Transistor Noise Lecture 10 High Speed Devices

Transcription

1 Transistor Noise 1

2 Transistor Noise A very brief introduction to circuit and transistor noise. I an not an expert regarding noise Maas: Noise in Linear and Nonlinear Circuits Lee: The Design of CMOS RFIC Gray-Mayer, chapter 9.

3 Transistor Noise Signal Source v sig v s,noise Transistor Amplifier A v, MSG, h 1 Source Amplifier All signal sources generates: Signal of interest Noise The circuit / transistor +Amplifies signal of interest -Amplifies source noise -Adds own noise to output 3

4 Noise - introduction v Noise is a randomly varying voltage/current Causes small fluctuations around a mean value v(t) t V 0 Noise is random: The instantaneous value v(t) can not be predicted The mean square value can usually be calculated for physical mechanism corresponding to dissipated noise power. This is the metric of interest! v = v t V 0 = 1 T න 0 T v t V 0 dt ഥi = i t I 0 = 1 T න 0 T i t I 0 dt 4

5 Correlation Several noise sources : we calculate the total root mean square value from each noise generator and add. v 1 (t) + v (t) = v 1 t + v t + v 1 t v t If v 1 and v are independent (uncorrelated) : v 1 t v t = 0 Typically uncorrelated Correlated at goals Uncorrelated Most physical noise sources can be considered independent v 1 (t) + v (t) = v 1 t + v t 5

6 Noise Spectral Density Each process that causes noise is associated with a noise spectral density S I (f) S V (f) (A /Hz) (V /Hz) i ω, T = T/ න T/ i t e jωt dt S I ω = lim T i ω, T T ഥi = S I f Δf In a small bandwidth Df around f: i = S I f Δf RMS value of i Equivalent to a sinusoidal current generator with amplitude i! Noise problems can be treated using ordinary, linear circuit analysis! Several noise sources : we calculate the total root mean square value from each noise generator and add. Square to the mean square value. Easy if the sources are uncorrelated. 6

7 Noise sources frequency domain v n (ω) In a small Δf around f we an treat each noise source as a sinusoidal generator P = 1 vi v n = v n v n Instantaneous noise power (unknown) v n = v n v n Time averaged noise power v 1 v = C 1, C 1, =0 if sources are uncorrelated v n R = ഥ i n R Time averaged noise power delivered to resistance Note that the instantaneous noise power is unknown! 7

8 Thermal Noise All resistive materials show Thermal Noise Random fluctuations in the kinetic energy of the carriers Essentially independent of current through the resistor Constant spectral density : white noise v = 4kTRΔf ഥi = 4kT R Δf R v ഥi R Increases linear with temperature Thevenin Norton Total noise power: P = v R = 4kTΔf Antennas also show thermal noise black body radiation 8

9 Shot Noise (Hagelbrus) Discrete nature of electron charge When a DC current flow (uncorrelated electrons) Non-degenerate electron gas (pn-junction, i C for HBTs) Low transmission (MOS oxide leakage i g ) Generation/recombination (i B for a HBT) Off-state current for a MOSFET (non-degenerate) Constant spectral density : white noise Resistances do not show shot noise ഥi = qi D Δf 9

10 1/f Noise (flicker noise) Semiconductor defects can cause trapping or electrons: Mobility fluctuations Variation in resistivity r(t) Carrier concentration fluctuations A DC current must be present for this r(t) variation to be transformed into noise FETs : drain current noise HBTs: base current noise Can also be present in ordinary resistors Shows a 1/f spectral density (Pink Noise) ഥi = K 1 I a f Δf a 0.5 K 1 is a measure of the quality of the device 10

11 Noise in a diffusive FET Source Drain Drift/Diffusion along the transistor channel Resistive elements thermal noise! ഥi ഥ i Integrating over the channel one obtains: For short-channel (velocity saturated): ഥi n = 4kTγg 0 4kT 3 g mδf 3 < γ < 5 Noise current causes channel potential fluctuations couples capacitive to the gate Modeled by assuming thermal noise from NQS channel resistance v n = 4kTR gs Δf (Approximative) 11

12 Noise in a (quasi-)ballistic FET E fs T E fd There are random fluctuations in the occupation f. δf 0 = f 0 (1 f 0 ) T<1 random scattering of electrons through the channel Leads to shot-noise Thermal noise in the contacts. For energies much below E fs : δf 0 0 due to the Pauli principle 1

13 Noise in a (quasi, 1D) ballistic conductor I DS = q h n න de T n E [f s f d ] ഥi n = S = e πħ n න de T n E [f s 1 f s + f d 1 f d ] + T n E 1 T n E f s f d n:1d subbands ഥi n = e πħ න E(0) def s (1 f s ) Single 1D subband Fully ballistic I 1D = q h [F 0 η fs ] δf S Occupation fluctuations. 1D fully ballistic current 13

14 Noise in a (quasi D) ballistic conductor 8x45 nm quantum well FET Fully Ballistic qi D S I 4kTg m

15 Noise in a (quasi D) quasi-ballistic conductor 8x45 nm quantum well FET T=0.7 qi D S I 4kTg m 1. 15

16 Noise in a (quasi, 1D) ballistic conductor 8x45 nm quantum well FET T=0.1 Very low Transmission Noise is set by shot noise limit qi D Typically more noisy S I 4kTg m.0 16

