mos analysis

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ecee.colorado.edu/~bart/book/book/chapter6/ch6_3.htm#6_3_1

Principles of Semiconductor Devices

Title Page - Table of Contents - Help B. Van Zeghbroeck, 2011

Chapter 6: MOS Capacitors

6.3. MOS analysis

6.3.1. Flatband voltage calculation6.3.2. Inversion layer charge6.3.3. Full depletion analysis6.3.4. MOS Capacitance

6.3.1. Flatband voltage calculation

If there is no charge present in the oxide or at the oxide-semiconductor interface, the flatband voltage simply

equals the difference between the gate metal workfunction, FM, and the semiconductor workfunction, FS.

(6.3.1)

The workfunction is the voltage required to extract an electron from the Fermi energy to the vacuum level.This voltage is between three and five Volt for most metals. The actual value of the workfunction of a metaldeposited onto silicon dioxide is not exactly the same as that of the metal in vacuum. Figure 6.3.1 providesexperimental values for the workfunction of different metals, as obtained from a measurement on a MOScapacitor, as a function of the measured workfunction in vacuum. The same data is also listed in Table 6.3.1.

Figure 6.3.1.: Workfunction of Magnesium (Mg), Aluminum (Al), Copper (Cu), Silver (Ag), Nickel (Ni) andGold (Au) obtained from I-Vand C-Vmeasurements on MOS structures as a function of theworkfunction of those metals measured in vacuum.

Table 6.3.1: Workfunction of selected metals as measured in vacuum and as obtained from a C-Vmeasurement on an MOS structure.

The workfunction of a semiconductor F re uires some more thou ht since the Fermi ener varies with the
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ecee.colorado.edu/~bart/book/book/chapter6/ch6_3.htm#6_3_1 2

doping type as well as with the doping concentration. This workfunction equals the sum of the electron affinity

in the semiconductor, c, the difference between the conduction band energy and the intrinsic energy dividedby the electronic charge in addition to the bulk potential. This is expressed by the following equation:

(6.3.2)

For MOS structures with a highly doped poly-silicon gate one must also calculate the workfunction of the gatebased on the bulk potential of the poly-silicon.

(6.3.3)

Where Na,poly and Nd,poly are the acceptor and donor density of thep-type and n-type poly-silicon gate

respectively.

For a pMOS capacitor, which has an n-type substrate with doping density Nd, the workfunction difference

equals:

(6.3.4)

The flatband voltage of real MOS structures is further affected by the presence of charge in the oxide or at thoxide-semiconductor interface. The flatband voltage still corresponds to the voltage, which, when applied tothe gate electrode, yields a flat energy band in the semiconductor. Any charge in the oxide or at the interfaceaffects the flatband voltage. For a charge, Qi, located at the interface between the oxide and the

semiconductor, and a charge density, rox, distributed within the oxide, the flatband voltage is given by:

(6.3.5)

where the second term is the voltage across the oxide due to the charge at the oxide-semiconductor interfaceand the third term is due to the charge density in the oxide.

The actual calculation of the flatband voltage is further complicated by the fact that charge can move withinthe oxide. The charge at the oxide-semiconductor interface due to surface states also depends on the positioof the Fermi energy.

Since any additional charge affects the flatband voltage and thereby the threshold voltage, great care has tobe taken during fabrication to avoid the incorporation of charged ions as well as creation of surface states.

Example 6.1 Calculate the flatband voltage of a silicon nMOS capacitor with a substrate doping Na= 1017

cm-3 and an aluminum gate (FM= 4.1 V). Assume there is no fixed charge in the oxide or at

the oxide-silicon interface.

Solution The flatband voltage equals the work function difference since there is no charge in the oxide

or at the oxide-semiconductor interface.

