Spin Hall Magnetoresistive Noise

Spin Hall Magnetoresistive Noise

Contents 1 Introduction 1 2 Theory 5 3 Experimental Setup 17 4 Angle-Dependent Resistive Noise of YIG Pt Heterostructures 27 5 Summary and Outlook 53 A Data Post-Processing 57 B FFT Spectrum Analyzer Implemented on a Computer 61 C Narrow Bandwidth Detected Noise at Gigahertz 67 References 73 Acknowledgment 77

Chapter 1 Introduction

Chapter 2 Theory 2.1 Spin Hall Magnetoresistance +1/2 1/2 = + = σ e ( µ + µ ). µ ( ) σ e = e = 2e ( ).

Pure Charge Current Spin-Polarized Current Pure Spin Current J J J J q J = = = J q J J J s Js Figure 2.1: = /(2e) = h/(2π) µ ( ) µ = µ µ = σ 2e 2 µ. s α ( = α ) 2e (a) SHE J q Js (b) ISHE J s J q Figure 2.2:

2.1 Spin Hall Magnetoresistance (a) 7 (b) n M s t M s j Figure 2.3: Spin and charge currents in a ferromagnetic insulator normal metal (FMI NM) heterostructure with spin orbit interaction leads to spin Hall magnetoresistance (SMR). An incident charge current Jin q gets partially converted into a spin current JSHE by the spin Hall effect (SHE). A gradient in spin accumus lation arises from JSHE with magnitude depending on the boundary condition at s the FMI NM interface. For magnetization M s (a) no angular momentum can be transfered by spin transfer torque (STT) into the FMI. The result is a spin ISHE diffusion current Jdiff s, which itself is partially converted back into a charge Jq by the inverse spin Hall effect (ISHE). The boundary condition M s (b) allows for STT and a finite spin current JSTT propagates into the FMI, thereby reducing s are decreased, resulting in enhanced and JISHE the spin accumulation. Hence Jdiff q s longitudinal resistivity. By rotating the magnetisation M, the STT mechanism at the FMI NM interface may continuously be activated or deactivated. and the direction of spin current is perpendicular to both initial current flow and spin current polarization. The SHE is visualized in Fig. 2.2(a). Here an electron flow with polarization into of the plane is deflected to the top. This leads to a spin accumulation on both sides (top/bottom) of the conductor, as optically observed by Kato et al. [16]1. The same scattering effect can also translate a propagating spin current into a charge current for this reason it is called inverse spin Hall effect (ISHE) (cf. Fig. 2.2(b))[30]. It is worth mentioning that spin currents can propagate in magnetically ordered materials even in the complete absence of mobile charge carriers. Then Js is carried by magnons [31]. However, the physics then is qualitatively different from what is discussed here. Let us now consider a film of NM in the j-t plane (indicated in Fig. 2.3) with spin orbit interaction which hence has a finite spin Hall angle αsh. An incident charge current density Jin q is applied as sketched in Fig. 2.3(a). Due to SHE, electrons are deflected towards the top or bottom of the film spin dependently2, generating Jin a spin current density JSHE q, s. A spin imbalance is caused. The gradient s. antiparallel to JSHE in spin accumulation µs leads to a diffusive spin current Jdiff s s 1 2 More precisely, Kato et al. used a semiconductor for their Kerr rotation microscopy experiment. The t-axis is used as quantization for the spin because only spin current along n is relevant to SMR.

, = /M µ ρ ρ, > ρ,. t w l α ρ ρ ρ (α) = ρ + ρ 2 (α) ρ (α) = ρ (α) (α). ρ ρ ρ ρ ρ ρ ρ ρ ρ I

V V ρ = E J ρ = E J = V I wt l = R wt l = V I t = R t. R = V /I R = V /I R = l ( ρ 1 + ρ ) 2 (α) wt ρ R = 1 t ρ (α) (α). R := l wt ρ R/R := ρ/ρ l/w R ρ/ρ 10 4 10 3 t =. 2.2 Introduction to Noise Φ(t) {Φ(t)} p(φ) p(φ)

