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Pis'ma v ZhETF, vol. 92, iss. 7, pp. 507 { 512

Shot noise measurements in a wide-channel transistor near pinch-o
Institute of Solid State Physics RAS, 142432 Chernogolovka, Russian Federation
Submitted 9 August 2010 Resubmitted 30 August 2010

c 2010 October 10

V. S. Khrapai, D. V. Shovkun

We study a shot noise of a wide channel gated high-frequency transistor at temperature of 4.2 K near pinch-o . In this regime, a transition from the metallic to the insulating state is expected to occur, accompanied by the increase of the partition noise. The dependence of the noise spectral density on current is found to be slightly nonlinear. At low currents, the di erential Fano factor is enhanced compared to the universal value 1/3 for metallic di usive conductors. We explain this result by the e ect of thermal uctuations in a nonlinear regime near pinch-o , without calling for the enhanced partition noise.

A current I owing in a two-terminal conductor placed in external electric circuit exhibits uctuations around it's mean value I . The second moment of the uctuations is related to the noise spectral density (I I )2 f SI f , where f is a measurement bandwidth. In the absence of current (I = 0) the noise is related to thermal uctuations of the occupation number of the electronic states, known as Johnson-Nyquist noise (JN-noise). In this case SI = 4kB T R 1, where kB ; T and R are, respectively, the Boltzman constant, the temperature and the resistance of the conductor. Away from the equilibrium, when the voltage drop across the conductor is high enough jeV j kB T , and in the absence of dissipation inside the conductor and spurious noises, the current uctuations are caused by the discreteness of the elementary charge e 1]. This noise is referred to as shot noise and for a voltage-biased conductor has a spectral density of SI = 2F jeI j, where F is called a Fano-factor. In a non-interacting system in the linear transport regime, the shot noise is caused by a partition of incident carriers, which can be viewed as quantum e ect 2]. In this case, the Fano-factor is determined by the distribution of the eigen-channel transparencies (Tn ) of the conductor F = Tn (1 Tn )=Tn 1 2]. The noise is strongest (F = 1) in the Poissonian regime, which is obtained when all Tn 1. In a quasi one dimensional metallic di usive conductor the distribution of Tn is universal 3] and F = 1=3, which has been experimentally con rmed 4]. The universality of the value F = 1=3 in metallic conductors has been proven to be independent of geometry 5]. Near the transition from the metallic to the insulating state one expects an increase of the partition noise to the Poissonian value 2], although this regime haven't been studied experimentally. We study the shot noise in a gated wide channel transistor near pinch-o , where the transition to the insulat92 . 7 { 8 2010

ing state is expected to occur. The dependence of SI on current is slightly non-linear in the shot noise regime. The di erential Fano-factor FD = (2jej) 1 jdSI =dI j is enhanced above the universal metallic value 1=3 < FD 0:5 for low jI j and is close to this value FD 1=3 for higher jI j. We nd that the enhancement of FD is not necessarily related to the partition noise. In contrast, it can be explained by a classical e ect of thermal uctuations in a strongly nonlinear transport regime near pinch-o . The sketch of the measurement is shown in Fig.1. The sample and the cryogenic ampli er (LTAmp) are
UCAL rf-generator CAL IN 45W 5W 10nF
Divider Lock-in

Vg G SD CS VD
D

4.2 K C CD L

~5 K 10 nF Z
0

Transistor
ET

LTAmp

Detector

Filter

RoomTAmps

to spectrum analyzer

Fig.1. The sketch of the setup. The low temperature parts are shown by dashed boxes. The parameters of the resonant LC circuit and the undesired stray capacitances CSD and CD (shown by dashed lines inside the 4.2 K box) are given in the text. The signal from the room temperature ampli ers (Room T Amps) is either sent to the lter and detector for a Lock-in measurement or to the spectrum analyzer (dashed line with an arrow) for wide range spectra acquisition

507


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V. S. Khrapai, D. V. Shovkun
10
100