The flatband voltages for nMOS and pMOS capacitors with an aluminum or a poly-silicon gatare listed in the table below.
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6.3.2. Inversion layer charge

The basis assumption as needed for the derivation of the MOSFET models is that the inversion layer chargeisproportionalwith the applied voltage. In addition, the inversion layer charge is zero at and below thethreshold voltage as described by:

(6.3.6)

The linear proportionality can be explained by the fact that a gate voltage variation causes a charge variationin the inversion layer. The proportionality constant between the charge and the applied voltage is thereforeexpected to be the gate oxide capacitance. This assumption also implies that the inversion layer charge islocated exactly at the oxide-semiconductor interface.

Because of the energy band gap of the semiconductor separating the electrons from the holes, the electronscan only exist if thep-type semiconductor is first depleted. The voltage at which the electron inversion-layerforms is referred to as the threshold voltage.

To justify this assumption we now examine a comparison of a numeric solution with equation (6.3.6) as shownin Figure 6.3.2.

Figure 6.3.2.: Charge density due to electrons in the inversion layer of an MOS capacitor. Compared arethe analytic solution (solid line) and equation (6.3.6) (dotted line) forNa

= 1017 cm-3 and tox20 nm.

While there is a clear difference between the curves, the difference is small. We will therefore use our basic

assum tion when derivin the different MOSFET models since it dramaticall sim lifies the derivation, be it
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while losing some accuracy.

6.3.3. Full depletion analysis

We now derive the MOS parameters at threshold with the aid of Figure 6.3.3. To simplify the analysis we makthe following assumptions: 1) we assume that we can use the full depletion approximation and 2) we assumethat the inversion layer charge is zero below the threshold voltage. Beyond the threshold voltage we assumethat the inversion layer charge changes linearly with the applied gate voltage.

The derivation starts by examining the charge per unit area in the depletion layer, Qd. As can be seen in

Figure 6.3.3 (a), this charge is given by:

(6.3.7)

Wherexd is the depletion layer width and Na is the acceptor density in the substrate. Integration of the charge

density then yields the electric field distribution shown in Figure 6.3.3 (b). The electric field in the

semiconductor at the interface, s, and the field in the oxide equal, ox:

(6.3.8)

The electric field changes abruptly at the oxide-semiconductor interface due to the difference in the dielectricconstant. At a silicon/SiO2 interface the field in the oxide is about three times larger since the dielectric

constant of the oxide (eox = 3.9 e0) is about one third that of silicon (es = 11.9 e0). The electric field in the

semiconductor changes linearly due to the constant doping density and is zero at the edge of the depletionregion, based on the full depletion approximation.

The potential shown in Figure 6.3.3 (c) is obtained by integrating the electric field. The potential at the

surface, fs, equals:

(6.3.9)

Figure 6.3.3: Electrostatic analysis of an MOS structure. Shown are (a) the charge density, (b) the electricfield, (c) the potential and (d) the energy band diagram for an nMOS structure biased indepletion.
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The calculated field and potential is only valid in depletion. In accumulation, there is no depletion region andthe full depletion approximation does not apply. In inversion, there is an additional charge in the inversionlayer, Qinv. This charge increases gradually as the gate voltage is increased. However, this charge is only

significant once the electron density at the surface exceeds the hole density in the substrate, Na. We

therefore define the threshold voltage as the gate voltage for which the electron density at the surface equalsNa. This corresponds to the situation where the total potential across the surface equals twice the bulk

potential, fF.

(6.3.10)

The depletion layer in depletion is therefore restricted to this potential range:

(6.3.11)

For a surface potential larger than twice the bulk potential, the inversion layer charge increases exponentiallywith the surface potential. Consequently, an increased gate voltage yields an increased voltage across theoxide while the surface potential remains almost constant. We will therefore assume that the surface potentialand the depletion layer width at threshold equal those in inversion. The corresponding expressions for thedepletion layer charge at threshold, Qd,T, and the depletion layer width at threshold,xd,T, are:

(6.3.12)

(6.3.13)

Beyond threshold, the total charge in the semiconductor has to balance the charge on the gate electrode, QMor:

