Noise in Materials and Components Low frequency noise as a diagnostic tool for reliability and quality assessment of devices L.K.J. Vandamme Eindhoven University of Technology (EH 9.13) 5600 MB Eindhoven, The Netherlands
[email protected]
Toulouse, 26 February 2004
2
I. Introduction 1. Objectives To give an introduction on definitions, noise measuring set-ups and to give an overview of noise sources such as: thermal-, shot-, generation recombination-, RTS noise and 1/f noise. A better understanding of different types of noise can learn us what types of noise are inevitable and what can be reduced. Some emphasis on sensitivity in the omnipresent conductance 1/f noise. The noise in devices: resistance, resistance- type (MOSTs), diodes, and diode–type devices is discussed. L.K.J. Vandamme / Noise / 26-02-2004
3
From the analysis of RTS-, 1/f- and thermal-noise in MOSTs we explain that faster devices are noisier. Explain why low frequency noise is a good diagnostic tool and show how current crowding will enhance 1/f noise. The shot noise, 1/f noise and generation-recombination (RTS-) noise are important for quality assessment in e.g., diode type devices like: solar cells, laser diodes, LEDs, avalanche photo diodes and bipolar transistors Applications are e.g.,: contacts, conductive adhesive joints, thick and thin film resistors, parasitic series resistance or parallel conduction paths in devices like in: submicron MOS-, MES-,and MODFET and poly silicon emitter BJT and HBT.
L.K.J. Vandamme / Noise / 26-02-2004
4
Why knowledge of the physical origin of noise is important? a. Stochastic fluctuations set a detection limit to measuring systems and telecommunication systems b. Noise can be used for reliability assessment of devices. c. Knowing the physical origin of noise can help to reduce noise: Thermal noise, Brownian motion (resistance-type devices: T↓, W↑) Shot noise, stochastic emission (diode –type devices: avoid microplasma due to non uniform fields in reverse biased (FET) junctions) Generation recombination-, and RTS noise (∆N→∆σ→∆R: avoid traps) 1/f noise (∆µ→ ∆σ →∆R: N or Neff not too low) L.K.J. Vandamme / Noise / 26-02-2004
5
2. Some definitions and remarks [1-4] Electrical noise is a real stochastic signal often described in terms of variance (σ2), rms values or standard deviation (σ), average absolute amplitudes (∆I ) or relative absolute amplitudes, e.g., ∆I / I, correlation function, C (τ), amplitude distribution function (pdf) or spectral noise density (Sx(f)). There is a difference between: amplitude spectrum [x] and power spectrum [x2], both are line spectra for periodic functions. The power density spectrum [x2 / Hz] of the noise is continuous . Power often means x2, not Watt. ”x” can be a fluctuation in time of voltage V, current I, resistance R, optical power P [Watt], magnetization (Ni / Fe), extinction coefficient of an optical fiber, or height along a line (surface roughness) [x2/(1/x)]. Spectral density, Sx (f) and correlation functions C(τ) of physical quantities are real and we use positive frequencies. L.K.J. Vandamme / Noise / 26-02-2004
6
Correlation functions: C2 (τ) is a two-point correlation function [x2] C2 (τ) = ,C(τ) = C2 (τ) and S (f) are two different representations of x (t). The cosine transform or the WienerKhintchine theorem or the so- called Fourier transform of C(τ) gives S(f): ∞ S(f ) = 4 ∫ C(τ) cos 2πfτ.dτ 0
Spectral density Sx (f) is also defined as the variance of a band pass filtered x (t), that becomes per bandwidth ∆f at frequency f Variance
∫
∞
0
S x ( f ) df = ( ∆ x ) 2 = σ 2 L.K.J. Vandamme / Noise / 26-02-2004
7
3. Analogue noise measurement set-up [5, 5a, 5b] A
B AC amplifier and sample in a cage of Faraday and output at A is monitored with oscilloscope for 50Hz, 150Hz parasitic, do we have “normal Gaussian noise”, clipping or oscillations? At B, the signal looks like amplitude modulated carrier at frequency f with random envelope if ∆f is small enough, ∆Vf (t). Variance of band pass filtered noise divided by ∆f gives the power spectral density: ∆V 2 ( t ) L.K.J. Vandamme / Noise / 26-02-2004
SV ≡
f
∆f
8
FFT Fast Fourier Transform (FFT) systems (Spectrum analyzer) are based on: “periodic” in a time block of duration T, it will give a line spectrum. sampling on x(t) Power spectrum / ∆f = power spectrum x T Î Power density spectrum in e.g., V2/Hz or V2s