ϕ(t) τ { Φ(t) t (0, τ ) ϕ(t) = 0 τ R + ϕ ϕ 2 := 1 ˆ τ ϕ 2 (t) t = 1 τ T 0 ˆ ϕ(t) 2 t = 1 ˆ F[ϕ](f) 2 f τ F[ϕ](f) := ˆ ϕ(t)e i2πft t. ϕ 2 ϕ t (0, τ ) ϕ R F[ϕ]( f) = F [ϕ](f) F[ϕ](f) 2 f Φ F[ϕ](f) 2 T ϕ 2 = 1 ˆ F[ϕ](f) 2 f = τ ˆ 0 2 τ F[ϕ](f) 2 f. ϕ 2 f := 2 τ F[ϕ](f) 2. ϕ 2 f S(f) S(f) V S V (f) / S V / 2.3 Thermal Noise and Quantum Fluctuations S V

R E(ω, T ) T ω S V (f) 4R = E(ω, T ). 1/(4R) R I 2 R = V 2 R (R + R ) 2 (R = R) V 2 /(4R) hf/2 E(ω, T ) = hf ) 2 + hf 1. e hf k T 1 = hf 2 ( hf 2k T hf k T hf k T hf k T S V (f) = 4k T R hf k T S V (f) = 2hfR R T f S V, (R, T ) = 4k T R. V 2 B V 2 = 4k T RB. hf = ev hf k T S V (f) = 2eV R = 2eIR 2 ev k T S I = S V /R 2 = 2eI

ˆ= 2.4 From Spin Hall Magnetoresistance to Spin Hall Magnetoresistive Noise S = S V,0 + S V 2 (α) S V,0 = 4k T l ρ wt S V = 4k T l wt ρ. 2 (α) S V /S V,0 = ρ/ρ 0 10 3 I ρ (I I )(τ) = I (t)i (t + τ) ˆ S I (f) = 2 ρ (I I )(τ)e 2πifτ τ.

V S V V 0 - V 2 1 /2 2 1 /2 V t F F T S V 0 f s f m a x Figure 2.4: V = 0 (V ) = V 2 S V 2.5 Quantifying White Noise by the Fast Fourier Transform V (t) V = 0 S V (f) S V (f) S V / = 1. V (t) S V,r n / = n. τ τ τ = τ n. f = 1/τ / = n

S V n τ S V (f) = 2 ˆ V (t)e i2πft t τ 2. t k = kτ /n f r = r/τ k, r = 0,..., n 1 t τ /n S V (f r ) V (t k ) S V (f r ) = 2τ n 2 F [V ](f r ) 2 F [V ](f r ) = n 1 k=0 2π i V (t k )e n rk. f = n /(2τ ) S V V k := V (t k ) S V,r := S V (f r ) S V,r = 2τ n 2 = 2τ n 2 ( n 1 kj ) ( n 1 ) V k e 2πi n rk V j e 2πi n rj k=0 j=0 V k V j e 2πir n (j k) k, j = 0,..., n 1 V V k V j V k V j = V 2 δ kj δ kj S V,r = 2τ n 2 V 2 k = 2τ n V 2 τ /n f = 1/τ

R T V 2 B = f = n /(2τ ) S V,r = 4k T R X (X) = X 2 X 2. SV,r 2 S 2 V,r = 4τ 2 n 4 kjmn V k V j V m V n e 2πir n (j k+n m). V k V j V m V n V k V j V m V n V k V j V m V n = V 2 2 (δ kj δ mn + δ km δ jn + δ kn δ jm ) S 2 V,r = 4τ 2 n 4 V 2 2 ( km 1 + kj e 2πir n (2j 2k) + kj 1 ) = 4τ 2 n 4 V 2 2 (n 2 + 0 + n 2 ) = 8τ 2 n 2 V 2 2 1 X / = X (X). S V,r / = 1.