(6.3.14)

where we define the charge in the inversion layer as a quantity, which needs to determined but should beconsistent with our basic assumption. This leads to the following expression for the gate voltage, VG:

(6.3.15)

In depletion, the inversion layer charge is zero so that the gate voltage becomes:

(6.3.16)

while in inversion this expression becomes:

(6.3.17)

the third term in (6.3.17) states our basic assumption, namely that any change in gate voltage beyond thethreshold requires a change of the inversion layer charge. From the second equality in equation (6.3.17), wethen obtain the threshold voltage or:

(6.3.18)

Example 6.2 Calculate the threshold voltage of a silicon nMOS capacitor with a substrate doping Na = 101

cm-3, a 20 nm thick oxide (eox= 3.9 e0

) and an aluminum gate (FM= 4.1 V). Assume there is

no fixed charge in the oxide or at the oxide-silicon interface.

Solution The threshold voltage equals:
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Where the flatband voltage was already calculated in example 6.1. The threshold voltagevoltages for nMOS and pMOS capacitors with an aluminum or a poly-silicon gate are listed inthe table below.

6.3.4. MOS Capacitance

6.3.4.1. Simple capacitance model6.3.4.2. Calculation of the flat band capacitance6.3.4.3. Deep depletion capacitance6.3.4.4. Experimental results and comparison with theory6.3.4.5. Non-Ideal effects in MOS capacitors

Capacitance voltage measurements of MOS capacitors provide a wealth of information about the structure,which is of direct interest when one evaluates an MOS process. Since the MOS structure is simple to fabricate

the technique is widely used.To understand capacitance-voltage measurements one must first be familiar with the frequency dependenceof the measurement. This frequency dependence occurs primarily in inversion since a certain time is neededto generate the minority carriers in the inversion layer. Thermal equilibrium is therefore not immediatelyobtained.

The low frequency or quasi-static measurement maintains thermal equilibrium at all times. This capacitanceis the ratio of the change in charge to the change in gate voltage, measured while the capacitor is inequilibrium. A typical measurement is performed with an electrometer, which measures the charge added perunit time as one slowly varies the applied gate voltage.

The high frequency capacitance is obtained from a small-signal capacitance measurement at highfrequency. The bias voltage on the gate is varied slowly to obtain the capacitance versus voltage. Under suchconditions, one finds that the charge in the inversion layer does not change from the equilibrium value at theapplied dc voltage. The high frequency capacitance therefore reflects only the charge variation in the

depletion layer and the (rather small) movement of the inversion layer charge.

In this section, we first derive the simple capacitance model, which is based on the full depletion approximationand our basic assumption. The comparison with the exact low frequency capacitance will reveal that thelargest error occurs at the flatband voltage. We therefore derive the exact flatband capacitance using thelinearized Poisson's equation. Then we discuss deep depletion as well as the non-ideal effects in MOScapacitors.

6.3.4.1. Simple capacitance model

The capacitance of an MOS capacitor is obtained using the same assumptions as those listed in section 6.3.3The MOS structure is treated as a series connection of two capacitors: the capacitance of the oxide and thecapacitance of the depletion layer.

In accumulation, there is no depletion layer. The remaining capacitor is the oxide capacitance, so that thecapacitance equals:

(6.3.19)

In depletion, the MOS capacitance is obtained from the series connection of the oxide capacitance and thecapacitance of the depletion layer, or:

(6.3.20)

wherexd is the variable depletion layer width which is calculated from:

(6.3.21)
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In order to find the capacitance corresponding to a specific value of the gate voltage, we also need to use therelation between the potential across the depletion region and the gate voltage, given by:

(6.3.16)

In inversion, the capacitance becomes independent of the gate voltage. The low frequency capacitanceequals the oxide capacitance since charge is added to and removed from the inversion layer. The highfrequency capacitance is obtained from the series connection of the oxide capacitance and the capacitance othe depletion layer having its maximum width,x

d,T. The capacitances are given by:

(6.3.22)

The capacitance of an MOS capacitor as calculated using the simple model is shown in Figure 6.3.4. Thedotted lines represent the simple model while the solid line corresponds to the low frequency capacitance asobtained from the exact analysis.

