Thermal equilibrium noise with 1/f spectrum from frequency independent dielectric losses in barrium strontium titanate

Thermal equilibrium noise with 1/f spectrum from frequency

independent dielectric losses in barrium strontium titanate

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Vandamme, L. K. J., Khalfallaoui, A., Leroy, G., & Vélu, G. (2010). Thermal equilibrium noise with 1/f spectrum from frequency independent dielectric losses in barrium strontium titanate. Journal of Applied Physics, 107(5), 053717-1/6. [053717]. https://doi.org/10.1063/1.3327446

DOI:

10.1063/1.3327446 Document status and date: Published: 01/01/2010

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Thermal equilibrium noise with 1 / f spectrum from frequency independent

dielectric losses in barium strontium titanate

L. K. J. Vandamme,1,a兲 A. Khalfallaoui,2G. Leroy,2and G. Vélu2

1

Department of Electrical Engineering (PT 9.13), Eindhoven University of Technology, 5600MB Eindhoven, Netherlands

2

Laboratoire d’Etude des Matériaux et des Composants pour L’Electronique, E.A. 2601, Université du Littoral Côte d’Opale, B.P. 717, 62 228 Calais, France

共Received 23 October 2009; accepted 23 January 2010; published online 10 March 2010兲 We investigated the dielectric losses of doped and undoped BaSrTiO₃ 共BST兲 from thermal noise measurements. The results are compared to impedance measurements. The value for the frequency independent loss angle is about tg ␦= 2⫻10−2_{in the range 100⬍ f共Hz兲⬍10}5_{. The thermal voltage} noise of the BST capacitor with losses has a 1/ f spectrum in agreement with 4kT R共Z兲 and a frequency independent tg ␦. The detection limits due to the low noise voltage amplifier are investigated and experimentally verified. The frequency range f_high, f_low, where the “1/ f thermal noise” is above the background noise is characterized by the ratio f_high/ f_low= tg2_␦共R

in/Reqw兲, with R_inthe input resistance of the low noise voltage amplifier and R_eqwthe frequency independent part of its equivalent noise resistance at high frequencies. © 2010 American Institute of Physics. 关doi:10.1063/1.3327446兴

I. INTRODUCTION

Barium strontium titanate共BST兲 is a promising dielec-tric for rf applications due to its high values of ␧r共200⬍␧r

⬍600兲 and the fact that the value decreases by applying a positive or negative bias voltage, i.e., the tunability.1,2 We have investigated the loss angle tg ␦for doped and undoped BST from thermal noise in the frequency range 10⬍ f共Hz兲 ⬍105_{. The result without ac excitation is compared to} im-pedance measurements.

The aim of this work is共i兲 to establish the detection limit for the noise measurements on capacitors with losses and共ii兲 to compare results from doped and undoped BST by thermal equilibrium and ac excitation measurements. The fluctuation dissipation theorem for these materials is applied.

Thermal equilibrium noise at low temperature 共T ⬍4 K兲 with a 1/ f spectrum was observed in the magnetic susceptibility of different materials.3,4 Their results were in agreement with the fluctuation dissipation theorem as usual. The loss tangent in dielectrics is often frequency inde-pendent over 8 decades in frequency.5The thermal 1/ f noise in dielectrics with losses was discussed in Ref.6and thermal current noise proportional to f was experimentally verified in silicon p-n junctions in Ref.7. For nanoparticle WO3 films with a high value of the loss angle共tg ␦= 1兲, the detection of thermal current and voltage noise was easy and the analysis turned out to be in agreement with the fluctuation dissipation theorem. The noise of biased WO3and BST dielectrics was used as a diagnostic tool for dielectric quality assessment.8,9 The losses in liquid crystals with tg ␦⬎0.1 were also suc-cessfully investigated from noise measurements.10

However, detection problems arise when the loss angle of the material under test is smaller than 0.1. Our BST di-electrics have relatively low loss angles of about 10−2, which

makes a discussion of the detection limit unavoidable in Sec. II. In order to check detection problems, our noise measure-ments on BST capacitors with capacitance C are compared to the background noise of a variable air capacitor with negli-gible small tg ␦with Cair= C.