Chapter 3 Experimental Setup 10 3

2D-VM brass cylinder and Al box #1 1 Al box #2 Hall probes R sample C sample 2 Ccoax Camp SR560 FFT SR760 2D-VM Figure 3.1: 3.1 Sample Fabrication and Characterization, d 60 t =.. w l. w. l. 5 5 2 t =. V (I ) R(T )

YIG Figure 3.2: w, w, l, l t, α (a) l w w 1 n l 1 Pt t t Pt GGG (111) j (b) n j t I = R(T ) = R 0 [(1 + A(T. )]. R =.. A = 7.9 10 4 / < T < A A = 3.9083 10 3 / A = 3.9673 10 3 / 4.9 R(T ) A = 7.9 10 4 (a ) V 1 2 (m V ) 1 0 0 8 0 6 0 4 0 2 0 0 Y IG (6 0 n m ) P t(2.7 n m ) 0 1 0 2 0 3 0 4 0 5 0 I q (µa ) (b ) R 1 2 (Ω) 2 0 0 0 1 9 9 0 1 9 8 0 Y IG (6 0 n m ) P t(2.7 n m ) 1 9 7 0 2 8 5 2 9 0 2 9 5 3 0 0 3 0 5 T (K ) Figure 3.3:. T.

3.2 Instruments and Devices 3.2.1 Shielding and Wiring Figure 3.4: 50 50 2 1 2 R C = C +C +C 2 3.2.2 Two Dimensional Vector Magnet

(a ) 2 9 8.5 t (h ) 0 1 2 3 4 5 (b ) T H a ll p ro b e (K ) R 1 2 (Ω) 2 9 8.0 2 9 7.5 2 9 7.0 2 0 2 0 2 0 1 8 2 0 1 6 1.4 K Y IG (6 0 n m ) P t(2.7 n m ) 2 0 1 6.9 2 2 0 1 6.8 8 1.6 Ω 0 1 2 3 4 5 t (h ) 1 j t.0 K 0 1 2 3 m in Figure 3.5:. T R µ H = α =... µ H α α =. µ H =. 40.

.. 1.0 T < 3.2.3 Low Noise Voltage Amplifier 10 3 3.2.4 Fast Fourier Transform Spectrum Analyzer f = n = 1024 τ = n /f = f = f /2 = f = 1/τ =. R =. T =.

1024 V (t k ) R t k = k/f k = 0,..., n 1 F [V ](f r ) n 1 k=0 V (t k )e 2πirk n. n = n /2 F [V ](f r ) f r = r f r = 0,..., n 1 / S V (f r ) = 2τ n 2 F [V ](f r ) 2 = 2 n 2 f F [V ](f r ) 2. B f = τ 1024 400 B = /a a N τ, = a B < f ±0.02. 0.2 0.1

0 0 V(t r ) V 1 (t r ) L-1 V 2 (t r ) D D+L+1 N-1 V nav (t r ) N-1-L N-1 Figure 3.6: N K L D n 3 10 4 τ = no n n V (t k ) L D D < L V (t k ) o = D/L = 50 D = L/2 18/11

n / = 18 n 11 50 2 / τ AV

Chapter 4 Angle-Dependent Resistive Noise of YIG Pt Heterostructures 4.1 How low noise are we? Setup Characterization 4.1.1 Limitations by Bandwidth and Background noise

1 0-1 3 1 0-1 4 f = 9 7 5 m H z f = 6 2.5 H z f = 6 2.5 H z S V (V 2 /H z ) 1 0-1 5 1 0 0 k Ω M F R f 3 d B 1 0-1 6 1 0-1 7 s h o rte d S R 5 6 0 3 0 0 K 0.0 1 0.1 1 1 0 1 0 0 f (k H z ) Figure 4.1: f f =. f f =.. f f =. C C C C C = C + C + C. C f = 1 2πRC. 400 f = S V f <. < f <. B = f =... B = f = f =.

Figure 4.2:. f =. n = 15 10 3 S V (V 2 /H z ) x 1 0-1 7 1 6 3.9 k Ω M F R f c e n te r = 3 0 k H z 1 2 8 3 0 0 K 1 1 0 1 0 0 f (k H z ) 13 10 13 / 15 f 35 S =. <. S R > S R < R(t =. ) R(t =. ) f 4.1.2 Experimental Limit to S/N Ratio X x i i = 1,..., n X ( ) S N = X (X) x s X f =.

