We report on a direct experimental evidence of shot noise in a linear
macroscopic resistor. The origin of the shot noise comes from the fluctuation
of the total number of charge carriers inside the resistor associated with
their diffusive motion under the condition that the dielectric relaxation time
becomes longer than the dynamic transit time. Present results show that neither
potential barriers nor the absence of inelastic scattering are necessary to
observe shot noise in electronic devices.Comment: 10 pages, 5 figure

A. van der Ziel

A. J. Strauss

C. Pennetta

G. Bertuccio

G. Ferrari

G. Gomila

K. M. van Vliet

L. Reggiani

M. Büttiker

M. Sampietro

W. Schottky

English

arXiv

Crossref

Shot Noise in Linear Macroscopic Resistors

We report on direct experimental evidence of shot noise in a linear macroscopic resistor. The origin of the shot noise comes from the fluctuation of the total number of charge carriers inside the resistor associated with their diffusive motion under the condition that the dielectric relaxation time becomes longer than the dynamic transit time. The present results show that neither potential barriers nor the absence of inelastic scattering are necessary to observe shot noise in electronic devices

G. GOMILA

M. SAMPIETRO

G. FERRARI

G. BERTUCCIO

PENNETTA, Cecilia

REGGIANI, Lino

Archivio Istituzionale della Ricerca- Università del Salento

Shot noise in linear macroscopic resistors

Gomila Lluch, Gabriel

Pennetta, C.

Reggiani, L. (Lino), 1941-

Ferrari, G.

Sampietro, M.

Bertuccio, G.