T 2 n 2 S ( f a ) ≈ (a n + bn ) with f a = " harmonics" 2 T On the next slide some FFT artefacts with rectangular windowing
L.K.J. Vandamme / Noise / 26-02-2004
9
FFT of a sine wave of 1Hz, T = 1s; 1.25s; 1,5s → f1=1, 0.8, 0.66 Hz
L.K.J. Vandamme / Noise / 26-02-2004
Thermal- and 1/f noise, in time- and frequency domain, decomposition
V = I.( R + ∆ R )
10
Top: thermal noise, bottom left and below: the 1/f noise in time and frequency domain, at high frequencies the thermal noise always becomes visible and can be used for calibration
∆Vf2 ( t ) SV ≡ ∆f
SV (V2/Hz) 10-11 -12
10
figuur 2
-13
10
V = V + ∆V + Vth
10-14 10-15 10-16 0 10
L.K.J. Vandamme / Noise / 26-02-2004
1
10
2
10
3
10
4
10
5
10
6
10
7
10
f (Hz)
11
II. Noise Sources [6-11] 1. Thermal noise
S v = 4kTR and
S I = 4kT / R
Johnson or Nyquist-noise: Phys Rev, 32 (1928)no 1, pp 97-113. R→ Real [Z] and 1/R→ Real [Y] white noise for 0 < f < 3x1012 Hz Physicists avoid problems at f → ∞ by replacing “kT ” with
hf
hf / kT e −1 2
Engineers and material scientists multiply SV by 1 /(1 + (f / f o ) )
with f0 = 1/2πRC; τ = RC is a circuit time constant or the dielectric time constant τdiel : τ diel = ε oε r / qµn with 0.1s > τdiel > 10-12 s for most dielectrics and for metals. We observe in a bandwidth ∆f a variance of the voltage or current fluctuations given by = 4kTR ∆f or = (4kT/R) ∆f L.K.J. Vandamme / Noise / 26-02-2004
12
Equivalent circuit for network analysis ( SPICE ) with test sources A noise free resistor with a noise voltage in series or a noise current in parallel. The open circuit resistance is not heated by the current noise source. The short circuited resistor is not heated by the noise voltage! Equivalent, means equal to a certain level !
e n1 = e n 2 = 0 e n1.e n 2 = 0 e n ≠ e n1 + e n 2 e =e +e 2 n
2 n1
2 n2
L.K.J. Vandamme / Noise / 26-02-2004
Independent noise sources are added in squarred values, not in nV/√Hz
13
Origin of thermal noise Brownian motion (1827) of free charge carriers (electrons 1897) Random motion of particles in a fluid (1827-1900) in plants, organic and inorganic material; due to light?; due to evaporation?; persist after a year!; smaller particles move faster; motion increases with temperature; molecular impact; In analogy: electrons within a conductor’s lattice make a random walk at T > 0K. Average kinetic energy of an electron:
1 E = m * vth2 = (3 / 2)kT 2
3kT v = ≈ 10 7 cm / s at T = 300 K m* 2 th
17nm < λ = vth τ < 300 nm for 50 < µ ( cm2 /Vs ) < 16000 L.K.J. Vandamme / Noise / 26-02-2004
14
For f < fc holds: for the short circuit current fluctuations SI = 4kT/R and for the open circuit voltage fluctuations SV = SI R2 = 4kTR Thermal noise is independent of dc or ac current passed through the resistor. A temperature raise due to power dissipation can be taken into account by adapting T in the above equations. Ohms law, and the simple SV = 4kTR holds if collision time τ is not influenced by the field (vdrift < vth). We can expect deviations, if there are no collisions, (e.g. at 0 K at f > fc (THz) in very short time intervals and for resistors with a length L < λ and transit time of electron τtransit < τ ( ballistic transport; not enough collisions in L)
τ transit
2
L = < τ collision µV
L.K.J. Vandamme / Noise / 26-02-2004
15
Remarks on thermal noise: Thermal noise is reduced by lowering the temperature, e.g., cryogenic applications in satellites. In MOSFET design, very often wide and short channels are chosen to reduce equivalent input thermal noise voltage. Thermal noise and resistance measurements are used to measure the temperature in hostile environments (neutron flux and other ionizing radiation). Equivalent voltage noise at the input of amplifier (SVin = SVout/ G2 ) are often expressed in an equivalent noise resistance(@ 289K) or equivalent noise temperature (has nothing to do with the real device temperature). Req and Teq are equivalent noise parameters defined by
S vin = 4kTReq ( f ) and
Teq ( f ) = T + ∆T =
L.K.J. Vandamme / Noise / 26-02-2004
S vsystem + 4kTR 4kR
>T
16
2. Shot noise [8] Stochastic emission of electrons is often like a Poisson process. If the average transit time is t0 = τ and G is the generation rate (s-1) of electrons at the cathode, then we have at the average = G. t0 crossing electrons underway. The noise is in the emission-time and also in the transit-time due to initial velocity fluctuations.
