Impedance field and transition from thermal to shot noise in Cd 1 − x Zn x Te semi-insulating Ohmic detectors

Within a drift-diffusion model we investigate the role of the self-consistent electric field in determining the impedance field of a macroscopic Ohmic linear resistor made by a compensated semi-insulating semiconductor at arbitrary values of the applied voltage. The presence of long-range Coulomb correlations is found to be responsible for a reshaping of the spatial profile of the impedance field. This reshaping gives a null contribution to the macroscopic impedance but modifies essentially the transition from thermal to shot noise of a macroscopic linear resistor. Theoretical calculations explain a set of noise experiments carried out in semiinsulating CdZnTe detectors.


I. INTRODUCTION
Shot noise is due to the discreteness of the electrical charge responsible for current.It is evidenced by a frequency independent current spectral density obeying the relation S I = ⌫2qI, where q is the absolute value of the charge unit determining the steady current I and ⌫ is the Fano factor.In the absence of correlation between carriers, ⌫ = 1.The presence of attractive or repulsive correlations between carriers leads to the enhancement ͑⌫Ͼ1͒ or suppression ͑⌫Ͻ1͒ from the full value 2qI of the shot noise.3][4] In other words, it is generally speculated that, in a macroscopic conductor, shot noise is washed out by inelastic interactions, which are unavoidable because of the macroscopic length of the active region. 5In a previous work, 6 it was noticed that the possibility to observe shot noise in a macroscopic conductor is related to the fact that the instantaneous number of free carriers inside the sample can fluctuate in time. 7Accordingly, this possibility can be accomplished when the dynamic transit time, T = L 2 / ͑V͒, becomes shorter than the dielectric relaxation time of the material, d = ⑀ 0 ⑀ r ͑here ⑀ 0 is the vacuum permittivity, ⑀ r is the relative static dielectric constant of the material, is its resistivity, L is the length of the device, is the carrier mobility, and V is the applied voltage͒.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.Numerical estimates indicate that this constraint is fulfilled when the choice of the sample is limited to highly resistive materials, such as semi-insulating semiconductors.Theoretical predictions, carried out for the case of low-frequency diffusion noise in the presence of Coulomb correlations, 8 were experimentally confirmed in semiinsulating CdTe, 6,9 thus proving that shot noise can be ob-served also in macroscopic conductors.In particular, the crossover between thermal and shot noise was found to exhibit an anomalous behavior because of the presence of longrange Coulomb interaction and/or of generationrecombination ͑GR͒ processes. 8The case of GR noise alone was investigated in Ref. 10 for the case of low frequency and in Ref. 11 in the full frequency region.However, the joint presence of diffusion and GR noises within a self-consistent approach which includes the role played by Coulomb interaction was not carried out.
The aim of this work is to fill this lack of knowledge by developing a theoretical approach which combines diffusion and GR noise and by investigating how the GR processes can have an indirect effect on the diffusion noise even when the GR noise is not directly observed.Accordingly, we will be in the position to study the transition from thermal to shot noise in semi-insulating linear resistors, usually adopted for room temperature radiation sensor application. 12For this purpose we make use of a standard impedance field method generalized to include the presence of the diffusion term in the stochastic equation. 13The impedance field method 14 is a powerful technique to calculate voltage and/or current fluctuations in two terminal devices.Within this scheme, the voltage ͑current͒ spectral density is obtained by convolving in a spatial integral the two basic ingredients, namely, the local impedance field and the local noise source.The impedance field is a physical valuable quantity in itself, and a rigorous calculation of this quantity involves the solution of a second-order stochastic differential equation. 13Here we detail the derivation of the impedance field for the case of a homogeneous resistor within the drift-diffusion model.In particular, we discuss the effect of the presence or the absence of long-range Coulomb interactions and their implications in the calculation of the macroscopic impedance and noise.Theoretical results will be validated with experiments carried out in semi-insulating CdZnTe linear detectors. 15he paper is organized as follows.Section II reports the theoretical approach developed for a two terminal linear re-sistor, where charge transport is determined by a driftdiffusion model under conditions of compensation of shallow impurities and in the presence of GR processes controlled by a deep trapping center for majority carriers.Section III provides an explicit formulation of the impedance field in the presence of long-range Coulomb interaction.Numerical results specialized for the case of semi-insulating CdZnTe are presented and discussed in Sec.IV.Major conclusions are drawn in Sec.V.The Appendix details analytical calculations of number-density fluctuations.