S /N 1 0 0 1 0 B (k H z ) 0.4 4 4 0 3.9 k Ω M F R 3 0 0 K A V = 4 m in 1 1 0 1 0 0 f (H z ) Figure 4.3: S V f τ = / f X / (X) x s X µ = 1 n x i n i=1 s x = 1 n 1 n (x i x) 2.. f =. i=1 S + S, (., ).,. < f < S V (f i ) X = S V S V (f i ) i = 1,..., 400 X(t) 400

Figure 4.4: S V n f =. τ / τ S /N 1 0 0 1 0 A V (s ) 0.1 6 1.6 3 1 6.3 3 1 6 3.3 5 3.9 k Ω M F R 3 0 0 K f = 6 2.5 H z 1 0 1 1 0 2 1 0 3 1 0 4 n A V U n ifo rm B M H F la tto p H a n n in g n τ = n τ = B = f τ = f = B = f f > n τ n B = n f =. τ = / = n / = τ f = n τ = n τ f = τ 1 / = n 1 0 0 3.9 k Ω M F R 3 0 0 K Figure 4.5: S V n f =. τ = / = n S /N 1 0 f = 6 2.5 H z A V = 4 m in 1 0 1 1 0 2 1 0 3 1 0 4 n A V

S /N 4 0 3 5 3 0 2 5 2 0 1 5 1 0 2 5 0 2 0 0 1 5 0 1 0 0 5 0 0 3.9 k Ω M F R 3 0 0 K f = 3.9 H z U n ifo rm B M H A V (s ) 0 2 0 4 0 6 0 8 0 1 0 0 o (% ) Figure 4.6: S V o f =. n = 1 000 τ τ B =. o = 90 τ = on τ 60 B = n = 15 10 3 f =. 10 s 10 s+1 / 10 2 10 4 10 3 400 25 / = 10 2 400 25 = 10 4 α.

4.2 Angle-Dependent Magnetoresistance R R µ H 1 R R R V I =. V R α = α = α = α = 2 (α) ρ α R 2 (α) R R R (2α) R R.. R R = l w R =. =. t =. µ H = 1 V /R =.

(a) V long V trans n t j (b) R long (Ω) 2019 2018 2017 2016 α( ) 0 90 180 270 360 R trans 0.1 0.0 R long -0.1 R trans (Ω) (c) V 34 n t (d) 1619 j R 34 (Ω) 1618 1617 0 90 180 270 360 α( ) Figure 4.7:. R R R R R α µ H = α = 0 ρ/ρ = 1.3 10 3 ρ =. R α R 1.0 10 3 l/w R R R ρ =. ρ/ρ = 1.3 10 3. ρ/ρ = 1.3 10 3 α = t = 2 (α) R 2 (α ) R = V /I V I R R = V /I I V

Figure 4.8: w = w = l = α δ R(δ) l δ l δ 1/ (δ) (a) l 1 (b) (c) const. V,I q (I) w R 1 (I) M α` δ (II) w R(δ) (III) t j J q w 1 R (III) l δ α = α = α = α = 1.0 10 3 R =. α = α α δ δ ρ ( ) (α, δ) = ρ + ρ 2 (α + δ). δ δ R(δ) δ R(δ) R(δ) = ρl δ A l δ l δ A A δ l δ l δ = w (δ).

I δ = V /R(δ) V ( ) I δ (δ) I δ 1 R(δ) 1 l δ (δ). (δ) I = B π/2 cos(δ) δ π/2 B = I /2 δ ˆ π/2 1 ρ ( ) (α) = ρ + ρ π/2 2 (δ) 2 (α + δ) δ. [ 1 ρ ( ) (α) = ρ + ρ 3 + 1 ] 3 (α ) 2 ρ 0 2 (α ) 1/3 2R ( ) = 2ρ 0 w /(t l ) =.. R. 1 + 0.33. ρ = 0.7 ρ. R R 1.0 10 3 77 ρ/ρ = 1.3 t. ρ >> ρ