placed in a 4 He gas chamber with the walls maintained at 4.2 K. The sample is below the LTAmp and is connected to it with a 20 cm long cable. A heat sink connects the LTAmp to a liquid 4 He bath. The actual temperature of the LTAmp (TLT A ) measured with a thermometer is about 5.3 K. The actual temperature of the sample is taken to be 4.2 K, consistent with the JN-noise measurement (see below). At the input of the setup (CAL IN) a 50 cable is connected to the transistor source (S) via a divider. This input is used both for driving a current I and external rf calibration. A transistor drain (D) is followed by an L C resonator, which is connected to a cryogenic ampli er (LTAmp) ( 20dB gain). The resonator serves to match a high impedance of the sample and a low input impedance Z0 = 50 of the LTAmp. The output of the LTAmp is connected via a second 50 cable to the input stage of the room-temperature low-noise ampli ers (total gain of 3 20 dB). Finally, the ampli ed signal is ltered with a 30 MHz bandpass lter at the resonance frequency f0 125 MHz and rectied by a detector (8473C by Agilent Technologies). The ac modulation of the recti ed signal thanks to a current modulation, gate voltage chopping or amplitude modulation of the external rf is measured with the lock-in. Alternatively, a spectrum analyzer can be used to analyze the frequency spectra at the output of the roomtemperature ampli ers. We study shot noise of a commercial AlGaAs/InGaAs/GaAs pHEMT ATF-35143 by Agilent with a gate length (width) of 0.5 (400) m. This transistor is known for low noise at room temperature and is used as an active element in cryogenic LTAmp's by us and other authors 6]. We preformed the resistance measurements in two such transistors (samples 1 and 2), and noise measurements and calibration only in sample 2. Negative gate voltage Vg < 0 is used to deplete the channel which results in increase of the linear-response S-D resistance R dV =dI jI !0 . The huge aspect ratio of the gate electrode allows to work near the pincho with high channel resistivity in the range of M = while keeping a reasonable R. The dependence R(Vg ) at 4.2 K is shown in the inset of g.2 for sample 1 (solid line) and for several states of sample 2 (symbols). In all cases, the behavior R(Vg ) is roughly exponential and is reproducible up to insigni cant random threshold voltage shifts (see caption). Hence, most likely, the current is homogeneously distributed across the transistor channel near pinch-o . Such a strong dependence R(Vg ) might indicate that we enter the insulating phase near the pinch-o 7]. The measured di erential resistance Rdi dV =dI is shown as a function of current in g.2a for three values of the gate voltage in sample

(a)

R (kW)

10 1 ­0.32 ­0.27 Vg (V)

Rdiff (kW)
1 ­6

­4

0 ­2 I (mA)

2

4

4 2

(b)

I (mA)

0 ­2 ­4 ­5 0 5 V (mV) 10

Fig.2. (a) Di erential resistance Rdi as a function of current for three values of the gate voltage in sample 2. Inset { linear response S-D resistance as a function of gate voltage for sample 1 (solid line) and three di erent states of sample 2 (shown by di erent symbols). The Vg axis for di erent sample states was compensated for random shifts on the order of 50 mV. (b) Experimental I -V curves calculated from the data of (a) shown by the same symbols as the corresponding data for Rdi in (a). Model ts (see text) are shown by dashed lines

2. These data are taken simultaneously with the shot noise measurements presented below. Rdi is maximum in the linear response (I = 0) and falls down at nite current. The reduction of Rdi with current is most pronounced at low jI j and for more depleted channel. Rdi is an asymmetric function of current, which is related to the capacitive population/depopulation of the channel at negative/positive bias. The nonlinear I -V curves numerically calculated from these data are plotted in g.2b.
92 . 7 { 8 2010


Shot noise measurements in a wide-channel transistor near pinch-o The nonlinearity is somewhat similar to the behavior of the I -V curves in the insulator breakdown regime 8], although much less pronounced, possibly because of the much higher temperature. The main ingredient of the shot noise measurements is the calibration of the setup gain and bandwidth described below. A recti ed voltage VDET at the output of the detector is proportional to the power P incident on the detector. The power P is the sum of contribuT tions PT proportional to the transistor noise SI , PZ0 proportional to the input current noise of the LTAmp Z SI 0 and a constant contribution coming from the input voltage noise of the LTAmp and noises of all other ampliers. The LTAmp's input current noise is dominated by Z the JN-noise of the input resistor SI 0 4kB TLT A=Z0 . The actual noise temperature can be somewhat higher owing to extra current noise from the active parts of the LTAmp. Small noises from the two resistors at the input divider ( g.1) are neglected. Below we are interested only in a di erential part of VDET measured with a lock-in, which depends on the transistor S-D current I and/or its (linear or di erential) S-D resistance. Hence, up to an unimportant constant one gets:
T nois VDETe = D(PT + PZ0 ) = SI R2 DG
Z Z

509
(a)

30

PCAL (mW)

20

R 0.26 kW 0.8 kW 2.8 kW 13.4 kW

10

0 50

100

150 f (MHz)

200

6

(mV) VD
noise ET

4

jkT j2 df +
(1)

Z2 + SI 0 Z0 DG jkZ0 (f ; R)j2 df ;