A. Physical system
We consider the case of a semi-insulating homogeneous semiconductor sample, which is of relevant interest to manufacturing radiation detectors.Accordingly, the system under study is a unipolar semiconductor containing shallow donors and acceptors with densities N D and N A , respectively, under compensating conditions and in the presence of traps ͑for electrons in the present case͒ with a single energy level E t and density N t .The trap level E t is assumed to be deep in the semiconductor band gap, so that traps are partially ionized and hence contribute ͑i͒ to the full ionization of shallow donors and acceptors, ͑ii͒ to charge transport ͑nominally of n type in the present case͒, and ͑iii͒ to the determination of noise properties.For a given temperature, the density of ionized traps is N t + .The shallow donors and acceptors contribute to the determination of charge neutrality condition, but they do not participate in the noise processes since they are assumed to be fully ionized.Effects of minority carriers ͑holes in the present case͒ are neglected.The sample, of crosssectional area A, is assumed to be thick enough to justify a one-dimensional electrostatic treatment.For the sake of simplicity we assume that the electron mobility is independent of the electric field and nondegenerate conditions apply, so that Einstein relation for mobility and diffusion holds.The contacts consist of metal plates, assumed to form Ohmic contacts, so that the voltage drop inside them is negligibly small.The semiconductor resistivity is assumed to be high enough so that the metal-semiconductor contacts can be treated within the diffusion approximation.

B. Physical model
The equations describing transport and noise properties of the system under study are ͑i͒ the continuity equations for the free charge density and for the fixed charge density, ͑ii͒ the current equation within the drift-diffusion approach, and ͑iii͒ the Poisson equation.These equations, together with the Langevin sources associated with velocity and GR fluctuations, are written below within a one-dimensional geometry: 16 where r n and g n are the recombination and generation rates of the traps, D is the electron diffusion coefficient, n is the local free-carrier concentration, E is the local electric field, ␥ is the Langevin source of GR noise, and J D is the Langevin source of velocity fluctuation ͑diffusion͒ noise.The total current density J͑t͒ is which, from the model equations, is independent of the spatial coordinate to satisfy instantaneous charge conservation.
The Langevin sources associated with diffusion and GR noises have zero mean and low-frequency spectral density of the standard form 1

͑7͒
where are the strengths of the diffusion and GR noise sources, respectively, k B is the Boltzmann constant, T is the lattice temperature, and the bar denotes time average.The boundary conditions for the model are given by

͑11͒
where is the carrier density at the contacts independent of time and bias, N C is the effective density of states in the conduction band, and bn is the height of the contact barrier.

C. Steady-state solution
We assume that by an appropriate choice of the contact parameters a homogeneous solution exists.This solution will read where the free-carrier and trap ionized densities satisfy The I-V characteristics corresponding to this homogeneous state are taken to be linear, with the Ohmic resistance R given by R = L Aqn ¯. ͑20͒

D. Low-frequency noise
In the present work we are interested in the low-frequency noise properties ͑the case of a frequency dependent GR noise was considered in Ref. 11͒.Accordingly, we neglect all time derivatives present in the model, which simplifies the calculations considerably.Therefore, the total current fluctuation can be calculated as By linearizing the current equation we obtain ␦J n ͑x,t͒ = q␦n͑x,t͒E ¯+ qn ¯␦E͑x,t͒ + qD ‫␦ץ‬n͑x,t͒ ‫ץ‬x thus giving for the fluctuation of the total current where we have introduced the fluctuations of the total number of free carriers and of the applied bias, and where we have used the boundary conditions ␦n 0 ͑t͒ = ␦n L ͑t͒ = 0.Under fixed bias conditions ͓␦V͑t͒ =0͔ the current fluctuation simply reads where we have introduced the average value of the total number of free carriers N ¯= n ¯AL.To evaluate the current fluctuations we then need to compute the fluctuations of the total number of free carriers in the system.This calculation is performed in the Appendix.The final result reads where the contributions related to GR and diffusion processes are given respectively by where and the carrier lifetime is defined from

͑35͒
Here n Ј is the lifetime of the free carriers, and t Ј is the lifetime of the ionized centers.Note that the Debye screening length in the presence of traps, ˜D, appears renormalized with respect to the Debye screening length without traps, L D .Thus, we can introduce a dimensionless quantity ␣ ജ 1 which describes the effect of screening on the excess noise due to the presence of ionized traps,

͑36͒
From the expression of number-density fluctuations we arrive at the final expression for current fluctuation,