II. EXPECTED THERMAL NOISE FROM A CAPACITOR WITH LOSSES

A. Equivalent circuit of capacitor with losses and its thermal voltage noise

The equivalent circuit of a capacitor with a dielectric without free electrons is represented as shown in Fig.1. For not too high frequencies, the circuit in Fig. 1共a兲 describes well the capacitor with losses. As a good approximation holds for frequencies below 105 _{Hz that C is frequency} in-dependent and R describing the losses is frequency depen-dent. The admittance and a more general presentation of the dielectric follow from Fig.1共a兲

a兲_{Electronic mail: l.k.j.vandamme@tue.nl.}

CV ù j C I = V/R R I = R I C I I= + IR Ic ùRC 1 C I R I tgä ä = = I V

a

b

V

FIG. 1. 共Color online兲 共a兲 Equivalent circuit of a capacitor with losses. 共b兲 The definition of␦and tg ␦.

Y = j␻C + 1 R⬅ j␻C

ⴱ₌ j␻␧0A

L 共␧

⬘

− j␧

⬙

兲,

with Cⴱ the complex capacitance, 1/R=␻ ␧0 ␧

⬙

A/L, and C =␧0 ␧

⬘

A/L.

Figure1共b兲 shows the definition of the loss angle␦ and tg ␦, which is the ratio between the current component in phase with the applied voltage V兩IR兩 and the current

compo-nent 90° out of phase兩IC兩. The total current I makes an angle

␦ with ICand tg ␦is

tg ␦= 1 RC␻=

␧

⬙

␧

⬘

. 共1兲

The tg ␦ is often frequency independent.5 It is denoted as the ratio of the imaginary and real part of the complex ca-pacitance Cⴱ. The tg ␦ frequency independent means R ⬀1/␻ if C is frequency independent or ␧

⬙

and ␧

⬘

are fre-quency independent. If ␧

⬙

and ␧

⬘

are proportional to f−⌬, then tg ␦ is still frequency independent.

The impedance Z, the real part of Z, R共Z兲, and the real part of Y, R共Y兲, are given by

Z = R 1 + j␻RC ⇒ R共Z兲 = R 1 +共␻RC兲2 = tg ␦ ␻C共1 + tg2␦兲 for tg ␦Ⰶ 1 ⇒ Re共Z兲 ⬇ tg ␦ ␻C , 共2兲 Y = 1/R + j␻C⇒ Re共Y兲 = 1/R =␻C tg ␦. 共3兲 The expected thermal voltage noise in agreement with the fluctuation dissipation theorem for a capacitor with losses SV_therm= 4kT R共Z兲 is SV_therm= 4kT tg ␦ ␻C共1 + tg2␦兲⬀ 1 f for C and tg ␦ frequency independent

SV_therm⬵ 4kT tg ␦ ␻C or SV_therm 4kT ⬵ R tg 2_{␦ for 共4兲} tg ␦Ⰶ 1.

Hence, thermal 1/ f voltage noise holds for C and tg ␦ fre-quency independent. If␧

⬘

and␧

⬙

are weakly dependent on frequency and proportional to f−⌬ with often ⌬⬍0.2, then still, tg ␦ is frequency independent but the thermal noise SV_thermis 1/ f like noise proportional to 1/ f共1−⌬兲.

The thermal current noise SI_therm= 4kT R共Y兲 is

SI_therm= 4kT R共Y兲 =

4kT

R = 4kT ␻C tg ␦⬀ f 共5兲 for C and tg ␦ frequency independent

SV_thermSI_therm

共4kT兲2 = tg2␦ 1 + tg2␦⬵ tg

2_{␦ for tg ␦Ⰶ 1. 共6兲}

Hence, the tg ␦ can be calculated from the thermal noise as shown in Eqs. 共4兲–共6兲. The ratio SV_therm/SI_therm gives the

squared value of the impedance

SV_therm SI_therm =R共Z兲 R共Y兲= Z2= R2 1 +共␻RC兲2= 1 共␻C兲2· 1 1 + tg2␦ ⬇ 1 共␻C兲2 for tg ␦Ⰶ 1. 共7兲

B. Detection limits as a frequency range flowand fhigh for voltage noise

The detection below the lowest frequency flowis limited by the thermal noise of the input resistance of the low noise voltage amplifier, and the detection above the highest fre-quency fhighis limited by the background noise of the ampli-fier with an ac short-circuited input. This is explained by using the equivalent circuit of a low noise voltage amplifier with input impedance Zin, which is the parallel connection of input resistance Rin and input capacitance Cin as shown in Fig. 2共a兲. It will be used to calculate the frequency range, where thermal voltage noise detection of the capacitor with losses C/ /R as shown in Fig. 1共a兲 is possible. Figure 2共a兲 shows a noise voltage source enat the input, an ideal共infinite

input impedance兲 noise-free amplifier with the same gain and bandwidth f_{h amp}– f_{l amp}as the real amplifier.