2 (b) 0 5 R (kW) 10

where D is the detector power to voltage conversion coe cient, G { total power gain of the ampli ers divided by the input resistance Z0 . kT and kZ0 (f ; R) denote the voltage transfer functions of the corresponding noise sources to the input of the LTAmp, which both depend on f and R. The transfer functions are set by the parameters of the circuit in g.1, which we determine via the following calibration procedure. We apply an rf signal of amplitude UC AL at a frequency f to the input C AL CAL IN (see g.1) and measure the contribution VDET to the output detector voltage. The detector signal is proportional to power PC AL incident on the detector: C AL 0 2 VDET = DPC AL = DjkT j2 jkGEN j2 G UC AL; (2) where kGEN is the voltage divider coe cient at the 0 CAL IN input of the circuit ( g.1). kT is the rf-voltage transfer function, which is related to the noise trans0 fer function kT from eq. (1) as jkT j2 = jkT j2 . Factor 2 f 2 R2 CSD accounts for the suppression of the 2 = 1+4 transistor voltage noise caused by a stray S-D capacitance CSD . The power PC AL measured with a spectrum analyzer is shown in g.3a as a function of f for a set of gate voltages (symbols). The quality factor of the LC -resonator in g.1 increases with R, which results in
92 . 7 { 8 2010

Fig.3. (a) Frequency response spectra of the setup at I = 0 acquired with a spectrum analyzer for a set of linear response resistances R indicated in the legend. Experimental data and ts are shown by symbols and lines, respectively. (b) Calibration of the setup via equilibrium noise measurement (see text). Symbols and line: measured chop amplitude of the detector voltage and t, respectively

narrower peak for more depleted channel in g.3a. Solid lines represent the best ts to the data used to accurately determine the values of L 300 nH, C 1:5 pF, the drain-ground stray capacitance CD 3:9 pF and CS D 0:2 pF. We nd that CD is dominated by a stray capacitance of the hand-made inductor (L in g.1) and the value of CSD is close to an intrinsic parameter of the transistor. The quality of the ts is almost perfect, apart from small oscillations presumably caused by resonances in the rf-tract. These discrepancies are not important as they occur beyond the bandwidth used for noise measurements.


510

V. S. Khrapai, D. V. Shovkun We measure the dc contribution to the detector voltage nois caused by nite S-D current VDETe . This is achieved via a lock-in measurement of the derivative dVDET =dI and subsequent numeric integration. The integration constant is obtained via the equilibrium noise measurement ( g.3b). The (arbitrarily o set) result is shown in g.4 for one value of Vg (see caption). Here, the dashed line
VD
noise ET

Fitting the data of g.3a with eq. (2) returns the value of the product jkGEN j2 G, whereas a separate knowledge of G is required for noise measurements. This is achieved via a measurement of the equilibrium noise of the setup, which depends on R (see eq. (1)). We chop the transistor gate voltage between the two values, corresponding to nearly zero ( 10 ) and nite R and measure the rst harmonic ac component of the detecnoiT tor voltage VDEse with a lock-in. The result is plotted in g.3b as a function of R (symbols). The noise signal nois VDETe increases as the transistor is depleting, which reects the increase of the JN voltage noise of the transistor. The overall dependence is caused by the interplay of the R-dependent bandwidth of the resonant circuit and a (negative) contribution from the chopped LTAmp's input current noise. The experimental behavior is well captured by the dashed line t in g.3b. The tting parameters include G, the LTAmp's input noise temperature of TLT A 4:7 K and the sample temperature of T = 4:2 K. Consistently, an independent thermometry returned the value of 5.3 K for the temperature of the resistor Z0 ( g.1). Under assumption of TLT A = 5:3 K the best t to the data of g.3b would be obtained for a sample temperature of 4:8 K. This discrepancies represents a possible systematic error in our calibration and shot noise measurements, so that the Fano factor values given below may actually be within 10% higher. As follows from g.3, at I = 0 the rf-response of the transistor and its equilibrium noise are successfully described by a single stray capacitance parameter CSD . We nd that this is not the case under non-linear transport conditions. Presumably, the reason is the inhomogeneous electron density distribution below the gate at I 6= 0, which can change, e.g., distributed gate-drain and gate-source capacitances. Instead of introducing more tting parameters at I 6= 0, we calibrate the noise transfer function kT in-situ, with the help of the rf generator. According to eq. (2) integration of the frequency response of the setup to the external rf-signal gives:
DG
Z

10 mV

(mV) VD
ET

T VDET

VD ­4

Z0 ET

R = 3.3 kW

­3

­2

­1

0 1 I (mA)