I =
Nq
τ
∆I =
∆ Nq
(∆I ) 2 = (∆N ) 2
τ
(∆I ) = N 2
Poisson ⇒ ( ∆ N ) = N 2
q2
τ
2
q2
τ2
=I
C(t) = 0 for t > t0 (no overlap in populations) and for 0 < t < τ
C ( t ) = ( ∆ I ) (1 − t / τ ) 2
L.K.J. Vandamme / Noise / 26-02-2004
q
τ
17
∞
S I = 4 ∫ C (t ) cos ω t.dt ⇒ C (τ ) = 0 for t > τ o
τ
sin(πfτ ) (1 − ) cos ω t.dt = 2qI SI = ∫ τ 0 τ π f τ 4qI
t
2
for f < 1 / πτ
S I ( f ) = 2 qI
Log [ SI /2qI ] Fano- factor
1
SI/2qI 1.0
4/л2 2
0.8
0.1
0.6
0.04
0.4
0.016 0.01
0.2
fτ
0.0 0.0
0.5
1.0
1.5 2.0
2.5
3.0
Log [ f τ ] 0.001 0.01
L.K.J. Vandamme / Noise / 26-02-2004
0.1
0.5
1 2 3
18
3. Generation Recombination Noise [10]
∆ N (t ) = ∆ N ( 0 ) e − t / τ N is number of free electrons, not concentration, τ is lifetime of ∆N
Conduction band
elect ener
G
R
bandgap traps
Generation-recombination from traps
4τ S n ( f ) = (∆N ) and 2 1 + ( 2π f τ ) 2
(∆N ) = N ? 2
∞
4τ ∫0 1 + ( 2π f τ ) 2 df = 1
Is the smallest value of N, the number of full traps and the number of empty traps. It depends on EF. L.K.J. Vandamme / Noise / 26-02-2004
19
One single trap (∆N=1) and low N ==> RTS-noise; ∆I/I = ∆G/G = ∆N/N = 1/N 10
Lorentzian-spectrum SN / (4τ ∆N 2 )
τ can be read from the spectrum
10
−6
< τ [ s ] < 10
1 0.5
−2
f -2
10-1
1.6 105 > fc = 1/2πτ [Hz] >16 10-2
2πfτ 10-2
10-1
L.K.J. Vandamme / Noise / 26-02-2004
1
10
20
4. Burst noise, popcorn noise, RTS noise [9,11-14] RTS-noise is a special case of generation recombination noise with one single trap, ∆N = 1, if τe = τc then = ¼ , has its highest value
Two level noise and superimposed 1/f noise in time domain and its amplitude propability density function (pdf) L.K.J. Vandamme / Noise / 26-02-2004
21
Pure RTS without 1/f noise component has a Lorentzian spectrum.
4τ p S I Sv S N = 2 = 2 =K ⇒ 1/τ p = 1/τ e +1/τ c 2 2 1 + (2πfτ p ) I V N τp τ eτ c 1 ∆N 2 = 2 K ≡ 2 ⇒ or K = 2 . 2 τ τ + N N ( τ + τ ) N e c e c RTS is a problem of submicron devices with traps, e.g., in diode type devices with dislocations in sensitive areas and submicron MOSFETs with a low number of carriers. RTS is a poor (traps) device indicator. For strong asymmetric noise holds 2 → 0. Symmetric traps can become asymmetric in a MOSFET, by applying switching bias, but asymmetric ones can become symmetric 1 2 < ∆N > = 2 +τ e /τ c +τ c /τ e L.K.J. Vandamme / Noise / 26-02-2004
22
Analysis of RTS in time and frequency domain [13,14] 8⋅104
V(t)
pdf
LED#4k pdf examples
6⋅104
a
ϑi V0 t
0 -b
Solid line: Id=2.15⋅10-6 A Doted line: Id=1.53⋅10-4 A
4⋅104
2⋅104
τi
0
The waveform of the measured noise displayed by the oscilloscope
U, Volt -10-4
-5⋅10-5
0
5⋅10-5
10-4 2⋅10-4
The probability density function of the raw noise
L.K.J. Vandamme / Noise / 26-02-2004
V(t)
Detection of noise in noise
23
a
ϑi V0 t
0
Total noise: V(t) = V1/f(t) + VRTS(t) State “1” (V(t) > V0): State “0” (V(t) < V0):
-b
τi
V1/f(t) = V(1)(t) – a ; VRTS(t) = +a V1/f(t) = V(0)(t) + b ; VRTS(t) = −b
V0 threshold voltage to be found using the standard signal detection theory in the noise background L.K.J. Vandamme / Noise / 26-02-2004
24
Noise reconstruction The pdf of the raw noise: 8⋅104
pdf
WV (V ) =
LED#4k pdf examples
