TLM

Graphical description of the transfer length method (TLM)

To determine the specific contact resistivity ${\displaystyle \rho _{C}}$ 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.^[6][7]

In the image to the right the distance between the pads ${\displaystyle d_{i}}$ increases from the bottom to the top. Therefore, when the resistance between adjacent pads is measured the total resistance ${\displaystyle R_{Tot}}$ increases accordingly as it is indicated in the graph beneath the depiction of the metal pads. In this graph the abscissa represents the distance ${\displaystyle d}$ between two adjacent metal pads while the circles represent measured resistance values. The total resistivity ${\displaystyle R_{Tot}}$ 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 ${\displaystyle {\frac {R_{S}}{Z}}d_{i}}$, whereas ${\displaystyle R_{S}}$ represents the sheet resistance of the semiconductor substrate and ${\displaystyle Z}$ the width of the metal pads. The other component that contributes to the total resistance is denoted by ${\displaystyle 2R_{C}}$ 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 ${\displaystyle d}$ as independent variable:

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

If the contribution of the metal layer itself is neglected then ${\displaystyle R_{C}}$ 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).^[8] 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 ${\displaystyle {\sqrt {\frac {\rho _{C}}{R_{S}}}}}$ which is defined as the transfer length ${\displaystyle L_{T}}$.^[1][4][8] 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 ${\displaystyle Z}$. 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 ${\displaystyle w}$ is much larger than the transfer length) that a relation between ${\displaystyle R_{C}}$ and ${\displaystyle \rho _{C}}$ can be stated:^[3][4]

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

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

Circular TLM

Pad structure for circular transmission line measurements (c-TLM)

The original TLM method as described above has the drawback, that the current does not just flow within the area given by ${\displaystyle d_{i}}$ times ${\displaystyle Z}$. This means that the current density distribution also spreads to the vertical sides of the metallic pads in the figure in the TLM section, a phenomenon that is not considered in the derivation of the formula describing ${\displaystyle R_{Tot}}$.^[4][1] To account for this geometrical issue instead of rectangular metallic pads, circular pads with radius ${\displaystyle r_{i}}$ are used which are separated from a holohedral metallic coating by a distance ${\displaystyle d=r_{o}-r_{i}}$ (see figure to the right). When the total resistance between circular pad and holohedral coating is measured three distinguishable components contribute to the measured value, namely the gap resistance ${\displaystyle R_{Gap}}$ and the contact resistances at the inner and outer end of the gap area (${\displaystyle R_{i}}$ and ${\displaystyle R_{o}}$). This is expressed in the following formula:

${\displaystyle R_{Tot}=R_{Gap}+R_{i}+R_{o}}$

As will be derived in the theory section an expression for ${\displaystyle R_{Tot}}$ that allows the extraction of ${\displaystyle \rho _{C}}$ from experimental data as long as ${\displaystyle r_{i}}$ is much larger than ${\displaystyle L_{T}}$:

${\displaystyle R_{Tot}\approx {\frac {R_{S}}{2\pi }}\left[\ln \left({\frac {r_{o}}{r_{o}-d}}\right)+L_{T}\left({\frac {1}{r_{o}-d}}+{\frac {1}{r_{o}}}\right)\right]}$

Similar to the TLM method ${\displaystyle R_{S}}$ and ${\displaystyle \rho _{C}}$ can be obtained with a multiple linear regression analysis utilizing data-pairs of ${\displaystyle R_{Tot}(d)}$ and ${\displaystyle d}$.^[1]

TLM

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 ${\displaystyle R_{C}}$ with the specific contact resistivity ${\displaystyle \rho _{C}}$ 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 (${\displaystyle {\frac {R_{S}d_{i}}{Z}}}$) 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 (${\displaystyle {\frac {R_{S}\Delta x}{Z}}}$) represent the resistance due to the semiconductor substrate and the vertical resistor elements (${\displaystyle {\frac {\rho _{C}}{Z\Delta x}}}$) 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 ${\displaystyle \Delta x}$ 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.^[1]

By using Kirchhoff's circuit laws the following 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:

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

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

By taking the limit ${\displaystyle \Delta x\to 0}$ the following two differential equations are obtained:^[9]

