CMOS Device Model Objective CMOS transistor models Hand calculations for analog design Efficiently and accurately simulation CMOS transistor models Large signal model Small signal model Simulation model Noise model

Large Signal Model Nonlinear equations for solving dc values of device currents given voltages Level 1: Shichman-Hodges (VT, K', g, l, f, and NSUB) Level 2: with second-order effects (varying channel charge, short-channel, weak inversion, varying surface mobility, etc.) Level 3: Semi-empirical short-channel model Level 4: BSIM models. Based on automatically generated parameters from a process characterization. Good weak-strong inversion transition.

Transconductance when VDS is small

Transconductance when VDS is small

Transconductance when VDS is small

Effect of changing VDS for a large VGS

Effect of changing VDS for a given VGS

Effect of changing VDS for a given VGS

Effect of changing VDS for various VGS VGS<=VT

Effect of changing VDS for various VGS

Effect of changing VDS for various VGS

MOST Regions of Operation Cut-off, or non-conducting: VGS <VT ID=0 Conducting: VGS >=VT Saturation: VDS > VGS – VT Triode or linear or ohmic or non-saturation: VDS <= VGS – VT

With channel length modulation

Capacitors Of The Mosfet

CBD and CBS include both the diffusion-bulk junction capacitance as well as the side wall junction capacitance. They are highly nonlinear in bias voltages. C4 is the capacitance between the channel and the bulk. It is highly nonlinear and depends on the operation of the device. C4 is not measurable from terminals.

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Gate related capacitances

Small signal model

Typically: VDB, VSB are in such a way that there is a reversely biased pn junction. Therefore: gbd ≈ gbs ≈ 0

In saturation: But

In non-saturation region

High Frequency Figures of Merit wT AC current source input to G AC short S, D, B to gnd Measure AC drain current output Calculate current gain Find frequency at which current gain = 1. Ignore rs and rd,  Cbs, Cbd, gds, gbs, gbd all have zero voltage drop and hence zero current Vgs = Iin /jw(Cgs+Cgb+Cgd) ≈ Iin /jwCgs Io = − (gm − jw Cgd)Vgs ≈ − gmVgs |Io/Iin| ≈ gm/wCgs

At wT, current gain =1 wT ≈ gm/(Cgs+Cgd)≈ gm/Cgs or

High Frequency Figures of Merit wmax AC current source input to G AC short S, D, B to gnd Measure AC power into the gate Assume complex conjugate load Compute max power delivered by the transistor Find maximum power gain Find frequency at which power gain = 1.

wmax: frequency at which power gain becomes 1 PL=

BSIM models Non-uniform charge density Band bending due to non-uniform gate voltage Non-uniform threshold voltage Non-uniform channel doping, x, y, z Short channel effects Charge sharing Drain-induced barrier lowering (DIBL) Narrow channel effects Temperature dependence Mobility change due to temp, field (x, y) Source drain, gate, bulk resistances

“Short Channel” Effects VTH decreases for small L Large offset for diff pairs with small L Mobility reduction: Velocity saturation Vertical field (small tox=6.5nm) Reduced gm: increases slower than root-ID

Threshold Voltage VTH Strong function of L Process variations Use long channel for VTH matching But this increases cap and decreases speed Process variations Run-to-run How to characterize? Slow/nominal/fast Both worst-case & optimistic

Effect of Velocity Saturation Velocity ≈ mobility * field Field reaches maximum Emax (Vgs-Vt)/L reaches ESAT gm become saturated: gm ≈ ½mnCoxW*ESAT But Cgs still 2/3 WL Cox wT ≈ gm/Cgs = ¾ mnESAT /L No longer ~ 1/L^2

Threshold Reduction When channel is short, effect of Vd extends to S Cause barrier to drop, i.e. Vth to drop Greatly affects sub-threshold current: 26 mV Vth drop  current * e 100~200 mV Vth drop due to Vd not uncommon  100s or 1000 times current increase Use lower density active near gate but higher density for contacts

Other effects Temperature variation Normal-Field Mobility Degradation Substrate current Very nonlinear in Vd Drain to source leakage current at Vgs=0 Big concern for static power Gate leakage currents Hot electron Tunneling Very nonlineary Transit Time Effects

Consequences for Design SPICE (HSPICE or Spectre) BSIM3, BSIM4 models Accurate but inappropriate for hand analysis Verification (& optimization) Design: Small signal parameter design space: gm, CL (speed, noise) gm/ID, ID (power, output range, speed) Av0= gmro (gain) Device geometries from SPICE (table, graph); may require iteration (e.g. CGS)

Intrinsic voltage gain of MOSFET Sweep V1 Measure vgs Intrinsic voltage gain = gm/go = Dvds/Dvgs for constant Id

Electronic Noise Noise phenomena Device noise models Representation of noise (2-ports): Motivation Output spectral density Input equivalent spectral density Noise figure Sampling noise (“kT/C noise”) SNR versus Bits Noise versus Power Dissipation Dynamic range Minimum detectable signal

Noise in Devices and Circuits Noise is any unwanted excitation of a circuit, any input that is not an information-bearing signal. • External noise: Unintended coupling with other parts of the physical world; in principle, can be virtually eliminated by careful design. • Intrinsic noise: Unpredictable microscopic events inherent in the device/circuit; can be reduced, but never eliminated. Noise is especially important to consider when designing low-power systems because the signal levels (typically voltages or currents) are small.