For low noise voltage amplifiers with an open input holds SV_open= 4kT R共Zin兲+具en

2_{典. For good low noise} amplifi-ers with open input holds at f⬇ fl amp, SV_open= 4kT Rin. The noise of an amplifier with short-circuited input is often white above a characteristic frequency f0共f0ⱕ300 Hz兲 and propor-tional to 1/ f below f0. Therefore, the voltage noise with short-circuited input 具en2典=SV_short is often characterized by

Req, the equivalent noise resistance of an amplifier at room temperature as n in in C e R n in in C e R C R a b

FIG. 2. 共a兲 Equivalent circuit for a low noise amplifier. 共b兲 Equivalent cir-cuit of low noise amplifier with capacitor C and losses R.

SV_sh= 4kTReq= 4kTReqw

冉

1 + f0 f

冊

⇒ SV_short 4kT = Reqw

冉

1 + f0 f

冊

, 共8兲

with R_eqw representing the white part of the spectrum SV_short

and hence, Reqwis the value of Reqat fⰇ f0. For high quality low noise amplifiers holds 共i兲 RinⰇReqw共1+ f0/ f兲; 共ii兲 for SV_open SV_open⬵ 4kTRin 1 +共␻RinCin兲2 + 4kTReqw⇒ SV_open 4kT = Rin 1 +共␻R_inC_in兲2+ Reqw. 共9兲 The voltage noise of the amplifier with open input SV_open in

Eq. 共9兲 is approximated in three regions: low, f⬍ f1; me-dium, f1⬍ f ⬍ f2; and high frequency f⬎ f2 as, respectively,

SV_open= 4kTRin for fl amp⬍ f ⬍ f1= 1 2␲R_inC_in, SV_open= 4kT Rin共␻Cin兲2 for f1⬍ f ⬍ f2= f1

冑

R_in Reqw ,

SV_open= 4kTReqw for f2⬍ f ⬍ fh amp. 共10兲

Hence, for open input关Eq. 共9兲兴, two corner frequencies can be distinguished

f1= 1/共2␲Rin Cin兲 and f2= 1/关2␲共Rin Reqw兲1/2Cin兴 = f1共Rin/Reqw兲1/2.

The background noise for voltage noise measurements of capacitors with losses is calculated from Fig.2共b兲by replac-ing C by an air capacitor with tg ␦= 0 and a value Cair= C and R→⬁. For calibration purposes and the calculation of the detection limits, a spectrum is calculated and measured from an air capacitor C_air= C with negligible losses. The spectrum SV C_air with Cair and tg ␦⬇0 is the background noise for noise based tg ␦ measurements. The relation for SV C_air is given by Eq. 共9兲, where Cair is added to Cin. The corner frequencies are now f1共Cair兲and f2共Cair兲and are a factor

共1+Cair/Cin兲 lower than f1 and f2 given in Eq. 共10兲. The background noise can be approximated in analogy with Eq.

共10兲by three frequency regions

SV C_air= 4kTRin for fl amp⬍ f ⬍ f1共Cair兲=

f1 1 + Cair/Cin , 共11兲 SV C_air= 4kT

R_in关␻共C_in+ C_air兲兴2 for f1共Cair兲⬍ f ⬍ f2共Cair兲

= f2 1 + Cair/Cin

, 共12兲

SV C_air= 4kTReqw for f2共Cair兲=

f2 1 + Cair/Cin

⬍ f

⬍ fh amp. 共13兲

The noise SV_C//R of a capacitor C with losses is evaluated

from the total impedance Ztat the input in Fig.2共b兲. The real

part of the total impedance R共Zt兲 is given by Eq.共2兲, where

C is replaced by共C+Cin兲 and R is replaced by 共R/ /Rin兲 that is given by 共R//Rin兲 = R_inR Rin+ R = Rin 1 +␻RinC tg ␦ . 共14兲

From Eq.共14兲we define the corner frequency f_low flow=

1 2␲RinC tg ␦

. 共15兲

Below flow, the detection of losses共tg ␦兲 from thermal noise measurements is impossible. The parallel connection of losses and input resistance 共R/ /Rin兲 is approximately equal to Rin. The input impedance of the amplifier is the limiting factor.

