Shot Noise in Linear Macroscopic Resistors

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.

Nyquist noise and shot noise are the two prototypes of current noise displayed by electronic devices. Nyquist noise, originally found by Johnson in resistors [1], is displayed by all electronic devices at thermal equilibrium and is associated with the equilibrium thermal fluctuations.The spectral density of the current fluctuations of Nyquist noise is white and given by [2] S N yquist where k B is the Boltzmann constant, T the temperature and R the linear resistance. Shot noise, originally found in saturated vacuum tubes, [3] is displayed under non-equilibrium conditions and is associated with the discreteness of the electric charge. Its current spectral density at low frequency is usually given by [3] S shot I = 2qI (2) where I is the average current and q the carrier charge. Shot noise is routinely found in many solid-state electronic devices like tunnel and Schottky diodes, p-n junctions [4], and more recently in mesoscopic structures. [5] The existing claiming is that shot noise can be observed in electronic devices provided there exists an internal potential energy barrier [4] or in the absence of inelastic scattering. [5][6][7] Here, we report on a direct experiment evidence of shot noise in a linear macroscopic resistor, which does not satisfy the above requeriments.
In a macroscopic linear resistor, the possibility to observe shot noise is conditioned to the fact that the electrical charge can pile up inside the device [8][9][10] or, conversely, to the fact that the instantaneous number of free carriers inside the sample can fluctuate in time.
[11] This possibility can be accomplished when the dielectric relaxation time of the material, i.e. the time required for a charge fluctuation to vanish, τ d = ǫǫ 0 ρ, becomes longer than the dynamic transit time, i.e. the time a particle lasts to cross the sample at its drift velocity, Here ǫ 0 is the vacuum permittivity, ǫ r the relative static dielectric constant of the material, ρ its resistivity, L its length, µ its mobility, and V the applied voltage. Under such a condition the long range Coulomb interaction does not induce correlations between current fluctuations, thus, charge neutrality of the device can be violated and shot noise can be observed. In order to fulfil this constraint (τ T ≪ τ d ), numerical estimates indicate that the choice of the sample is limited to highly resistive materials like semi-insulating semiconductors. In relatively good conductors, like metals or highly doped semiconductors, the above constraint is hard, if not impossible, to be achieved under realistic experimental conditions.
In the present work we have considered a resistor made of a 2 mm thick semi-insulating CdTe semiconductor embedded between two gold plates of 2 × 10 mm 2 . The choice of CdTe has been motivated by the fact that this semiconductor material allows for a high degree of compensation [12], thus presenting the desired semi-insulating property . Moreover, it displays a linear velocity-field characteristics up to several kV/cm at room temperature, [12] thus allowing to apply considerable high voltages without the presence of hot-electron effects. Furthermore, the use of metal-semiconductor contacts is motivated by the fact that metals on semi-insulating materials exhibit the required nearly perfect ohmic behaviour in a wide range of voltages, since the carrier density at the interface imposed by the contact is of the same order of magnitude as the free carrier density of the semi-insulating material. [13] This fact avoids the presence of spurious space charge effects. Taken, for example, at T =323 K, the device displays an almost symmetric linear current-voltage (I − V ) characteristics between 50 and + 50 V. The forward characteristics is shown in Fig. 1

in a log-log scale
showing the linear behaviour from the lowest to the largest voltage bias. A best fit to the experiments gives a resistance R 323K = 0.233 GΩ, which implies a resistivity ρ = 0.233 GΩ cm. From these parameters, and by using [12] ǫ r = 12, the dielectric relaxation time for the material is τ d ∼ 0.3 ms. Therefore, by extracting a mobility corresponding to holes µ = 50 cm 2 /(Vs) from the meaurement of the cut-off frequency in the noise spectra in the shot noise region [14] (the sample is known to be p-type [15]), and for an applied bias of 10 V, the transit time is τ T ∼ 0.08 ms , thus making accessible the shot noise condition (τ T ≪ τ d ) for applied bias above a few tens of Volts.
Current noise experiments have been performed on the CdTe resistor by means of the correlation technique implemented on a state of the art noise spectrum analyzer able to probe noise levels as low as 10 −30 A 2 /Hz and to reach frequencies up to 10 5 Hz at room temperature [16]. The characteristics of the measurement set up allow reaching the extremely low current noise levels present in semi-insulating materials and to cover a wide enough range of frequencies to get rid of the 1/f contribution that may hide the presence of the shot noise plateau. Figure 2 reports the spectral density of the current fluctuations measured at T = 323 K for different values of the current. At thermal equilibrium (i.e. zero current) the spectrum is white and takes a value in agreement with Nyquist noise, Eq.
At increasing currents the spectrum becomes current and frequency dependent. It displays a 1/f region at low frequencies, followed by a plateau at intermediate frequencies, and a cut-off region at the highest frequencies.
The values of the plateaux for the different current values extracted by a best fit function that includes the plateaux and the cut-off regions are reported as a function of current in where s D,ex Here, τ D = L 2 /D is the diffusion transit time, i. e. the time required for a particle to travel a distance L due to diffusion, with D being the diffusion coefficient, related to the mobility through Einstein's relation, D/µ = k B T /q. By considering the right hand side of Eq.(3), we note that the first term corresponds to the expected Nyquist noise contribution, while the second term is due to finite size effects controlled essentially by the interplay between the values of the dynamic transit time and the dielectric relaxation time [10]. The continuous line in Fig. 3 corresponds to the theoretical results obtained from the model after using the parameters of the device under test. The agreement between theory and experiments is within experimental uncertainty, what is remarkable in view of the absence of any adjustable parameters. In particular, the anomalous cross-over between Nyquist and shot noise found in the experiments is well reproduced by the theory. Therefore, we conclude that the shot noise observed in our sample is due to the inelastic drift and diffusion of the carriers under the condition that the long range Coulomb interaction is not affecting the current fluctuations, and is not related to a potential energy barrier or to the absence of inelastic scattering. According to the present view, the fact that shot noise is not observed in macroscopic resistors made of good conductors (e.g. metals) is due to the strong correlations induced by the long range Coulomb interaction in these samples that inhibits the pile up of charge carriers under realistic experimental conditions, rather than due to the presence of inelastic scattering processes, as sometimes claimed in the literature. [5][6][7] It is worth noting that the absence of long range Coulomb correlations is also at the basis of the presence of shot noise in vacuum tubes and ballistic diodes under saturation [17] and Schottky barrier diodes [18] or number fluctuations in non-degenerate Fermi gases [19].
The strong dependence on temperature of the resistivity of semi-insulating CdTe [12] allows us to perform experiments for different sample resistivities by simply varying the sample temperature. Figure