mathrm {d} ^{2}V2}{\mathrm {d} x^{2}}}={\frac {R_{S}}{\rho _{C}}}V(x)$

${\frac {\mathrm {d} ^{2}I}{\mathrm {d} x^{2}}}={\frac {R_{S}}{\rho _{C}}}I(x)$

Both differential equations have solutions of the form $f(x)=Acosh(\alpha x)+Bsinh(\alpha x)$ whereat $A$ and $B$ are constants which need to be determined by using appropriate boundary conditions and $\alpha$ is given by ${\sqrt {R_{S}/\rho _{C}}}$. Two boundary conditions can be obtained by defining the voltage as well as the current at the beginning of a metallic pad area as $V_{0}$ and $I_{0}$ respectively. In a formal manner this means that $V(x=0)=V_{0}$ and $I(x=0)=I_{0}$ 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 ${\frac {\mathrm {d} V(x=0)}{\mathrm {d} x}}=-{\frac {R_{S}}{Z}}I_{0}$ and ${\frac {\mathrm {d} I(x=0)}{\mathrm {d} x}}=-{\frac {R_{S}}{Z}}V_{0}$ which is the inverse of the previously defined transfer length $L_{T}$. Eventually two equations, describing the voltage and the current as a function of distance $x$ are obtained by using the four stated boundary conditions:

$V(x)=V_{O}\cosh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}x\right)}-{\frac {I_{0}{\sqrt {R_{s}\rho _{C}}}}{Z}}\sinh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}x\right)}$

$I(x)=I_{O}\cosh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}x\right)}-{\frac {V_{0}Z}{\sqrt {R_{s}\rho _{C}}}}\sinh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}x\right)}$

When a measurement is performed, it can be assumed that no current is flowing at the opposing end of each metallic pad, which in turn means that $I(x=w)=0$. This allows a further refinement of the equation describing the voltage when using the relation $\sinh(x-y)=\sinh x\cosh y-\cosh x\sinh y$:

$V(x)={\frac {I_{O}{\sqrt {R_{S}\rho _{C}}}}{Z}}{\frac {\cosh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}(w-x)\right)}}{\sinh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}w\right)}}}$

The last equation describes the voltage drop across the region covered by a metallic pad (compare with the figure in this section). By realizing that the resistance value $R_{C}$ can be expressed with ${\frac {V_{0}}{I_{0}}}$ and by setting $x=0$ in the last formula an expression can be found that relates $R_{C}$ to the specific contact resistivity $\rho _{C}$:

$R_{C}={\frac {V_{0}}{I_{0}}}={\frac {\sqrt {R_{S}\rho _{C}}}{Z}}{\frac {\cosh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}w\right)}}{\sinh {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}w\right)}}}={\frac {\sqrt {R_{S}\rho _{C}}}{Z}}\coth {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}w\right)}$

The last equation allows the calculation of $\rho _{C}$ by utilizing experimental data. Since $\coth {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}w\right)}$ goes to 1 as $d$ increases (and is significantly larger than the transfer length $L_{T}={\sqrt {\frac {\rho _{C}}{R_{S}}}}$) often the estimation $R_{C}\approx {\frac {\sqrt {R_{S}\rho _{C}}}{Z}}$ is used instead of the strictly derived equality. This is identical to what was stated in the general description section. In summary the voltage as well as the current as a function of distance in the region of a metallic pad has been derived by utilizing a model that is similar to the telegrapher's equations. This enabled to find an expression that allows the calculation of the specific contact resistivity $\rho _{C}$ of the metal-semiconductor junction by using the experimentally found quantities $R_{C}$ and $R_{S}$ and the width $Z$ of a metallic pad.

See also

• Four-terminal sensing – Method of measuring electrical impedance
• Van der Pauw method – An accurate way to measure resistance and hall voltage

Further reading

• Gary Tuttle, Dept. of Electrical and Computer Engineering, Iowa State University. "Contact resistance and TLM measurements" (PDF).{{cite web}}: CS1 maint: multiple names: authors list (link)