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Advanced Electrical Characterization of Charge Trapping in MOS Transistors

5.2 Capacitance-Voltage Measurements

Depending on the type of measurements that can be performed on the DUT and setup, different methods of defect parameter extraction are available. In general, trap densities can be extracted both from the capacitance and the conductance. In fact, they are related to each other by the Kronig-Kramers relation [4]. The effect of defects at the interface or in the oxide on CV measurements depends on the charge transition times of the defects in relation to the temporal parameters of the measurement. The following cases can be distinguished and are illustrated in Figure 5.8:

  • Fixed charges or very slow defects, which do not change their charge state during the measurement will add a constant voltage offset to the theoretical defect-free CV curve. This is due to the additional voltage which has to be supplied to the gate to compensate these charges.

  • Defects fast enough to react to the gate bias sweep, but too slow to react to the AC frequency. These defects will charge or discharge during the measurement, at which point part of the change in gate bias is required to supply the opposing charge to the gate. This modifies the relation between gate voltage and surface potential, which appears as a stretch-out in the CV curve where such defects (dis)charge.

  • Defects fast enough to react to the AC measurement signal. In addition to the stretch-out of the CV curve, these defects cause additional small-signal current and thus add to the measured capacitance.


Figure 5.8: Schematic CV curves of a pMOS transistor, affected by defects located close to the conduction band. (a) Defect free CV curve. (b) Fixed charges or defects with \( \tau f\sub {sweep} > 1 \) shift the CV curve along the voltage axis. (c) Defects fast enough to react to the DC sweep but too slow to react to the AC frequency cause a stretch-out of the CV curve. (d) Defects fast enough to react to the AC frequency cause additional capacitance in the CV curve. In general, a superposition of these effects will be obtained, depending on the chosen measurement parameters.

5.2.1 Capacitance Methods

The general shape of the differential capacitance obtained from a MOSCAP or MOSFET depends on measurement conditions. To understand why, consider the differential capacitance of the device as viewed from the gate:

(5.38) \{begin}{`} C = \frac {\intd {Q\sub {G}}}{\intd {V\sub {G}}}. \{end}{`}

We can replace the gate charge with the opposing semiconductor (\( Q\sub {s} \)) and interface (\( Q\sub {it} \)) charges. Further, the gate voltage can be written as the sum of the flatband (\( V\sub {FB} \)) and oxide (\( V\sub {ox} \)) voltages and the surface potential (\( \phi \sub {s} \)) [105]:

(5.39) \{begin}{`} C = \frac {-\intd (Q\sub {s}+Q\sub {it})}{\intd (V\sub {FB}+V\sub {ox}+\phi \sub {s})} \{end}{`}

By rearranging the equation and replacing the semiconductor charge by the sum of hole charge (\( Q\sub {p} \)), bulk charge in the space charge region (\( Q\sub {b} \)) and electron charge (\( Q\sub {n} \)), we obtain:

(5.40) \{begin}{`} C &= - \left ( \frac {\intd V\sub {ox}}{\intd Q\sub {s}+\intd Q\sub {it}} + \frac {\intd \phi \sub {s}}{\intd Q\sub {p} + \intd Q\sub {b} + \intd Q\sub {n} +
\intd Q\sub {it}} \right )^{-1} \nonumber \\ &= \left ( \frac {1}{C\sub {ox}} + \frac {1}{C\sub {p} + C\sub {b} + C\sub {n} + C\sub {it}} \right )^{-1}. \{end}{`}

This is shown as an equivalent circuit in Figure 5.9a for an nMOS device.


Figure 5.9: Equivalent circuits for CV measurements in the different regimes. (a) General equivalent circuit. (b) Schematic CV-measurments for varying measurement condi- tions. (1–4) Accumulation, depletion, low-frequency inversion and high-frequency inversion equivalent circuits. To obtain the low- and high-frequency curves (blue and orange), the DC sweep rate has to be slow enough for the minority carriers to follow, otherwise the deep depletion curve (red) is obtained. If in addition the AC frequency is low enough for the minority carriers to respond, the low-frequency curve is obtained.