The voltage noise SV_C//Ris given by

SV_C//R=

4kT共R//R_in兲

1 +关␻共R//R_in兲共C + C_in兲兴2+ 4kTReqw 共16兲 and Eq. 共16兲 is approximated for medium frequencies flow ⬍ f ⬍ fhighas SV_C//R 4kT ⬵ R 1 +关␻R共C + Cin兲兴2 =

冉

tg ␦ C␻

冊

1 tg2␦+共1 + Cin/C兲2 共17兲 for tg ␦Ⰶ1, and C_in/CⰆ1 holds 共SV_C//R/4kT兲⬵共tg ␦/C␻兲.

The observed noise above the corner frequency f_highwill be white due to the term 4kTR_eqw in Eq.共16兲and the detec-tion of the losses by noise is impossible. This occurs at f ⬎ fhighdefined as SV_C//R 4kT = tg ␦ ␻C = Reqw⇒ fhigh= tg ␦ 2␲ReqwC . 共18兲

Summarizing the measured thermal voltage noise SV_C//Rwith

a 1/ f spectrum equals the expected value given in Eq. 共4兲 only for the medium frequencies

SV_C//R= 4kT tg ␦ ␻C for flow= 1 2␲RinC tg ␦ ⬍ f ⬍ fhigh= tg ␦ 2␲R_eqwC. 共19兲

For a frequency independent loss angle and C, the spectrum is proportional to 1/ f.

SV_C//R= 4kTReqw for f⬎ fhigh= tg ␦

2␲ R_eqwC. 共20兲 The tg ␦ can be extracted from thermal voltage noise mea-surement in the frequency range

flow= 1 2␲RinC tg ␦ ⬍ f ⬍ fhigh= tg ␦ 2␲ReqwC . 共21兲

From Eq. 共21兲 follows the ratio of the detection limits f_high/ f_lowas f_high flow = tg2␦ Rin Reqw . 共22兲

The use of an amplifier with a high ratio Rin/Reqwis a nec-essary condition to detect tg ␦ from thermal 1/ f voltage noise over several decades in f. Typical values for low noise voltage amplifiers are R_in= 108 _{⍀, R}

eqw= 70 ⍀, and Cin = 15 pF共Brookdeal 5003兲. To demonstrate the importance of a high ratio R_in/R_eqw, we calculated spectra expressed as SV/4kT 共equivalent noise resistance兲 and the detection limits.

The results for two amplifiers are shown in Figs. 3 and 4

with the values for Rin, Cin, Reqw, and f0as indicated on top of the diagram. The detection limits flow and fhighare indi-cated by arrows. The circles show Req versus f. The dia-monds show R versus f. The triangles show the expected thermal noise with a 1/ f spectrum Eq.共4兲from C = 800 pF, with a frequency independent tg ␦= 3.2⫻10−2_{. The crosses} show the calculated noise with a loss-free air capacitor and nonideal amplifier. The calculated and experimentally ob-served background noise shows the typical 1/ f2 proportion-ality between a high and low plateau level given by Rinand Reqw, respectively. The squares show the calculated noise of a capacitor with losses.

The detection of thermal current noise from capacitors with loss angles tg ␦⬍2⫻10−2 _{is below the detection level} of most commercially available low noise current amplifiers with an open input current noise of about 2⫻10−28 _A2_/Hz and will, therefore, not be discussed here.

III. EXPERIMENTAL RESULTS A. Sample preparation

The sol-gel technique was used to deposit undoped and Mn-doped and K-doped BST films. The precursors used in the preparation of the solution are barium acetate, strontium acetate, manganese acetate, and kalium acetate. Acetic acid and isopropanol were used as the solvent. Barium acetate and strontium acetate were dissolved in hot acetic acid. Barium and strontium are in the concentration of 0.5 mol of the site A in the perovskite structure. The dopant precursor was in a concentration between 2.5 and 10 mol % of the substituted site. After getting a clear solution, titanium iso-propoxide was added to obtain the final precursor solution. After total dissolution ethylene glycol was added to improve the stability.

The precursor solution was spin coated on platinum coated silicon共100兲 substrates by a spinner at 3000 rpm for 30 s. The samples were heated for 30 s on a hot plate at 300 ° C. Afterwards, in order to crystallize the films in the perovskite phase, thermal annealing at 750 ° C was used in a tubular furnace. Finally, gold was evaporated through a me-tallic shadow mask to realize the upper circular electrodes with diameters ranging from 150 ␮m to 2 mm. The thick-ness of the undoped layers was 600 and 430 nm for the Mn-doped and 400 nm for the K-doped BST. The Pt/Ti layer acts as the bottom electrode. More details on the BST films can found in Ref.11.