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

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

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

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

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

Both differential equations have solutions of the form ${\displaystyle f(x)=Acosh(\alpha x)+Bsinh(\alpha x)}$ whereat ${\displaystyle A}$ and ${\displaystyle B}$ are constants which need to be determined by using appropriate boundary conditions and ${\displaystyle \alpha }$ is given by ${\displaystyle {\sqrt {R_{S}/\rho _{C}}}}$. Two boundary conditions can be obtained by defining the voltage as well as the current at the beginning of a metallic pad area as ${\displaystyle V_{0}}$ and ${\displaystyle I_{0}}$ respectively. In a formal manner this means that ${\displaystyle V(x=0)=V_{0}}$ and ${\displaystyle I(x=0)=I_{0}}$ 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 ${\displaystyle {\frac {\mathrm {d} V(x=0)}{\mathrm {d} x}}=-{\frac {R_{S}}{Z}}I_{0}}$ and ${\displaystyle {\frac {\mathrm {d} I(x=0)}{\mathrm {d} x}}=-{\frac {R_{S}}{Z}}V_{0}}$ which is the inverse of the previously defined transfer length ${\displaystyle L_{T}}$. Eventually two equations, describing the voltage and the current as a function of distance ${\displaystyle x}$ are obtained by using the four stated boundary conditions:^[4]

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

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

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 ${\displaystyle I(x=w)=0}$. This allows a further refinement of the equation describing the voltage when using the relation ${\displaystyle \sinh(x-y)=\sinh x\cosh y-\cosh x\sinh y}$:

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

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 ${\displaystyle R_{C}}$ can be expressed with ${\displaystyle {\frac {V_{0}}{I_{0}}}}$ and by setting ${\displaystyle x=0}$ in the last formula an expression can be found that relates ${\displaystyle R_{C}}$ to the specific contact resistivity ${\displaystyle \rho _{C}}$:

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

The last equation allows the calculation of ${\displaystyle \rho _{C}}$ by utilizing experimental data. Since ${\displaystyle \coth {\left({\sqrt {\frac {R_{S}}{\rho _{C}}}}w\right)}}$ goes to 1 as ${\displaystyle w}$ increases and is significantly larger than the transfer length ${\displaystyle L_{T}={\sqrt {\frac {\rho _{C}}{R_{S}}}}}$ often the estimation ${\displaystyle R_{C}\approx {\frac {\sqrt {R_{S}\rho _{C}}}{Z}}}$ is used instead of the strictly derived equality. This is identical to what was stated in the general description section.^[3][8][4] 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 ${\displaystyle \rho _{C}}$ of the metal-semiconductor junction by using the experimentally found quantities ${\displaystyle R_{C}}$ and ${\displaystyle R_{S}}$ and the width ${\displaystyle Z}$ of a metallic pad.

Circular TLM

Infinitesimal resistor network for the derivation of the c-TLM differential equations

The physical idea of deriving differential equations for the c-TLM method is the same as for TLM but polar coordinates are used instead of cartesian coordinates. This changes the resistor network that describes the metal covered area as can be seen in the figure to the right. Like for TLM by using Kirchhoff's circuit laws two coupled differential equations are obtained.

${\displaystyle {\frac {\mathrm {d} V}{\mathrm {d} r}}=-{\frac {R_{S}}{2\pi r}}I(r)}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} r}}=-{\frac {2\pi r}{\rho _{C}}}V(r)}$

When the current ${\displaystyle I}$ is eliminated a different equation for the voltage ${\displaystyle V}$ is obtained:^[10]

${\displaystyle {\frac {\mathrm {d} ^{2}V}{\mathrm {d} r^{2}}}+{\frac {1}{r}}{\frac {\mathrm {d} V}{\mathrm {d} r}}-{\frac {R_{S}}{\rho _{C}}}V=0}$

A general solution to this type of differential equations is given as follows, whereat ${\displaystyle A}$ and ${\displaystyle B}$ are unspecified constants and ${\displaystyle \alpha }$ is ${\displaystyle {\sqrt {R_{C}/\rho _{C}}}}$. The functions ${\displaystyle I_{0}}$ and ${\displaystyle K_{0}}$ are zero-order modified Bessel functions of the first and second kind respectively.^[11]

${\displaystyle V(r)=AI_{0}(\alpha r)+BK_{0}(\alpha r)}$

By utilizing the coupled differential equations above and the differentiation rules for modified Bessel functions (${\displaystyle I_{0}'(x)=I_{1}(x)}$, ${\displaystyle K_{0}'(x)=-K_{1}(x)}$)^[11] an expression for the current can be obtained. The functions ${\displaystyle I_{1}}$ and ${\displaystyle K_{1}}$ are first-order Bessel functions of the first and second kind respectively.