As can be seen from the figure, depending on the measurement conditions, any of three principal types of CV curves can be obtained. Common among all of them are the accumulation and depletion regions. In accumulation, the majority carriers are at the interface and \( C\sub {p} \) is large, leading to \( C\approx C\sub {ox} \). Towards depletion, a space charge region forms which—together with the interface charges—compensates the gate charge by varying its thickness. At higher voltages, an inversion layer will form if minority carriers can be generated or supplied fast enough to follow the sweep frequency. If this is not the case, the depletion layer will widen and the device will operate in deep depletion. If an inversion layer forms, a low- or high frequency curve is obtained4. If the measurement frequency is too high for the inversion layer charge density to follow the AC signal, these charges will be compensated by the width of the space charge region. Since the DC part will be compensated by the inversion layer, the small signal capacitance will stay essentially constant. Only if the AC signal is slow enough for the inversion layer charges to follow, a low-frequency curve will be obtained. This is usually the case for MOSFETs, if the source/drain contacts are connected to ground, as minority carriers can be quickly supplied from there, but usually not for MOSCAPs, as very low measurement frequencies are needed to allow sufficient time for minority carrier generation in the bulk. The capacitance in the low-frequency regime approaches \( C\sub {ox} \), the situation is analogous to that in accumulation.

4 Note that the meanings of high- and low-frequency curves are ambiguous. Depending on the context, they might refer to measurements where either the interface defects or the inversion charges are (un)able to respond to the measurement frequency.

Low frequency method

Low frequency here refers to measurements where both inversion charges and interface traps can respond. As shown in Figure 5.9, the low frequency capacitance can be written as

(5.41) \{begin}{`} C\sub {lf} = \left ( \frac {1}{C\sub {ox}} + \frac {1}{C\sub {s}+C\sub {it}} \right )^{-1}, \{end}{`}

with \( C\sub {s}=C\sub {b}+C\sub {n} \) the semiconductor surface capacitance in depletion and inversion. From this, the interface trap density can be calculated with \( D\sub {it} = C\sub {it}/qA \) as

(5.42) \{begin}{`} \label {eqn:ex:cv:ditlf} D\sub {it} = \frac {1}{qA} \left ( \frac {C\sub {ox} C\sub {lf}}{C\sub {ox}-C\sub {lf}} - C\sub {s} \right ). \{end}{`}

The semiconductor capacitance \( C\sub {s} \) necessary to calculate this can be obtained by either TCAD simulation or analytically [105] with the surface potential required for this calculation obtained using Berglund’s method [102]:

(5.43) \{begin}{`} \phi \sub {s} = \int _{V\sub {G_1}}^{V\sub {G_2}} (1-C\sub {lf}/C\sub {ox}) \intd {V\sub {G}} + \Delta , \{end}{`}

with \( \Delta \) the surface potential at \( \vg =V\sub {G_1} \). The difficulty in this method lies in obtaining the correct semiconductor capacitance, as even small deviations from its real value may lead to relatively large spurious defect densities. If the prime interest of a study lies in finding the changes to the defect density, e.g. due to electrical stress, a variant of this method can be used wherein an initially recorded low frequency curve is used as a reference, e.g.:

(5.44) \{begin}{`} \label {eqn:ex:cv:deltaditlf} \Delta D\sub {it} = D\sub {it}-D\sub {it,ref} = \frac {1}{qA} \left ( \frac {C\sub {ox} C\sub {lf}}{C\sub {ox}-C\sub {lf}} - \frac
{C\sub {ox} C\sub {lf,ref}}{C\sub {ox}-C\sub {lf,ref}} \right ). \{end}{`}

This method has been used by the author in [BSC3] to measure the temporal evolution of near-interface defects under BTI stress to study the permanent component of the degradation using the hydrogen-release model (see Section 3.1.5, [BSC4])

High-low frequency method

Another way of obtaining \( C_s \) required for Equation (5.42) is to measure a high-frequency curve at an AC frequency too fast for the interface traps to respond. The interface trap density can then be expressed as:

(5.45) \{begin}{`} D\sub {it} = \frac {1}{qA} \left ( \frac {C\sub {ox} C\sub {lf}}{C\sub {ox}-C\sub {lf}} - \frac {C\sub {ox} C\sub {hf}}{C\sub {ox}-C\sub {hf}} \right ). \{end}{`}

This method is limited in range to the onset of inversion [105].

Terman method

A high frequency curve, as measured for the high-low frequency method above still contains information about interface defects. While the defects will not respond to the AC frequency, they will be charged during the DC sweep and cause a stretch out of the curve along the gate voltage axis, as additional charge has to be supplied to the gate in order to reach the same surface potential as in an ideal device.