The electrical properties were determined at room tem-perature in the frequency range 100 Hz–1 MHz as function of dc bias with the HP 4284A impedance analyzer.

101 102 103 104 105 106 107 108 100 ₁₀1 ₁₀2 ₁₀3 ₁₀4 ₁₀5 5003: R in=10 8 W, C_in=15 pF, R eqw=100W, f0=300 Hz; C = 800 pF, tgd = 3.2x10-2 R Req SV-Cair SV-C Re [Z] f [Hz] f low f high R (W) S V/4kT

FIG. 3. 共Color online兲 R vs f for a capacitor C=800 pF and frequency independent tg ␦= 3.2⫻10−2_{. The equivalent noise resistance of the} ampli-fier Reqvs f with f0= 300 Hz and Reqw= 100 ⍀ and the expected thermal noise proportional to R共Z兲 vs f. The calculated background noise with Cair= C and Rin= 108is labeled by crosses and show the typical 1/ f2 propor-tionality between the levels Rinand Reqw. The calculated noise of the capaci-tor with losses is denoted by squares and takes into account the effects of the nonideal low noise amplifier 共f0= 300 Hz and Reqw= 100 ⍀; Rin= 108 ⍀ and Cin= 15 pF兲. The detection frequency range fhighand flowis indicated by arrows. 101 102 103 104 105 106 107 108 100 101 102 103 104 105 5004 : R in= 5x10 6 W, C in= 50 pF, Reqw= 50W, f 0= 300 Hz; C=800 pF, tgd=3.2x10 -2 R Req SV-Cair SV-C Re [Z] R (W) S V/4kT f [Hz] f low f high

FIG. 4. 共Color online兲 The same as in Fig. 3 but now for an amplifier characterized by f0= 300 Hz and Reqw= 50 ⍀; Rin= 5⫻106 ⍀ and Cin = 50 pF.

B. Thermal noise voltage

The spectra are measured with a Brookdeal 5003 low noise voltage amplifier with Rin= 108 ⍀, Cin= 15 pF, and Reqw= 70 ⍀. The detection is limited in the frequency range flowand fhighand is calculated with Eqs.共15兲and共18兲.

The results of an undoped BST sample with a diameter of 500 ␮m are shown in Fig. 5. The spectrum SV_C//R

cor-rected for SV C_air is used to calculate tg ␦. In the frequency

range between flow= 90 Hz and fhigh= 4⫻104 Hz, reliable values for tg ␦ are obtained. The loss angle tg ␦⬇2 ⫻10−2_{and is constant over 3 decades in frequency as can be} seen from the dotted line versus the right hand scale.

The results for K-doped BST with a diameter of 500 ␮m are in Fig.6. K-doped BST shows a slightly higher ␧

⬘

than the undoped BST. The open squares show the

back-ground noise with air capacitor SV C_air. The experimentally

observed background noise shows the typical 1/ f2 propor-tionality between a high and low plateau level given by Rin and Reqw, respectively. The full squares show SV_C//R. The full

line shows SV_C//R– SV C_airwith the typical 1/ f proportionality.

The calculated tg ␦ versus the right hand scale results in reliable values between the detection limits flow= 50 Hz and fhigh= 4⫻104 Hz.

Figure7 shows the results of a 234 pF Mn-doped BST sample with a diameter of 250 ␮m. The open squares show the background noise with an air capacitor, SV C_air. The full

line shows SV_C//R– SV C_air. The dotted line represents tg ␦

ver-sus the right hand scale.

Figure8shows the comparison between the results from thermal voltage noise 共open symbols兲 and impedance mea-surements 共full symbols兲. Experimental artifacts are visible below 10 kHz due to the impedance measurement system. Hence, an increase in tg ␦ with a decrease in f is errone-ously suggested.