${\displaystyle I(r)=-{\frac {2\pi r}{R_{S}}}\left[A\alpha I_{1}(\alpha r)-B\alpha K_{1}(\alpha r)\right]}$

Modified Bessel functions of the first kind, I_α(x), for α = 0, 1, 2, 3

Modified Bessel functions of the second kind, K_α(x), for α = 0, 1, 2, 3

Now after having obtained expressions for the current as well as for the voltage, expressions for the contact resistances corresponding to the inner and outer boundary of the gap area have to be found (compare with the schematic illustration of the measurement metallization in the general section). The contact resistance at the inner boundary is given by ${\displaystyle R_{i}={\frac {V(r_{i})}{|I(r_{i})|}}}$ and during a measurement the current in the middle of the circular metallic pad is zero (${\displaystyle I(r=0)=0}$). Since the modified Bessel function ${\displaystyle K_{0}(r)}$ tends to infinity when ${\displaystyle r}$ tends to zero (see figures to the right), the constant ${\displaystyle B}$ has to be zero because the voltage can not be infinite. Considering this, the contact resistance at the inner boundary of the gap area equates to:

${\displaystyle R_{i}={\frac {V(r_{i})}{|I(r_{i})|}}={\frac {R_{S}}{2\pi r_{i}\alpha }}{\frac {I_{0}(\alpha r_{i})}{I_{1}(\alpha r_{i})}}}$

In a similar manner an expression for the contact resistance at the outer boundary of the gap area can be found when ${\displaystyle r_{i}}$ is replaced with ${\displaystyle r_{o}}$ (compare with the drawing in the general section). Here, also a boundary condition for the current can be given, namely ${\displaystyle I(r=\infty )=0}$. This means that A has to be zero because the function ${\displaystyle I_{0}(r)}$ tends to infinity (see figure to the right) as ${\displaystyle r}$ goes to infinity. In turn this means that the contact resistance at the outer boundary of the gap area is given by:

${\displaystyle R_{o}={\frac {V(r_{o})}{|I(r_{o})|}}={\frac {R_{S}}{2\pi r_{o}\alpha }}{\frac {K_{0}(\alpha r_{o})}{K_{1}(\alpha r_{o})}}}$

The resistance due to the gap area itself ${\displaystyle R_{Gap}}$ can be found by considering the horizontal differential resistor ${\displaystyle {\frac {R_{S}dr}{2\pi r}}}$ in the figure in this section and by integrating from ${\displaystyle r_{i}}$ to ${\displaystyle r_{o}}$. By adding ${\displaystyle R_{i}}$, ${\displaystyle R_{o}}$ and ${\displaystyle R_{Gap}}$ an expression for the total resistance can be given:

${\displaystyle R_{Tot}={\frac {R_{S}}{2\pi }}\left[ln\left({\frac {r_{o}}{r_{i}}}\right)+{\frac {L_{T}}{r_{i}}}{\frac {I_{0}(r_{i}/L_{T})}{I_{1}(r_{i}/L_{T})}}+{\frac {L_{T}}{r_{o}}}{\frac {K_{0}(r_{o}/L_{T})}{K_{1}(r_{o}/L_{T})}}\right]}$

When the outer and the inner radius are much larger than the transfer length ${\displaystyle L_{T}}$ the quotients of the modified Bessel functions are approximately one.^[1] This means that when ${\displaystyle r_{i}}$ is substituted with ${\displaystyle r_{o}-d}$ the same formula for ${\displaystyle R_{Tot}}$ as given in the general section is found, which can be used for extracting ${\displaystyle R_{S}}$ and ${\displaystyle \rho _{C}}$ from experimental data:

${\displaystyle R_{Tot}\approx {\frac {R_{S}}{2\pi }}\left[\ln \left({\frac {r_{o}}{r_{o}-d}}\right)+L_{T}\left({\frac {1}{r_{o}-d}}+{\frac {1}{r_{o}}}\right)\right]}$