Diposit Digital de la Universitat de Barcelona

P H Y S I C A L R E V I E W L E T T E R S week ending4 JUNE 2004VOLUME 92, NUMBER 22Shot Noise in Linear Macroscopic ResistorsG. Gomila,1 C. Pennetta,2 L. Reggiani,2 M. Sampietro,3 G. Ferrari,3 and G. Bertuccio31Department d’Electronica and Research Centre for Bioelectronics and Nanobioscience, Universitat de Barcelona,Edifici Modulari, Josep Samitier 1-5, 08028 Barcelona, Spain2INFM-National Nanotechnology Laboratory and Dipartimento di Ingegneria dell’Innovazione, Universita` di Lecce¨,via Monteroni, 73100 Lecce, Italy3Dipartimento di Elettronica ed Informazione, Politecnico di Milano, Pzza Leonardo da Vinci 32, 20133 Milano, Italy(Received 15 May 2003; published 2 June 2004)226601-1We report on direct experimental evidence of shot noise in a linear macroscopic resistor. The origin ofthe shot noise comes from the fluctuation of the total number of charge carriers inside the resistorassociated with their diffusive motion under the condition that the dielectric relaxation time becomeslonger than the dynamic transit time. The present results show that neither potential barriers nor theabsence of inelastic scattering are necessary to observe shot noise in electronic devices.DOI: 10.1103/PhysRevLett.92.226601 PACS numbers: 72.70.+m, 05.40.–a, 73.50.Tdthan the dynamic transit time, i.e., the time a particlelasts to cross the sample at its drift velocity, T from the lowest to the largest voltage bias. A best fit tothe experiments gives a resistance R323K  0:233 G,Nyquist noise and shot noise are the two prototypes ofcurrent noise displayed by electronic devices. Nyquistnoise, originally found by Johnson in resistors [1], isdisplayed by all electronic devices at thermal equilibriumand is associated with the equilibrium thermal fluctua-tions. The spectral density of the current fluctuations ofNyquist noise is white and given by [2]SNyquistI 4kBTR; (1)where kB is the Boltzmann constant, T is the temperature,and R is the linear resistance. Shot noise, originally foundin saturated vacuum tubes [3], is displayed under non-equilibrium conditions and is associated with the dis-creteness of the electric charge. Its current spectraldensity at low frequency is usually given by [3]SshotI  2qI; (2)where I is the average current and q is the carrier charge.Shot noise is routinely found in many solid-state elec-tronic devices such as tunnel and Schottky diodes, p-njunctions [4], and more recently in mesoscopic structures[5]. The existing claim is that shot noise can be observedin electronic devices provided there exists an internalpotential energy barrier [4] or in the absence of inelasticscattering [5–7]. Here, we report on direct experimentevidence of shot noise in a linear macroscopic resistor,which does not satisfy the above requirements.In a macroscopic linear resistor, the possibility to ob-serve shot noise is conditioned to the fact that the elec-trical charge can pile up inside the device [8–10] or,conversely, to the fact that the instantaneous number offree carriers inside the sample can fluctuate in time [11].This possibility can be accomplished when the dielectricrelaxation time of the material, i.e., the time required fora charge fluctuation to vanish, d  0, becomes longer0031-9007=04=92(22)=226601(4)$22.50 L2=V. Here 0 is the vacuum permittivity, r is therelative static dielectric constant of the material,  itsresistivity, L its length,  its mobility, and V the appliedvoltage. Under such a condition, the long range Coulombinteraction does not induce correlations between currentfluctuations; thus, charge neutrality of the device can beviolated and shot noise can be observed. In order to fulfillthis constraint (T  d), numerical estimates indicatethat the choice of the sample is limited to highly resistivematerials such as semi-insulating semiconductors. Inrelatively good conductors, such as metals or highlydoped semiconductors, the above constraint is hard, ifnot impossible, to be achieved under realistic experimen-tal conditions.In the present work, we have considered a resistor madeof a 2 mm thick semi-insulating CdTe semiconductorembedded between two gold plates of 2 10 mm2.The choice of CdTe has been motivated by the fact thatthis semiconductor material allows for a high degree ofcompensation [12], thus presenting the desired semi-insulating property. Moreover, it displays linear velocity-field characteristics up to several kV=cm at roomtemperature [12], thus allowing one to apply considerablehigh voltages without the presence of hot-electron effects.Furthermore, the use of metal-semiconductor contacts ismotivated by the fact that metals on semi-insulatingmaterials exhibit the required nearly perfect Ohmic be-havior in a wide range of voltages, since the carrierdensity at the interface imposed by the contact is of thesame order of magnitude as the free carrier density of thesemi-insulating material [13]. This fact avoids the pres-ence of spurious space charge effects. Taken, for example,at T  323 K, the device displays almost symmetriclinear current-voltage (I-V) characteristics between 50and 50 V. The forward characteristics is shown inFig. 1 in a log-log scale showing the linear behavior2004 The American Physical Society 226601-1T=323 Kf (Hz)101 102 103 104S I(f)[A2 /Hz]10-2910-2810-2710-2610-2510-240 pA, 12.6 pA1.56 nA7.9 nA23.7 nA67.0 nA194 nAI = 235 nAFIG. 2. Spectral density of the current fluctuations as a func-tion of frequency for different values of the electric current atT  323 K.I [A]10-12 10-11 10-10 10-9 10-8 10-7S Iplateau[A2 /Hz]10-2910-2810-2710-2610-25T=323KI=0 A2qI4kBT/R323K2qI coth[qV/(2kBT)]FIG. 3. Experimental values of the current spectral density atthe white plateaux as a function of the current at T  323 K(filled circles). The continuous line corresponds to the theo-retical prediction of the unipolar drift-diffusion model. Thedotted line corresponds to the standard cothlike behavior.V [V]0.01 0.1 1 10 100I[A]10-1110-1010-910-810-710-6T=323 KR323K = 0.233 GΩFIG. 