17 NQS Noise Model extrinsic resistances v g = 4kTR B Δf R G v d = 4kTR B Δf R D R GD R GS R S v S = 4kTR B Δf All physical resistances have thermal noise R G, R D and R G The output conductance is NOT a physical resistance no noise 17

18 Channel Noise v g = 4kTR B Δf R G v d = 4kTR B Δf R D R GD R GS R S v S = 4kTR B Δf ഥi d = 4kTγg m Δf + KI D a f Δf Diffusive transistor: channel has thermal noise roughly equal to 4kTγg m Δf With γ 3 4. Quasi-ballistic transistor has thermal noise from reservoirs and (suppressed shot noise). Similar γ 3 1. Also some amount of 1/f-noise. 18

19 Induced gate noise v g = 4kTR B Δf R G v d = 4kTR B Δf R D v id R GD = 4kTR GD Δf R GS v is = 4kTR GS Δf R S v S = 4kTR B Δf ഥi d = 4kTγg m Δf + KI D a f Δf Channel potential fluctuations (scattering) can couple to the gate terminal. Typically modeled by allowing for R GS and R GD to produce thermal noise. Not well understood for quasi-ballistic transistors. v is, v id and ഥ i d are typically (slightly) correlated. 19

20 Induced gate noise v g = 4kTR B Δf R G v d = 4kTR B Δf R D i g = qi G Δf R GD v i = 4kTR GD Δf R GS v i = 4kTR GS Δf R S v S = 4kTR B Δf ഥi d = 4kTγg m Δf + KI D a f Δf Gate leakage can cause shot-noise Full noise model is really quite complicated -> numerical modeling required for detailed work. Measured noise data is essential for accurate circuit design. The i d source is approximate. Use g as a fitting parameter. For low transmission better use i d = qi DS Δf 0

21 Noise representation It is possible to represent the noise sources of the transistor with two sources at the input Noisy circuit v n : generates noise produced (at the output) when input is short circuited i n : generates noise produced (at the output) when input is open v n ഥi n Noiseless circuit These are typically partly correlated since they are originating from the same sources! 1

22 Noise representation v n General noise model v n -i n model ഥi n Noiseless circuit v total model Voltage source with source impedance v total in series with v signal and v ng v signal Z s = R s + jx s v n,g v total Noiseless circuit v total = v n + Z s i n z 1 + z z 1 + z = z 1 + z + Re(z 1 z ) v total = v n + Z ഥ s in + Re(v n i n Z s )

23 Signal to noise ratio / Noise Figure All sources sees the same R s and transistor input impedance we can work with maximum available source power: P signal = v signal 4R s SNR 1 = P signal P g,noise P g,noise = v n,g = 4kTR s 4R s 4R s = kt SNR = P signal P g,noise + P amp,noise P amp,noise = v total 4R s Noise figure (or noise factor): How much is the SNR degraded by the transistor! F = SNR 1 SNR = (P amp,noise+p g,noise ) P gnoise NF = 10log(F) = 1 + P amp,noise kt F in db is typically called the noise figure =1+Amplifier noise available at input/kt Note that P g,noise and P amp,noise have the same Δf on all sources they cancel! 3

24 Minimum Noise Figure / Optimal Source Impedance F = 1 + v n + Z s ഥ in + Re v n i n Z s 4kTR s Note that F varies with Z s = R S + jx s By minimizing F with respect to R s and X s, we obtain: F min = kT v n ഥ i n Im v n i n + Re vn i n Z opt = R opt + jx opt = v n ഥi n Im v ni n ഥi n j Im v ni n ഥi n 4

25 Simple FET Noise Model R G Simplest NQS transistor model C gd =0, C M =0, No R s /R d. R i C GS g m v in i n1 = 4kTγg m Δf We evaluate the v n i n -model to: The two sources become correlated: v n = 4kTR + i n1 1 + jωrc = 4kT R + γ 1 + ωrc g m g m ഥi n = i n1 g m v n i n = 4kTγg m jωc = 4kTγ g m ωc 1 + jωcr g m jωc g m Unless we are calculating noise figure multiply all terms with Δf 5

26 Simple Transistor Noise Model F min = 1 + Rγg m ω ω T + g m γ R ω4 ω T 4 + γg m R ω ω T ω T = g m C gs F min 1 + Rγg m ω ω T 1 + γ ω ω T Low F min : Small R g and R i. Large f T. Small g. ω ω T R = R i + R G X opt = 1 ωc Inductor R opt = R + R γg m ω T ω R Optimal source impedance for minimum noise is not the same as for maximum gain! 6

27 Simple Transistor Noise Model F min = 1 + Rγg m ω ω T + g m γ R ω4 ω T 4 + γg m R ω ω T ω T = g m C gs F min 1 + Rγg m ω ω T 1 + γ ω ω T ω ω T R = R i + R G Low F min : Small R. Large g m. Small C gs. Small g. R n = v n 4kT F = F min + 4R n Z 0 = R + γ g m 1 + ωrc Noise resistance Γ s Γ sopt 1 Γ s 1 + Γ sopt R n is a measure on how rapidly F min varies for non-optimal source impedance! Should be small Large g m! 7