Figure 6.3.4 : Low frequency capacitance of an nMOS capacitor. Shown are the exact solution for the lowfrequency capacitance (solid line) and the low and high frequency capacitance obtained withthe simple model (dotted lines). Na

= 1017 cm-3 and tox= 20 nm.

6.3.4.2. Calculation of the flat band capacitance

The simple model predicts that the flatband capacitance equals the oxide capacitance. However, thecomparison with the exact solution of the low frequency capacitance as shown in Figure 6.3.4 reveals that theerror can be substantial. The reason for this is that we have ignored any charge variation in thesemiconductor. We will therefore now derive the exact flatband capacitance.

To derive the actual flatband capacitance we first linearize Poisson's equation. Since the potential across thesemiconductor at flatband is zero, we expect the potential to be small as we vary the gate voltage around theflatband voltage. Poisson's equation can then be simplified to:

(6.3.23)

Charge due to ionized donors or electrons were ignored, since neither are present in ap-type semiconductoraround flatband. The linearization is obtained by replacing the exponential function by the first two terms of its

Taylor series expansion. The solution to this equation is:

(6.3.24)
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s .

potential enables the derivation of the capacitance of the semiconductor under flatband conditions, or:

(6.3.25)

The flatband capacitance of the MOS structure at flatband is obtained by calculating the series connection ofthe oxide capacitance and the capacitance of the semiconductor, yielding:

(6.3.26)

Example 6.3 Calculate the oxide capacitance, the flatband capacitance and the high frequencycapacitance in inversion of a silicon nMOS capacitor with a substrate doping Na

= 1017 cm-3

a 20 nm thick oxide (eox = 3.9 e0) and an aluminum gate (FM = 4.1 V).

Solution The oxide capacitance equals:

The flatband capacitance equals:

where the Debye length is obtained from:

The high frequency capacitance in inversion equals:

and the depletion layer width at threshold equals:

The bulk potential, fF, was already calculated in example 6.1

6.3.4.3. Deep depletion capacitance


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Deep depletion occurs in an MOS capacitor when measuring the high-frequency capacitance while sweepingthe gate voltage "quickly". Quickly means that the gate voltage must be changed fast enough so that thestructure is not in thermal equilibrium. One then observes that, when ramping the voltage from flatband tothreshold and beyond, the inversion layer is not or only partially formed. This occurs since the generation ofminority carriers cannot keep up with the amount needed to form the full inversion layer. The depletion layertherefore keeps increasing beyond its maximum thermal equilibrium value, xd,T resulting in a capacitance

which further decreases with voltage.

The time required to reach thermal equilibrium can be estimated by taking the ratio of the total charge in theinversion layer to the thermal generation rate of minority carriers. A complete analysis should include both the

surface generation rate as well as generation in the depletion layer and the quasi-neutral region. A goodapproximation is obtained by considering only the generation rate in the depletion region and the quasi-neutral region. This yields the following equation:

(6.3.27)

where the generation in the depletion layer was assumed to be constant. For materials with a long diffusionlength one can also ignore the generation in the depletion region. The rate of change required to observedeep depletion is then obtained from:

(6.3.28)

This equation predicts that deep depletion is less likely at higher ambient temperature, since the intrinsiccarrier density ni increases exponentially with temperature. The intrinsic density also decreases exponentially

with the energy bandgap. Therefore, MOS structures made with wide bandgap materials (for instance 6H-SiCfor which Eg = 3 eV), have an extremely pronounced deep depletion effect.