Noise vs random process variations Variations from one device to another For any device, it is fixed after fabrication Noise Unpredictable variations during operation Unknown after fabrication Remains unknown after measurement during operation May change with environment

Time domain description of noise

What is signal and what is noise?

Signal and noise power:

Physical interpretation If we apply a signal (or noise) as a voltage source across a one Ohm resistor, the power delivered by the source is equal to the signal power. Signal power can be viewer as a measure of normalized power. power

Signal to noise ratio SNR = 0 dB when signal power = noise power Absolute noise level in dB: w.r.t. 1 mW of signal power

SNR in bits A sine wave with magnitude 1 has power = 1/2. Quantize it into N=2n equal levels between -1 and 1 (with step size = 2/2n) Quantization error uniformly distributed between +–1/2n Noise (quantization error) power =1/3 (1/2n)2 Signal to noise ratio = 1/2 ÷ 1/3 (1/2n)2 =1.5(1/2n)2 = 1.76 + 6.02n dB or n bits

C=0: n1 and n2 uncorrelated C=1: perfectly correlated

Adding uncorrelated noises Adding correlated noises

For independent noises

Frequency domain description of noise Given n(t) stationary, its autocorrelation is: The power spectral density of n(t) is: For real signals, PSD is even.  can use single sided spectrum: 2x positive side ↑ single sided PSD

Parseval’s Theorem: If If x(t) stationary,

Interpretation of PSD Pxf1 = PSDx(f1) PSDx(f)

Types of “Noise” “man made” “intrinsic” noise Interference Supply noise … Use shielding, careful layout, isolation, … “intrinsic” noise Associated with current conduction “fundamental” –thermal noise “manufacturing process related” flicker noise

Thermal Noise Due to thermal excitation of charge carriers in a conductor. It has a white spectral density and is proportional to absolute temperature, not dependent on bias current. Random fluctuations of v(t) or i(t) Independent of current flow Characterization: Zero mean, Gaussian pdf Power spectral density constant or “white” up to about 80THz

Thermal noise dominant in resisters Example: R = 1kΩ, B = 1MHz, 4µV rms or 4nA rms

HW Equivalently, we can model a real resistor with an ideal resistor in parallel with a current noise source. What rms value should the current source have? Show that when two resistors are connected in series, we can model them as ideal series resistors in series with a single noise voltage source. What’s the rms value of the voltage source? Show that two parallel resistors can be modeled as two ideal parallel resistors in parallel with a single noise current source. What’s the rms value of the current source?

Noise in Diodes Shot noise dominant – DC current is not continuous and smooth but instead is a result of pulses of current caused by the individual flow of carriers. It depends on bias, can be modeled as a white noise source and typically larger than thermal noise. − Zero mean – Gaussian pdf – Power spectral density flat – Proportional to current – Dependent on temperature

Example: ID= 1mA, B = 1MHz, 17nA rms

MOS Noise Model

Flicker noise –Kf,NMOS 6 times larger than Kf,PMOS –Strongly process dependent −when referred to as drain current noise, it is inversely proportional to L2

BJT Noise

Sampling Noise • Commonly called “kT/C” noise • Applications: ADC, SC circuits, … R von C Used:

Filtering of noise x(t) y(t) H(s) |H(f )|2 = H(s)|s=j2pf H(s)|s=-j2pf

Noise Calculations 1) Get small-signal model 2) Set all inputs = 0 (linear superposition) 3) Pick output vo or io 4) For each noise source vx, or ix Calculate Hx(s) = vo(s) / vx(s) (or … io, ix) 5) Total noise at output is 6) Input Referred Noise: Fictitious noise source at input:

Example: CS Amplifier Von=(inRL +inMOS)/goT VDD goT = 1/RL + sCL RL M1

wo=1/RLCL

Some integrals

HW In the previous example, if the transistor is in triode, how would the solution change? HW If we include the flicker noise source, how would that affect the computation? What do you suggest we should modify? HW In the example, if RL is replaced by a PMOS transistor in saturation, how would the solution change? Assume appropriate bias levels.