The largest frequency range in order to observe reliable tg ␦ values from thermal noise with an amplifier

character-10-18 10-17 10-16 10-15 10-4 10-3 10-2 10-1 100 101 102 103 104 105 S_V[V2/Hz] tgd f [Hz] undoped 840 pF; D = 500mm corrected: S V C//R- SV Cair

FIG. 5. 共Color online兲 The full line is SV_C//R– SV C_airwith the 1/ f propor-tionality typical for a frequency independent loss angle. The dashed line is a guide for the eye with 1/ f slope. The dotted line represents the tg ␦vs f with the right hand scale. The calculated value for tg ␦from thermal 1/ f noise measurement and Eq. 共19兲is about 2⫻10−2 _{and is reliable in the} frequency range flow= 95 Hz and fhigh= 5.4⫻104 Hz.

10-18 10-17 10-16 10-15 10-4 10-3 10-2 10-1 100 101 102 103 104 105 K-doped; 1035 pF; D = 500mm S V[V 2 /Hz] tgd f [Hz] corrected: S_{V C//R}- S_{V Cair} S V Cair; Cair= 1035 pF S_{V C//R}

FIG. 6. 共Color online兲 The results from a K-doped BST sample with a diameter D = 500 ␮m and C = 1035 pF. The dotted line represents tg ␦ cal-culated from the thermal 1/ f noise given by the full line 共SV_C//R– SV C_air兲. Reliable results for tg ␦are between flow= 40 Hz and fhigh= 7⫻104 Hz. The SV_C//R is denoted by full squares and the background noise SV C_airis shown by open squares.

10-18 10-17 10-16 10-15 10-5 10-4 10-3 10-2 10-1 102 103 104 105 S V[V 2 /Hz] tgd f [Hz] S V Cair; Cair= 234 pF Mn doped 234 pF corrected: S V C//R- SV Cair

FIG. 7.共Color online兲 The results from a Mn-doped sample of 234 pF with a diameter of 250 ␮m. The full line is SV_C//R– SV C_airwith the 1/ f propor-tionality typical for a frequency independent loss angle. The open squares show the background noise SV C_air. The dotted line represents tg ␦. The reliable values for tg ␦= 2⫻10−2 _{are between f}

low= 400 Hz and fhigh = 105 _{Hz. The smaller C共smaller diameter兲 compared to the C of samples in} Figs.5and6results in higher values for f_lowand f_high关see Eqs.共15兲and

共18兲兴. 10 100 103 104 105 Mn K undo ped Mn undo ped K 0,01 0,1 1 tgd f (Hz) full symbols: impedance

open symbols: thermal noise 1

0.1

0.01

FIG. 8.共Color online兲 Comparison between tg ␦from impedance共full sym-bols兲 and noise measurements 共open symbols兲.

ized by Rin= 108 ⍀, Cin= 15 pF, f0= 300 Hz, and Reqw = 70 ⍀ is for samples with a capacitance of about 1000 pF. IV. DISCUSSION AND CONCLUSION

To get reliable values for the loss angle from thermal noise measurements, a correction for the background noise obtained with an air capacitor without losses is a necessary condition especially at frequencies around f_lowand f_high. Ig-noring the correction at low frequencies leads to an errone-ous increase in tg ␦ with decreasing frequency, and at high frequencies an erroneous increase in tg ␦ with increasing frequency is the result; overcorrection leads to opposite re-sults.

The observed voltage noise at high frequencies 共f ⬎ fhigh兲 of a capacitor with losses is larger than 4kT R共Z兲 with Z the impedance of the capacitor under investigation. At high frequencies the low noise voltage amplifier has an ac short-circuited input and the voltage noise of the amplifier must be subtracted.

The loss angle 共tg ␦兲 can be calculated from the thermal voltage noise in a limited frequency range 1/共2␲RinC tg ␦兲⬍ f ⬍tg ␦/共2␲ ReqwC兲.

From the thermal noise measurement we observe reli-able values for tg ␦ for 103_{⬍ f共Hz兲⬍10}5_{. The loss angle is} frequency independent with an average value of about 2 ⫻10−2 _{for all BST samples. The K-doped have slightly} higher tg ␦ values and the undoped slightly lower. Dielec-trics with high␧

⬘

values that are strongly voltage dependent are better measured from the thermal noise in equilibrium especially at frequencies below 104 _Hz.

In general, the possible link between the 1/ f thermal voltage noise and the 1/ f noise due to resistance fluctuations should be distinguished carefully.6 Kleinpenning12,13 made an exception where the 1/ f noise in tunnel junctions was related to a constant loss tangent of the insulator in between the two metal electrodes. Then the transparency factor of the barrier is modulated by the thermal 1/ f noise 共if tg ␦and C are frequency independent兲 with 1/ f noise in the conduction as a result.

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