2

3

4

Z0 noi T Fig.4. Integrated detector voltage VDEse = VDET + VDET T as a function of S-D current (dashed line). Contributions T from the transistor noise (VDET ) and input current noise Z0 ) are shown by solid and dotted lines, of the LTAmp (VDET respectively. The data are taken for the same gate voltage as the traces shown by squares in g.2 (linear-response resistance indicated in the gure)

2 jkT j2 df = ( jkGEN j2 UC AL) 1 V
R

Z

C AL DET df (3)

The quantity KC AL DG jkT j2 df obtained in this way accounts for the I -dependent gain and bandwidth of the shot noise measurement in the nonlinear regime. Using equations (1) and (3) the measurement of the T shot noise spectral density SI is straightforward. In the nonlinear regime the voltage noise of the transistor is determined by the di erential resistance Rdi which substitutes R in eq. (1) and expression for factor . The LTAmp's noise transfer function kZ0 (f ; R) is evaluated with the known I = 0 circuit parameters and R = Rdi .

nois is the experimental VDETe . The evaluated contribution Z0 of the LTAmp's input current noise VDET is shown by Z0 dots. VDET is not a constant, thanks to the Rdi dependence on I in the nonlinear regime, which slightly modi es the impedance connected to the input of the LTAmp. As seen from g.2, Rdi changes stronger for I < 0, which results in a corresponding asymmeZ0 try of VDET as a function of I in g.4. The di erence Z0 T nois VDET = VDETe VDET is the contribution thanks to transistor shot noise shown by solid line. The functional T dependence of VDET on I is related to that of the noise T T 2 spectral density as SI = VDET =(Rdi KC AL). T In gs.5a, b and 5c the noise spectral density SI is plotted as a function of I for three values of the gate voltage (symbols). At I = 0 the noise spectral density is minimum and equals the JN value. At T I 6= 0, SI increases as a function of jI j and demonstrates a nearly linear behavior at high enough currents. We nd that for all experimental traces, in the
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Shot noise measurements in a wide-channel transistor near pinch-o
0.8 R = 1.2 kW 0.6 0.4 0.2 (a) 0 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 0.6 R = 3.3 kW

511

0.4

­24

rents the di erential Fano-factor is enhanced compared to 1/3, which is most pronounced for the data in g.5b with FD 0:5 for jI j 1 A. This discrepancy is beyond the experimental uncertainty and can be explained by the e ect of thermal uctuations in the nonlinear transport regime, as we propose below. The general result for shot noise spectral density of a two-terminal conductor is usually express in terms of the energy-dependent 1D eigen-channel transparencies Tn (E ) 2]. For the case of wide channel transistor, it is convenient to express the same result in terms of the P energy-dependent conductance (E ) = e2 =h Tn(E ), where h is the Planck's constant, and the Fano-factor of the partition noise F = Tn (1 Tn )=Tn : (E )dE ffL (1 fL) + (4) +fR (1 fR ) + F (fL fR )2 g Here, fi = (1 + exp(E i )=kB T ]) 1 are the Fermi distributions of the left (i = L) and right (i = R) reservoirs, with respective electrochemical potentials of i = jejV =2. The rst two terms in the integrand of eq. (4) represent the thermal noise of the reservoirs, while the last term stands for the shot noise. Eq. (4) provides a phenomenological description of the shot noise behavior in the nonlinear regime. We nd that the experimental enhancement of the di erential Fano-factor FD > 1=3 ( g.5a and 5b) is not necessarily related to energy dependence of F . Below we assume that the partition noise Fano-factor in the last term of eq. (4) has a universal value F = 1=3 for metallic di usive conductors independent of energy. (E ) The energy dependent conductance is directR related to the transport current as ly I = jej 1 (E )(fL fR )dE and can be obtained by tting the experimental I -V curves ( g.2b). We model (E ) by a step-like functional dependence on energy (E ) = 0 (1 + tanh (E E0 jejV )= ]). Parameters 0 10 2 1 and E0 ; 1 meV de ne the shape of the conductance step, whereas the parameter 0:1 accounts for the asymmetry of the I -V curves ( g.2b) thanks to capacitive e ects of nite bias. This conductance model was chosen for its analogy to the step-like behavior of the density of states near the metal-insulator transition in two dimensions 7]. Fig.2b demonstrates that the model provides good ts (dashed lines) to the experimental nonlinear I -V curves (symbols). The so-obtained (E ) and eq. (4) predict the behavior of the noise spectral density, which is shown by solid lines in g.5. The enhancement of the di erential Fano-factor in a less depleted channel at small I (symbols in gs.5a
SI = 2
Z

SI (10

A /Hz)

