2.2.1 Basic Electrical Characterizations and Parameters
The basic C-V characteristic of a metal-oxide-semiconductor capacitor and the relative 1/C2-V curve are shown in Fig. 2-1. Both the high-frequency C-V (HFCV) and quasi-static C-V (QSCV) curves are included. The capacitance equivalent thickness (CET) was calculated from the capacitance at accumulation mode. If the physical thickness of oxide layer is thick enough to suppress the quantum phenomenon, the CET is equal to the equivalent oxide thickness (EOT) and can be obtained fromC=εox/EOTataccumulation mode. Flat-band voltage (Vfb) is extracted as the gate bias to achieve the flat-band capacitance (Cfb) which is determined by :
Cox is the oxide capacitance and CD is the capacitance contributed by the Si substrate as biased at Vfb. εSi is the silicon permittivity, K is the Boltzman constant, T is the absolute temperature, and q is electron charge. LD is Debye length dependent upon substrate concentration NA, which is evaluated from the slop of 1/C2 -Vg plot at the depletion region. The interface states (Dit) between oxide and
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gate dielectric can be calculated from the capacitance difference between HFCV and QSCV, and is determined by :
( QS ) The Jg-Vg characteristic is shown in Fig. 2-2 (a). The corresponding Fowler-Nordheim (FN) plot is shown in Fig. 2-2 (b). As the conduction through the oxide is due to FN tunneling, the FN plot is a straight line. The slope is determined by:
, where h is Planck constant and m* is effective mass of electron in dielectric. The barrier height (ΦB) can be extracted from the parameter S. To get accurate ΦB of metal gate, the tunneling current should injected from the gate electrode at accumulation mode and the oxide electric field is calculated from (Vg −Vfb) /EOT.
2.2.2 Work Function Extraction
The work function extracted from the electrical characteristic is always called effective work function (Φeff). Effective work function is electrically extracted commonly using the Vfb versus EOT plot and sometime using the HFCV curves and FN plots. In some cases, effective work function can be extracted from single C-V measurement as the theoretical C-V curve is known. We can calculate the theoretical C-V curve of a MOS capacitor from the following equations [4]:
1 1 1
where Vox is the voltage drop accross oxide and VSi is the voltage drop accross Si substrate. CSi is the capacitance contributed by the Si substrate. VSi also represents for the surface potential ψs. Vox is the product of applied electrical field (Fox) and the oxide thickness (EOT). To evaluate the Fox and CSi regardless of quantum mechanics, we should first introduce an abbreviation:
1 1 1 2/
The electrical field is hence determined by:
2 ( , po ) equilibrium densities of electrons and holes, respectively, in the bulk of semiconductor, and β=KT q/ as the substrate dopant is p-type. The CSi is
As the substrate concentration and oxide thickness are known and given a value of ψs, the relative C and Vg are obtained from equations (4)-(7). A whole theoretical curve with a Vfb=0 can be obtained as all ψs are assigned.
To extract effective work function from the measured C-V curves, we first obtained the theoretical curve and then shift the theoretical curve to fit the measured curve, as shown in Fig. 2-3. Regarding the offset (∆Vfb), at Vfb the band structure of silicon substrate is flat for both theoretical curve and measured curve;
the ∆Vfb is hence contributed by the effective work function difference between the gate electrode and Si substrate. We can obtain the effective work function of the gate electrode from the value of the sum of the offset and the substrate work
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function which is obtained according to the substrate concentration. If the oxide is charged, the effective work function is further minus Qeff/Cox of which Qeff is the effective oxide charges.
The extraction of effective work function from the FN plot is based on the extraction of ΦB as described in the last section. The effective work function is then determined by:
,
m eff B χox
Φ = Φ + , ……….. (8) where χox is the electron affinity of oxide.
The extraction of effective work function from the Vfb-EOT plotis based on the equation:
, / /
fb m eff s eff ox m s eff ox
V = Φ − Φ −Q C = Φ − Φ −Q EOT ε , ……….. (9) The extrapolation of Vfb at EOT=0 give the value of Φm,eff-ΦSi, as shown in Fig.
2-4. If the ΦSi is known, we can obtain the effective work function. The hypothesis is based on that the substrate concentration is constant and the oxide charges are the same and fixed at the interface for all capacitors with different EOT so that the linear equation is assured. A general formula of Vfb should be express as:
0 where ρ(x) is the charge distribution in gate oxide. As the bulk charge is not low enough, eq. (9) cannot be applied to extraction Φm,eff because of the nonlinear behavior. For example, high dielectric-constant (high-k) material such as HfO2 has a high density of oxide charges in bulk and a thick interfacial layer at the interface between HfO2 and Si substrate. These result in the failure of the conventional work-function extraction from the Vfb-EOT plot. To solve this issue, an intentional, thick silicon-dioxide is grown before high-k film deposition [5]. The intentional
oxide layer can suppress the interfacial layer growth between high-k film and Si substrate during the high-temperature annealing such that the structure is thermally stable. We can easily fabricate a set of stable dielectric stacks with fixed HfO2 layerand varied SiO2 layer or otherwise. If the charge distribution is uniform in bulk and the interface charge is a sheet function as shown in Fig. 2-5, the Vfb
could be rewritten as respectively. Qh.k. and Qox are the interface charges. In general, the amount of bulk oxide charges of SiO2 is small enough to be ignored. The terms of ρox hence are eliminated from eq. (11). With a fixed high-k thickness, the first bracket term is constant and eq. (11) becomes a linear function of EOTtot
(EOTtot =EOTh k. .+EOTox). Eq. (11) is simplifies to be
According to eq. (12), the effective work function extracted from linear extrapolation is counting for an offset due to charges in high-k layer. As the EOTh.k.
is small enough, the offset can be ignored. Hence eq. (12) is the same as eq. (9), and the evaluated extracted work function on high-k layer is more accurate.
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