(V + b) 2 (V − a) 2 p + exp − exp − 2 2 2 2 2σ 2σ 2πσ 2πσ q
6⋅104
Solid line: Id=2.15⋅10-6 A Doted line: Id=1.53⋅10-4 A
4⋅104
p = /( +)
2⋅104
q = /( +)
U, Volt
0 -10
-4
-5
-5⋅10
0
-5
5⋅10
10-4 2⋅10-4
Likelihood relation:
Λ =1
(V − a) 2 Λ(V ) = p exp − 2 2σ
Probabilities for states “1” and “0”
(V + b) 2 q exp − 2 2σ
a −b σ a V0 = ln + 2 a+b b
V(t)
2
L.K.J. Vandamme / Noise / 26-02-2004
a
ϑi V0 t
0 -b
τi
25
Obtained results after detection reconstruction V
10-12
SV, V2/Hz
RTS
a V0 t
0
RAW
10-13
1/f
-b
10-14
V 1/f
If 1/f and RTS have a different dependence on bias, then there is a different physical origin
t
0
10-15 a
0
-b
V RTS
t
10
f, Hz
-16
10
102
103
104
Raw noise spectrum and its decomposition in a 1/f and RTS L.K.J. Vandamme / Noise / 26-02-2004
26
5. 1/f Noise [15-19] For metal-, semiconductor-, organic- samples with perfect contacts, holds an empirical relation for the 1/f conductance fluctuations. Spectra are 1/fγ with 0.8< γ 10 14 detection problems, always 4kTR. L.K.J. Vandamme / Noise / 26-02-2004
SP dBc = 10 Log 2 Pc
∆f = 10kHz
28
5. ∆ρ is not due to ∆T: because samples with a negligible small temperature coefficient have the same 1/f noise as samples with a normal ∆R/R∆T=∆ρ/ρ∆T=-∆µ/µ∆T
µ ∝T
−δ
∆ρ
⇒ log µ = −δ log T + A ⇒ ρ
=
δ .∆T T
⇒ δ → 0 ⇒ ∆ρ → 0
6. omnipresent as a bulk phenomenon (SV /V2 ∝ 1/N) in: metals (solid-liquid), semiconductors, polymers (homogeneous, contacts) in dielectrics like optical fibers as a 1/f fluctuation in the attenuation coefficient, in magnetization fluctuations in magnetoresistive sensors (NiFe) and in devices like: (photo) diodes, laser diodes, BJT, HBT, JFET, MESFET, MODFET and MOSFET. Exist also in non electronic systems: loudness in music; heartbeat fluctuations; electro encephalic-graphs during sleep or in the state of attention L.K.J. Vandamme / Noise / 26-02-2004
29
7. Hooge’s empirical relation, [19] “1/f noise is no surface effect”…Au samples, no oxide, no traps, no ∆ N G = qµnA / l = qµN / l 2 ∆G ∆µ = µ G
or
∆G ∆N = ? G N
∆R / R = −∆G / G ⇒ (∆R / R) = (∆G / G ) 2
2
Experimental results on homogeneous samples submitted to homogeneous fields are often well described by
SV S R C α SG S I = 2 = = = 2 = 2 2 V R f Nf G I 1
N = nΩ
2
1
2
by applying a current source by applying a voltage source
is the number of carriers with Ω volume of the sample L.K.J. Vandamme / Noise / 26-02-2004
30
About the 1/N dependence, N in the empirical relation does not suggest number fluctuations → noise source is distributed uniformly N in the volume ∆σ ∆N ∆µ = + ⇒ ∆N or ∆µ or both ? ⇒ G ∝ µi N µ σ i =1 Mobility fluctuations
∑
G ∝ Nµ
2
∆G ∆N G ∝ Nµ ⇒ = G N Poisson, or sub-Poissonian in bulk ∆N = pN with p = 1 or
∆N 2 = pN [p < 1]
N ∆µ 1 ∆µ i ∆G = = 2 N µ G N µ
∆G 2 ∝ N ∆µ i2
Number fluctuations
2
2
2
2 i 2
2
2
p ∆G = N G
Or traps at interface, Fermi 1 1 1 with = + + 2 N t N Pt ∆N
L.K.J. Vandamme / Noise / 26-02-2004
1
2
2 G ∆ N ∆ = 2 N G
31
About Neff [17, 20,21] 1/f noise is only detectable above thermal noise for N < 1014 [71] (MOST; tox = 2nm, L = 0.2µm, W = 2 µm, VG* = 46mV ÎN = 2x103 ) Inhomogeneous current density (current crowding) in homogeneous material: N must be replaced by Neff