Figure 4.9: f =.. α = µ H = S V (V 2 /H z ) x 1 0-1 7 1 0 8 6 4 Y IG (6 0 n m ) P t(2.7 n m ) 2 9 9 K 2 5 k H z 1 1 0 1 0 0 f (k H z ) 4.3 Angle-Dependent Resistive Noise Measurements α α =. f f =... < f <.. < f <. 400 f

n = 396 S V, (,. ) + S =. α 25 / 10 4 < f < n = 29 α. τ = B = n = 15 10 3 f =. < f <. 10 3 S V,i (f, α) i i = 1,..., n f n α S V,i (f, α) α S =. S V (α) S V (α) = 1 n S V,i (f, α) S. n n i=1 f n n n 400 f n σ (SV ) = σ ( SV ) = σ ( SV ) n n 1 n (S V (α) n n 1 S V,i (f, α)) 2 i=1 =. f

(a) n t (b) n t j j (c) S V,long (V 2 /Hz) S V,long 3.298 3.296 3.294 3.292 x10-17 YIG(60 nm) Pt(2.7 nm) 295.8 K V long 0 90 180 270 360 α( ) 2019 2018 2017 2016 2015 R long (Ω) Figure 4.10: S V, R S V, R α µ H = α = 0. n = 29 29 ±. R S V, S V, R S V, = 4k (. )R S V /S V,0 = (1.4 ± 1) 10 3 R/R = 1.3 10 3 S V,0 =. R =.

σ ( SV ) n = 29 n = 396 11484 S V, (α) S V, (α) α α = 2 (α) S V, =. S V /S V,0 = (1.4 ± 0.1) 10 3 S V, R S V, 2400 2182 R S V, y S V, = 4k T R T =. T, =. T, =. S V, R T = S =.

Figure 4.11: f =. µ H = α = < f < S V (V 2 /H z ) x 1 0-1 7 1 0 8 6 4 Y IG (6 0 n m ) P t(2.7 n m ) 2 9 8 K 2 5 k H z 1 1 0 1 0 0 f (k H z ) (a) n t (b) V 34 n t S V,trans j j (c) S V,trans (V 2 /Hz) 2.670 2.669 2.668 2.667 x10-17 YIG(60 nm) Pt(2.7 nm) 298.6 K 1619 1618 1617 0 90 180 270 360 α( ) R 34 (Ω) Figure 4.12: S V, S V, R α µ H = 17 ±. n = 54 S V, R T =. α α S V /S V,0 = (7±1) 10 4 R/R = 1.0 10 3 S V,0 =. R =.

f > α = α = α = α = n = 54 (31.3115 ± 0.125) (41.0625 ± 0.125) σ ( SV ) =. σ (SV ) = 1 389 59 σ ( SV ) =.. S V, α = α = T =. T, =. T, =. σ (SV ) S V /S V,0 = 7 10 4 R/R = 1.0 10 3 S V, R 34 t = 2.2 S V, n = 19 f =. ±. ± R S V, R T =.. 2 (α) S V, S V /S V,0 = 1.1 10 3 R/R = 1.0 10 3 t = 2.2

(a) n t (b) n t j j S V,long (c) 6.882 S V,long (V 2 /Hz) 6.879 6.876 6.873 6.870 x10-17 YIG(60 nm) Pt(2.2 nm) 291.5 K V long 4276 4274 4272 4270 4268 R long (Ω) 0 90 180 270 360 α( ) Figure 4.13: t =. S V, R S V, R α α n = 19 T =. S V /S V,0 = 1.1 10 3 S V,0 =. R/R = 1.0 10 3 R =.

2D-VM Hall probes 2D-VM R sample C sample 9V 1 220 kω 2 Figure 4.14: t =. 4.4 Biased ADRN. R = R 20 R 1 I = V /R I = I = R =.

1 0-1 3 u n b ia s e d, n o filte r b ia s e d, h ig h p a s s 1 0-1 4 1 0 0 H z - 6 d B /o c t S V (V 2 /H z ) 1 0-1 5 1 0-1 6 1 0-1 7 1 0-1 8 f H P 2 5 k H z 3.9 k Ω M F R 0.0 1 0.1 1 1 0 1 0 0 f (k H z ) Figure 4.15:. V =. f f = (7.92±0.02) 10 17 / (7.80 ± 0.02) 10 17 / S V S V < f < (7.92±0.02) 10 17 / (7.80±0.02) 10 17 / T = T = 6.6 f ±. t =. I =.