1. Current-voltage characteristics of the CdTe resistor atT  323 K (filled circles). A best linear fit to the experiments(dashed line) gives a resistance R323K  0:233 G.P H Y S I C A L R E V I E W L E T T E R S week ending4 JUNE 2004VOLUME 92, NUMBER 22which implies a resistivity   0:233 Gcm. From theseparameters, and by using [12] r  12, the dielectricrelaxation time for the material is d  0:3 ms. There-fore, by extracting a mobility corresponding to holes  50 cm2=V s from the measurement of the cutoff fre-quency in the noise spectra in the shot noise region [14](the sample is known to be p type [15]), and for anapplied bias of 10 V, the transit time is T  0:08 ms,thus making accessible the shot noise condition (T d) for applied bias above a few tens of volts.Current noise experiments have been performed on theCdTe resistor by means of the correlation technique im-plemented on a state of the art noise spectrum analyzerable to probe noise levels as low as 1030 A2=Hz and toreach frequencies up to 105 Hz at room temperature [16].The characteristics of the measurement setup allowreaching the extremely low current noise levels presentin semi-insulating materials and to cover a wide enoughrange of frequencies to get rid of the 1=f contribution thatmay hide the presence of the shot noise plateau. Figure 2reports the spectral density of the current fluctuationsmeasured at T  323 K for different values of the current.At thermal equilibrium (i.e., zero current), the spectrumis white and takes a value in agreement with Nyquistnoise, Eq. (1), as it should be. At increasing currents,the spectrum becomes current and frequency dependent.It displays a 1=f region at low frequencies, followed by aplateau at intermediate frequencies, and a cutoff region atthe highest frequencies.The values of the plateaux for the different currentvalues extracted by a best fit function that includes theplateaux and the cutoff regions are reported as a functionof current in Fig. 3. The dependence of the current spec-tral density at the plateaux on current is found to exhibitthree distinct behaviors. At the lowest currents, the spec-tral density keeps a constant value corresponding toNyquist noise. At intermediate currents, the spectral den-226601-2sity exhibits a sharp increase (crossover region) untilapproaching an asymptotic linear increase with currentthat merges with the full shot noise behavior. We note thatthe onset of shot noise occurs at a current around 100 nA,which corresponds to a voltage 20 V, in agreement withthe voltage estimated above. These experiments prove thata linear macroscopic resistor can display shot noise underthe condition that the dynamic transit time is shorter thanthe dielectric relaxation time of the sample. It is alsoworth remarking that the crossover between Nyquistand shot noise is found to depart from the standard coth-like behavior observed in many solid-state devices (dot-ted line in Fig. 3).In order to show that the shot noise displayed by theCdTe resistor is not related to the presence of some sort ofpotential energy barrier or to the absence of inelastic226601-2V [V]10-2 10-1 100 101 102I[A]10-1010-910-810-7 T=323KR323K = 0.23 GΩT=300KR300K = 1.6 GΩT=280KR280K = 10 GΩFIG. 4. Current-voltage characteristics for the CdTe resistorat three different temperatures: T  280 K (filled triangles),T  300 K (filled circles), and T  323 K (filled squares). Thelines correspond to a best linear fit to the experiments: con-tinuous line T  323 K, dashed line T  300, and dotted lineT  280.P H Y S I C A L R E V I E W L E T T E R S week ending4 JUNE 2004VOLUME 92, NUMBER 22scattering, we provide a quantitative theoretical interpre-tation of the experiments by means of a unipolar drift-diffusion noise model recently developed by two of theauthors [10]. The model assumes that charge transportand current fluctuations are due to the drift and diffusionof free carriers in a spatially homogeneous (in average)electric field. The fact that the resistor is macroscopic isimplicitly accounted for by assuming that the free car-riers are in local thermal equilibrium with the lattice atthe bath temperature due to the energy exchange with thelattice phonons. In the model, the only source of noise isthat related to the diffusive motion of the carriers Theanalytical solution of the model can be obtained, givingfor the low frequency current spectral density the follow-ing relation [10]:SI  4kBTR 1 sexI ; (3)wheresD;exI 22  21e1  1e2  122122e1  e22 	2e2  1e1  1  1e1  1e2  1;(4)with1;2   12DT11 4 2TDds : (5)Here, D  L2=D is the diffusion transit time, i.e., thetime required for a particle to travel a distance L due todiffusion, with D being the diffusion coefficient, relatedto the mobility through Einstein’s relation, D= kBT=q. By considering the right-hand side of Eq. (3),we note that the first term corresponds to the expectedNyquist noise contribution, while the second term is dueto finite size effects controlled essentially by the interplaybetween the values of the dynamic transit time and thedielectric relaxation time [10]. The continuous line inFig. 