In silicon MOS capacitors, one finds that the occurrence of deep depletion can be linked to the minority carrielifetime. Structures with a long (0.1 ms) lifetime require a few seconds to reach thermal equilibrium, which

results in a pronounced deep depletion effect at room temperature. Structures with a short (1 ms) lifetime donot show this effect.

Carrier generation due to light will increase the generation rate beyond the thermal generation rate andtherefore reduces the time needed to reach equilibrium. Deep depletion measurements are therefore done inthe dark.

6.3.4.4. Experimental results and comparison with theory

As an example, we present the measured low frequency (quasi-static) and high frequency capacitance-voltag

curves of an MOS capacitor. The capacitance was measured in the presence of ambient light as well as in thedark as explained in the caption of Figure 6.3.5.
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gure . . : ow requency quas-s a c an g requency capac ance measuremen o a pcapacitor. Shown are, from top to bottom, the low frequency capacitance measured in thepresence of ambient light (top curve), the low frequency capacitance measured in the dark,the high frequency capacitance measured in the presence of ambient light and the highfrequency capacitance measured in the dark (bottom curve). All curves were measured fromleft to right. The MOS parameters are Na

- Nd= 4 x 1015 cm-3 and tox

= 80 nm. The device

area is 0.0007 cm2.

Figure 6.3.5 illustrates some of the issues when measuring the capacitance of an MOS capacitance. First, onshould measure the devices in the dark. The presence of light causes carrier generation in thesemiconductor, which affects the measured capacitance. In addition, one must avoid the deep depletion

effects such as the initial linearly varying capacitance of the high frequency capacitance measured in the darkon the above figure (bottom curve). The larger the carrier lifetime, the slower the voltage is to be changed toavoid deep depletion.

The measured low frequency capacitance is compared to the theoretical value in Figure 6.3.6. The highfrequency capacitance measured in the presence of light is also shown on the figure. The figure illustrates theagreement between experiment and theory.

Figure 6.3.6: Comparison of the theoretical low frequency capacitance (solid line) and the experimentaldata (open squares) obtained in the dark. Fitting parameters are Na

- Nd= 3.95 x 1015 cm-3

and tox = 80 nm.

6.3.4.5. Non-Ideal effects in M OS capacitors

Non-ideal effects in MOS capacitors include fixed charge, mobile charge and charge in surface states.Performing a capacitance-voltage measurement can identify all three types of charge.

Fixed charge in the oxide simply shifts the measured curve. A positive fixed charge at the oxide-semiconductor interface shifts the flatband voltage by an amount, which equals the charge divided by the

oxide capacitance. The shift reduces linearly as one reduces the position of the charge relative to the gate
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electrode and becomes zero if the charge is located at the metal-oxide interface. A fixed charge is caused byions, which are incorporated in the oxide during growth or deposition.

The flatband voltage shift due to mobile charge is described by the same equation as that due to fixedcharge. However, the measured curves differ since a positive gate voltage causes any negative mobile chargto move away from the gate electrode, while a negative voltage attracts the charge towards the gate. Thiscauses the curve to shift towards the applied voltage. One can recognize mobile charge by the hysteresis inthe high frequency capacitance curve when sweeping the gate voltage back and forth. Sodium ionsincorporated in the oxide of silicon MOS capacitors are known to yield mobile charges. It is because of thehigh sensitivity of MOS structures to a variety of impurities that the industry carefully controls the purity of thewater and the chemicals used.

Charge due to electrons occupying surface states also yields a shift in flatband voltage. However as theapplied voltage is varied, the Fermi energy at the oxide-semiconductor interface changes also and affects theoccupancy of the surface states. The interface states cause the transition in the capacitance measurement tobe less abrupt. The combination of the low frequency and high frequency capacitance allows a calculation ofthe surface state density. This method provides the surface state density over a limited (but highly relevant)range of energies within the bandgap. Measurements on n-type andp-type capacitors at differenttemperatures provide the surface state density throughout the bandgap.

Boulder, December 200

mos analysis

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