S V (V 2 /H z ) 8 7 6 5 x 1 0-1 7 Y IG (6 0 n m ) P t(2.7 n m ) u n b ia s e d, n o filte r b ia s e d, h ig h p a s s 1 0 0 H z, 6 d B /o c t 2 5 k H z 1 1 0 1 0 0 f (k H z ) Figure 4.16: f =. I =. µ H = f = f =. n = 15 10 3.. T =... α V V S ( ) V, I R R

I =. f =.. ± ± 389 S ( ) V, α α = α = ± n = 19 S ( ) V, α = α = 2 (α) R T =.... S V /S V,0 = 8 10 4 S V,0 =. ρ/ρ = 1.3 10 3 ±1.6 10 4 S ( ) V, 2 (α) R S V

YIG(60 nm) Pt(2.7 nm) (a) n t j R long (Ω) α ( ) 0 90 180 270 360 2019 (b) 0.1 2018 2017 0.0 R trans (Ω) (c) V trans 2016-0.1 V long (d) x 10-17 3.342 (e) 303.3 K 1995 (f) S V,long (g) S V,trans V long (V2 /Hz), S (I) 3.340 3.338 x 10-17 2.710 (h) 299.0 K 1994 1993 1992 1641 R long (Ω) (i) V long V 34 V trans (V2 /Hz), S (I) 2.708 2.706 1640 1639 1638 0 90 180 α ( ) 270 360 R 34 (Ω) Figure 4.17: α µ H = R R S ( ) V, I =. S ( ) V, R R S ( ) V, I =. S ( ) V, R R n = 19 S ( ) V, n = 30 S ( ) V, ±1.6 10 4 S ( ) V = 4k T R T =. S ( ) V, T =. S ( ) V, S V /S V,0 = 8 10 4 ρ/ρ = 1.3 10 3 S V,0 =. R =. S V /S V,0 = ρ/ρ = 1.0 10 3 S V,0 =. R =.

I =. n = 30 ±. 2 (α ) S ( ) V, n = 19 n = 30 S α = α =. T =. n = 30 α = α = S ( ) V, α = α = R I =. R S ( ) V, =. R =. T =. T =... S ( ) V, S V /S V,0 = (1.0 ± 0.2) 10 3 ρ/ρ = 1.0 10 3 (α) (α) 2 (α )

. S V /S V,0 S V,0 ρ/ρ 0 R 0 ( 10 3 ) ( 10 18 / ) ( 10 3 ) ( ) 1.4 ± 0.1 32.9 1.3 2.02 0.7 ± 0.1 26.7 1.0 1.62 0.8 ± 0.2 33.4 1.3 1.99 1.0 ± 0.2 27.1 1.0 1.64 Table 4.1: S V /S V,0 S V,0 ρ/ρ 0 R 0. ±. ±6 10 5 R ±1 10 5 ±2 10 4 2 (α) S ( ) V, S ( ) V, 2 (α ) R ±2 10 4 6 10 5 f < R 2 (α) 2 (α)

Chapter 5 Summary and Outlook 5.1 Summary / = n I < T < 10 3. ρ/ρ =

1.3 10 3 ρ/ρ 23... ±2 10 4 6 10 5 5.2 Outlook Detecting SMN up to MHz

B τ / =. 1 + τ B τ τ. B B. 1.88 / = 6.4 10 4 ±( / ) 1 = 1.5 10 5 6 10 5 B Noise Detected Magnetization Dynamics at Gigahertz τ = τ/τ 1 τ τ = n /f

Appendix A Data Post-Processing / 510 1 2 3 f,i B,i i = 1,..., s 4 w j j = 1,..., 400 S V (f j ) w j {0, 1} {{f,1, B,1 },..., {f,s, B,s }} (a) x10-18 (b) x10-18 S V (V 2 /Hz) 49 48 47 20 25 30 35 40 45 f(khz) S V (V 2 /Hz) 48.2 48.1 48 0 10 20 30 t(h) Figure A.1:

(a) x10-18 48.3 (b) 48.18 x10-18 S V (V 2 /Hz) 48.2 48.1 S V (V 2 /Hz) 48.16 48.14 48.0 0 90 180 270 360 α( ) 48.12 0 90 180 270 360 α( ) Figure A.2: 3 0 nf,1 B,1 f j nf,1 + B,1, w j = nf,s B,s f j nf,s + B,s, n N 1. S V (f j ) w j S V (f j ) w j = 0 f =. B = n = 400 j=1 w j n = 396 α 29 σ / n ±σ / n n α α = 0 σ / n n

Appendix B FFT Spectrum Analyzer Implemented on a Computer 10 9 ± S V τ 0 f f 2

(a ) S V (V 2 /H z ) 1 0-1 5 (b ) 1 0-1 6 3 2 k H z 1 0-1 7 2 9 7 K 1 0-1 8 1 0 0 k Ω M F R 1 0-1 9 3.9 k Ω M F R s h o rte d S R 5 6 0 1 0-2 0 0.1 1 1 0 1 0 0 1 0 0 0 f (k H z ) S /N 1 0 0 1 0 3.9 k Ω M F R 2 9 7 K f = 7 6.3 H z A V = 4 m in 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 n A V Figure B.1:. <. 10 3 n. f f =. τ = / = n n f = f n. R <. S =. S V (f) = 4k T R G(f) 2 + S G(f) 2 = 1 1 + (f/f ) 2 f = 1 2πRC. G(f) C R =. <.

S V f >.. f >. f f = n n = 2 17. τ = τ = n n /f f / = n n < 10 4 f < n n n / = n n n τ B < f < B =

2 (n ) f ( ) 200 100 50 25 10 5 2 1 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 Table B.1: τ n f τ < 1 1 < τ < 10 τ > 10 f B = n f = n n n τ = n n τ (1 + τ ) = n n (1 + τ ) f τ τ = τ/τ 1 τ τ τ = 0 / = B τ 1 + τ B = / = ( )τ f f = τ = 0.382 n = 2 24 B = / = ( )τ

13 10 3. R C = f. G 2 = 1 G B = 2 f G 2 =. f = τ = 1.88 n = 2 24 τ =. / = 6.4 10 4. 5 10 3 2 10 4 / = (2 10 4 ) 1

Appendix C Narrow Bandwidth Detected Noise at Gigahertz Z R > Z Γ Γ = (R Z )/(R+Z ) C R ω λ/2 Z (ω ) 1 R (ω C ) 2. R ω C Z = λ/2 R

(a) (b) sample 2 C k 1 λ/2 λ/2 Figure C.1: λ/2 λ/2 C 1 R =. f =. V = 4k T RB. k =. T R B P = V 2 4R = k T B. N T R T = N k B

DUT N o R T e R N o R Figure C.2: T ~ T 0 =290 K R Noisy network G, T e R P i =S i +N i P o =S o +N o Figure C.3: / / F := / / 1 F = 1 + T T N = k T B T := T N = k GB(T + T ), G G =. F =. N = k T B T = T N... B = f = N F T [ ] = (10 F [ ]/10 1)T [ ] = 0.968 T [ ] =

(a) (b) N e N o +N IMN R=50 Ω T 0 =290 K Kuhne G, T e Z 0 IMN N e +N imn Kuhne G, T e Z 0 N i N o Γ IMN (f) 1 f 0 f Figure C.4: R =. f = Γ(f) N (d B m ) -1 1 6-1 1 8-1 2 0-1 2 2-1 2 4-1 2 6 Γ 5 0 Ω IM N 3.5 4.0 4.5 5.0 5.5 6.0 6.5 f (G H z ) Figure C.5: N f Γ 0.9 0.6 0.3 0.0 Γ G[ ] = 10 G[ ]/10 = 1244. N [ ] = Gk B(T + T ) =. N [ ] = 10 (N [ ]) = N = Γ(f) N

N N = Gk B(T + T ) =. = N = k T B T = T N = Gk B(T + T ) =. = l =. 70 f = 0.7c/l =. c R =. N

References

Acknowledgements

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