3 corresponds to the theoretical results obtainedfrom the model after using the parameters of the deviceunder test. The agreement between theory and experi-ments is within experimental uncertainty, which is re-markable in view of the absence of any adjustableparameters. In particular, the anomalous crossover be-tween Nyquist and shot noise found in the experiments iswell reproduced by the theory. Therefore, we concludethat the shot noise observed in our sample is due to theinelastic drift and diffusion of the carriers under thecondition that the long range Coulomb interaction is notaffecting the current fluctuations, and is not related to apotential energy barrier or to the absence of inelasticscattering. According to the present view, the fact thatshot noise is not observed in macroscopic resistors madeof good conductors (e.g., metals) is due to the strong226601-3correlations induced by the long range Coulomb interac-tion in these samples that inhibits the pileup of chargecarriers under realistic experimental conditions, ratherthan due to the presence of inelastic scattering processes,as sometimes claimed in the literature [5–7]. It is worthnoting that the absence of long range Coulomb correla-tions is also at the basis of the presence of shot noise invacuum tubes and ballistic diodes under saturation [17]and Schottky barrier diodes [18] or number fluctuations innondegenerate Fermi gases [19].The strong dependence on temperature of the resistivityof semi-insulating CdTe [12] allows us to perform experi-ments for different sample resistivities by simply varyingthe sample temperature. Figure 4 reports the forwardmeasured I-V characteristics for the device under studyat T  280 and T  300 K (for an easier comparison theresults at T  323 K are also displayed). At the differenttemperatures, the device still exhibits a linear behavior ina broad range of applied bias (only at T  280 K minordeviations from linearity are observed at the highestvoltages probably due to some injection effects from thecontacts). The lines in Fig. 4 represent a best linear fit tothe experimental results, giving for the resistancesR300K  1:6 and R280K  10 G. The measurements ofthe current spectral density as a function of frequency atT  280 and T  300 K are found to exhibit the generalfeatures of the spectra reported in Fig. 2. The values ofthe spectra at the intermediate white plateaux are plottedin Fig. 5 as a function of the current (the results obtainedat T  323 K are also displayed for an easier compari-son). As seen in the figure, the results for the differentresistivities follow the general trends already describedfor T  323 K (see Fig. 3), and, in particular, they226601-3I [A]10-12 10-11 10-10 10-9 10-8 10-7S Iplateaux[A2 /Hz]10-3010-2910-2810-2710-2610-25 2qI4kBT/RI = 0AI = 0AFIG. 5. Experimental values of the current spectral density atthe white plateaux as a function of current at three differenttemperatures: T  280 K (filled triangles), T  300 K (filledcircles), and T  323 K (filled squares). The lines correspondto the theoretical predictions of the unipolar drift diffusionmodel: continuous line T  323 K, dashed line T  300, anddotted line T  280.P H Y S I C A L R E V I E W L E T T E R S week ending4 JUNE 2004VOLUME 92, NUMBER 22confirm the presence of shot noise in macroscopic resis-tors. The simple unipolar drift-diffusion model (lines inFig. 5) reproduces the main trends of the experimentalresults for the different temperatures in spite of theabsence of any adjustable parameter and of its extremesimplicity. The disagreements observed in the crossoverregion between Nyquist and shot noise are probably due togeneration-recombination noise neglected here [20].In summary, we have presented experimental evidenceof the existence of shot noise in linear macroscopic re-sistors, and proved that shot noise can be observed inelectronic devices even in the absence of an internalpotential energy barrier or in the presence of inelasticscattering processes. Present findings provide support forthe existence of an additional physical situation in whichshot noise can be present in electronic devices.Professor A. Cavallini of Bologna University isthanked for valuable information provided on the elec-trical properties of semi-insulating CdTe. Partial supportfrom the Spanish MCyT through the Ramo´n y Cajal226601-4Program, the European Comission through ProjectNo. IST2001-38899, and the Italy-Spain joint actionNo. HI02-40 is gratefully acknowledged.[1] J. B. Johnson, Phys. Rev. 32, 97 (1928).[2] H. Nyquist, Phys. Rev. 32, 110 (1928).[3] W. Schottky, Ann. Phys. (Leipzig) 57, 541 (1918).[4] A. van der Ziel, Noise in Solid-State Devices and Circuits(Wiley, New York, 1986).[5] Ya. M. Blanter and M. Bu¨ttiker, Phys. Rep. 336, 1 (2000).[6] A. Shimizu and M. Ueda, Phys. Rev. Lett. 69, 1403(1992).[7] M. J. M. de Jong and C.W. J. Beenakker, Phys. Rev. B 51,16 867 (1995).[8] K. M. van Vliet and J. Blok, Physica (Utrecht) 22, 231(1956).[9] M. Bu¨ttiker, in Noise in Physical Systems and 1/fFluctuations, edited by V. Bareikis and R. Katilius(World Scientific, Singapore, 1995), p. 35.[10] G. Gomila and L. Reggiani, Phys. Rev. B 62, 8068(2000).[11] R. Landauer, Phys. Rev. B 47, 16 427 (1993).[12] A. J. Strauss, Rev. Phys. Appl. 12, 167 (1977).[13] G. Gomila, I. R. Cantalapiedra, and L. Reggiani, J. Appl.Phys. 93, 375 (2003).[14] M. Sampietro, G. Ferrari, and M. Bertuccio, J. Appl.Phys. 87, 7583 (2000). We note that the value reported inthis reference for the mobility [25 cm2=Vs] was calcu-lated assuming that the cutoff frequency is equal to theinverse of the transit time. According to the calculationsin Ref. [10], the cutoff frequency is half the inverse of thetransit time, where the value of 50 cm2=Vs was usedfor the mobility in the present Letter.[15] A. Cavallini (private communication).[16] M. Sampietro, L. Fasoli, and G. Ferrari, Rev. Sci.Instrum. 70, 2520 (1999).[17] G. Gomila, I. R. Cantalapiedra, T. Gonza´lez, andL. Reggiani, Phys. Rev. B 66, 075302 (2002).[18] G. Gomila, L. Reggiani, and J. M. Rubı´, J. Appl. Phys. 88,3079 (2000).[19] C. M. Wilson, L. Frunzio, and D. E. Prober, Phys. Rev.Lett. 87, 067004 (2001).[20] G. Gomila and L. Reggiani, Appl. Phys. Lett. 81, 4380(2002).226601-4

Shot Noise in Linear Macroscopic Resistors

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