2 T

0.2 (b) 0 ­4 ­3 ­2 ­1 0 0.3 0.2 0.1 (c) 0 ­3 ­2 0 ­1 I (mA) 1 2 3 1 2 3 4

R = 10.7 kW

T Fig.5. Shot noise spectral density SI as a function of S-D current for three values of the gate voltage (linear-response resistance R shown in the gures). Experimental data in (a),(b),(c) are measured simultaneously with the data of g.2 and are shown by the respective symbols. Dashed lines are ts to the standard linear response shot noise formula 4]. Solid line are ts according to eq. (4) and a model of nonlinear transport described in the text

limit of high currents, the di erential Fano-factor is close to the universal value FD 1=3. For comparison, we plot SI expected for a metallic di usive conductor in the linear regime by dashed lines in g.5. These lines are drawn according to the standard for2 mula SI = 3 R 1 4kB T + jeV j coth(jeV j=2kB T )], where R is the experimental linear response resistance and V is the associated voltage drop V = I R 4]. The symbols in gs.5a and 5b (obtained for a less depleted transistor) are systematically above the dashed lines, i.e. for the same I the noise spectral density exceeds the one obtained with the above formula. Hence, at low cur92 . 7 { 8 2010


512

V. S. Khrapai, D. V. Shovkun result (dashed line). Note, however, that at low jI j this behavior might be an artefact caused by the uncertainty in the voltage noise suppression factor , which is most crucial for measurements at high Rdi & 3k . In summary, we performed the shot noise measurements in a commercial high-frequency transistor near pinch-o . The dependence of the shot noise on current is slightly nonlinear. The di erential Fano-factor is about FD 1=3 in the limit of high currents, and somewhat enhanced above the universal metallic value at lower currents 1=3 < FD < 0:5. The model of nonlinear transport near the pinch-o is suggested, which allows to explain the results in terms of classical e ect of thermal uctuation, without assuming the enhancement of the partition noise. We acknowledge the discussions with A.A. Shashkin, V.T. Dolgopolov and V.F. Gantmakher. Financial support by RFBR, RAS, Ministry of Science and the grant # MK-3470.2009.2 is gratefully acknowledged. VSK acknowledges support from the Russian Science Support Foundation.
1. 2. 3. 4. 5. 6. 7. 8. 9. W. Schottky, Ann. Phys. (Leipzig) 57, 541 (1918). Ya .M. Blanter, M. Buttiker, Phys. Rep. 336, 1 (2000). C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). M. Henny, S. Oberholzer, C. Strunk, and C. Schonenberger, Phys. Rev. B 59, 2871 (1999). Yu. V. Nazarov, Phys. Rev. Lett. 73, 134 (1994). L. Roschier and P. Hakonen, Cryogenics 44, 783 (2004). T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). A. A. Shashkin, V. T. Dolgopolov, G. V. Kravchenko et al., Phys. Rev. Lett. 73, 3141 (1994). K. E. Nagaev, Phys. Rev. B 52, 4740 (1995).

and 5b) is qualitatively captured by the ts according to eq. (4) (solid lines). Note, that thanks to assumption of F = 1=3 the eq. (4) reduces to SI = 2=3jeI j at T = 0. In other words, within our model the enhancement of FD > F = 1=3 is a nite temperature e ect. The nite temperature determines the thermal uctuations in the electron ow incident on the conductor. In the nonlinear regime, the contribution of thermal uctuations to the current noise increases as a function of I , thanks to the energy dependence of the conductance (E ). This results in enhanced di erential Fano-factor FD > F = 1=3. Note, that this nonlinear e ect is not related to a thermalization of the non-equilibrium carriers inside the conductor 9] or in the reservoirs 4]. The importance of the thermal uctuations is best illustrated in the ultimate case of thermally activated conductance in the insulating phase, which in our model is achieved for E0 ; T . This is expected to occur in a strongly depleted transistor. Here, the eq. (4) predicts FD = 1 at low jI j. In this case, the Poissonian value of the Fano-factor is caused by a classical reason that the occupation number of the current carrying states is small fi (E E0 ) 1 (i = L; R). This is fully analogous to the case of the shot noise for thermionic emission in a vacuum tube considered by Schottky 1], and is not related to particular properties of the model we used for (E ). In the limit of high currents, the eq. (4) predicts a crossover to the partition noise in the insulator breakdown regime with FD ! F . Unfortunately, we could not observe such a behavior experimentally. Fig.5c shows the noise spectral density for the lowest gate voltage (symbols), where the non-linearities are most pronounced (same symbols in g.2). Unlike the prediction of eq. (4) (solid line), FD 1=3 and the data falls close to the standard metallic linear-response

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. 7 { 8 2010