By comparing the obtained capacitances \( C\sub {hf} \) with the ones from a theoretical curve, the relation between \( \Delta V\sub {G} = V\sub {G}-V\sub {G,ideal} \) and \( \phi \sub {s} \) can be obtained. The interface trap capacitance can then be calculated as [105]

(5.46) \{begin}{`} D\sub {it} = \frac {C\sub {ox}}{qA}\left ( \frac {d\vg }{d\phi \sub {s}} \right ) - \frac {C\sub {s}}{qA} = \frac {C\sub {ox}}{qA} \frac {d\Delta V\sub {G}}{d \phi
\sub {s}}. \{end}{`}

Gray-Brown method

The Gray-Brown method [136] can be used to estimate the interface trap density close to the majority carrier band edge. For this, high-frequency CV curves are measured at various temperatures typically from room temperature to below 100K. The change in temperature causes a shift of the Fermi level between the measurements which in turn changes the interface trap occupancy. By comparing the corresponding flat band voltages, the interface trap capacitance can be extracted in a manner similar to the Terman method. To obtain true high frequency CV curves near flatband, measurement frequencies of more than 200 MHz [4] or even GHz [137] may be required. Lower frequency measurements may be used to obtain qualitative results.

5.2.2 Conductance Method

Unlike the capacitance methods, the conductance method developed by Nicollian and Goetzberger [138, 139] measures the loss occurring from (dis)charging the interface traps. Delayed responses to the AC voltage from interface traps with time constants in the frequency range of the voltage causes a real-valued current. By measuring at varying frequency and gate bias, this loss can be characterized and linked to interface trap density.


Figure 5.10: MOSCAP in depletion between low and high frequency regimes for a single interface defect. (a) Equivalent circuit with lossy interface defect represented by a series RC circuit. (b) Simplified circuit with equivalent parallel capacitance and conductance. (c) Behavior of C(math image) and G(math image) with varying measurement frequency. At low frequencies, \( C\sub {P} = C\sub {b} || C\sub {it} \) and at high frequencies \( C\sub {P} = C\sub {b} \), in both cases \( G\sub {P}/\omega \approx 0 \). At the corner frequency \( \omega \tau =1 \), however, \( G\sub {P}/\omega \) peaks at \( C\sub {it} \). This behavior is exploited in the conductance method [4].

Figure 5.10a shows the equivalent circuit for the situation with interface trap loss. This is commonly replaced with the circuit from Figure 5.10b, as it allows to write the equations in a symmetric form. For a single interface trap with \( \tau \sub {it} = R\sub {it}C\sub {it} \) this yields

(5.47–5.48) \{begin}{`} C\sub {p}-C\sub {b} &= \frac {C\sub {it}}{1+(\omega \tau \sub {it})^2} \\ \frac {G\sub {p}}{\omega } &= \frac {qA\omega \tau \sub {it}D\sub
{it}}{1+(\omega \tau \sub {it})^2}, \{end}{`}

plotted in Figure 5.10c. In measurements, many interface traps will be present, with their trap levels distributed in energy. With the assumption of traps distributed continuously throughout the band gap, the conductance can be written as [139]

(5.49) \{begin}{`} \frac {G\sub {p}}{\omega } = \frac {qAD\sub {it}}{2\omega \tau \sub {it}} \ln \left [ 1+ (\omega \tau \sub {it})^2 \right ]. \{end}{`}

The maximum of this expression is found at \( \omega \approx 2/\tau \sub {it} \), resulting in

(5.50) \{begin}{`} D\sub {it} \approx \frac {2.5}{qA} \left ( \frac {G\sub {p}}{\omega } \right )_{max}. \{end}{`}

The measurement thus works by recording capacitance curves at multiple frequencies to determine \( G\sub {p}/\omega \) for each voltage. The method is typically used from flatband to weak inversion [105]. To simplify the calculation, \( G\sub {p}/\omega \) can also be obtained from the equivalent parallel capacitances and conductances provided by many instruments using:

(5.51) \{begin}{`} \frac {G\sub {p}}{\omega } = \frac {\omega G\sub {m}C\sub {ox}^2}{G\sub {m}^2 + \omega ^2(C\sub {ox}-C\sub {m})^2}. \{end}{`}

The conductance method is generally considered more sensitive than the capacitance methods, but requires measurements over a wide range of frequencies to yield results.