͑37͒
The above expression decomposes the current fluctuation into three terms, the first related directly to diffusion noise and the other two related to number fluctuations ͑the former associated with GR processes and the latter with diffusion processes͒.It can be shown that the three terms are uncorrelated ͑see the Appendix͒.By computing the correlation functions of current fluctuations we arrive at the following expression for the low-frequency current spectral density: where arises from the first and third terms in the right-hand side of Eq. ͑37͒ and arises from the second term in the right-hand side of Eq. ͑37͒ ͑in spite of the minus sign in the expression for S I GR ͑0͒, this magnitude is always positive͒.Here we have defined the equilibrium thermal noise ͑Nyquist͒ as and the dimensionless parameter ␤, which describes the strength of the GR contribution, as where we have introduced the variance of the total number of carriers, and a renormalized Debye diffusion time with D the Debye diffusion time.Moreover, Note that s I D ͑ 1 , 2 ͒ and s I GR ͑ 1 , 2 ͒ are dimensionless and positive definite.
Both the diffusion and GR current spectral densities in Eqs.͑39͒ and ͑40͒ display two different contributions.The first contribution corresponds to what one would have obtained in a very long ͑infinite͒ sample.This can be shown explicitly since for very long samples it is 1 L → + ϱ, 2 L → −ϱ, from where s I GR ͑ 1 , 2 ͒, s I D ͑ 1 , 2 ͒ → 0. In this limit the current spectral density would simply read The second contribution, being determined by the terms proportional to s I D ͑ 1 , 2 ͒ and s I GR ͑ 1 , 2 ͒, is related to the finite size of the system.The most surprising effect concerning these terms is that while the contribution related to diffusion noise source increases the overall noise, that related to GR source decreases the overall noise. 10In other words, the minimum noise associated with diffusion is the Nyquist noise, while the maximum noise associated with GR is the standard excess noise 4⌬N 2 ͑I ¯/ N ¯͒2 .As a conclusive remark on this limit, it is confirmed that shot noise does not appear in infinite systems. 17e now consider the case in which the finite size of the sample becomes of relevance.This happens when the length of the sample L does not differ significantly from the effective Debye screening length L ˜D.To illustrate this case, Figs.1͑a͒ and 1͑b͒ report the current spectral density normalized to the Nyquist noise as a function of the applied voltage normalized to the thermal one for two values of the ratio L / L ˜D.In this universal form, the diffusion contribution depends only on L / L ˜D and the GR contribution on ␤.From Figs. 1͑a͒ and 1͑b͒ we note the following important results.͑i͒ The diffusion noise ͑gray lines͒ can give rise to an excess noise which at the highest voltages becomes shot noise.The transition from Nyquist to shot noise can be anomalous with respect to the standard behavior S I =2qI coth͑qV /2k B T͒. 1 Indeed, for L / L ˜D ӷ 1 and ␤ Ͻ 0.1 the crossover region exhibits a cubic dependence on current ͑or voltage͒ as found for the case of diffusion noise in the presence of space charge. 8͑ii͒ The GR noise for low and intermediate applied voltages is in cases proportional to the square voltage.However, at the highest voltages it saturates to a constant value.Accordingly, GR noise does not give rise to shot noise in homogeneous semiconductors, contrary to what is sometimes stated in the literature. 18͑iii͒ The competitive relevance of diffusion noise against GR noise is controlled by the parameter ␤.Depending on the value of ␤, one can find several behaviors of the overall current noise spectral density at increasing voltages ͓continuous lines in Figs.1͑a͒ and 1͑b͔͒.In particular, even for negligibly small values of ␤ ͑i.e., ␤ ഛ 0.1͒, when the direct contribution of GR noise ͓see Eq. ͑40͔͒ is negligible, the effect of GR processes can still be detected through the renormalization of the effective Debye screening length controlled by the parameter ␣.At the highest bias, shot noise is achieved from values below or above the full value S I =2qI.

III. IMPEDANCE FIELD IN THE PRESENCE OF SPACE CHARGE
The impedance field is a relevant physical quantity through which impedance and noise can be easily expressed ͑for a recent review see Ref. 14͒.In this section, following the procedure of Ref. 13 the impedance field is explicitly obtained for the sample under investigation.From the stan-dard theory, 14 the impedance field ٌZ͑x͒ can be extracted from the integral For the explicit calculation of ␦V͑t͒ we consider the same set of Eqs.͑1͒-͑4͒ without the Langevin terms, and in Eq. ͑3͒ we add a current perturbation at slice x, ␦I x ͑t͒.This set of equations is the same as that for the fluctuations by just taking ␥͑x , t͒ = 0 and ␦I x ͑t͒ = A␦J D ͑x , t͒.Accordingly, ␦V͑t͒ is obtained from Eq. ͑23͒ by using the following procedure.We impose a constant current condition, i.e., ␦I͑t͒ = 0, and substitute for ␦N͑t͒ the result reported in the Appendix as Eq.
͑A25͒, thus obtaining where Z is the macroscopic impedance.By comparing Eq. ͑48͒ with Eq. ͑49͒ we find By using Eq.͑A23͒ of the Appendix, the final result for the impedance field in the presence of space charge, ٌZ on , is In any case, both the above expressions for the impedance field satisfy the sum rule In other words, while the impedance field is sensitive to the presence or absence of long-range Coulomb interaction, the value of the impedance remains the same in both cases, as it should be.

IV. RESULTS AND DISCUSSION
We have applied the above theory to interpret experimental results performed on a CdZnTe homogeneous resistor with 3% of Zn, nominally n type, and with semi-insulating properties.In particular, current voltage ͑I-V͒ and current noise spectra have been measured on devices grown by high-pressure Bridgman modified ͑HPBM͒ technique, named HPBM1 and HPBM2, with a length of 6 mm and a cross section of 6 ϫ 20 mm 2 . 15Both devices exhibit linear I-V characteristics up to voltages of about 200 V ͑HPBM1͒ and 1000 V ͑HPBM2͒, respectively.In the linear voltage regions, the current noise spectrum at frequency around 10 kHz exhibits a plateau at the value of shot noise softly suppressed at low voltages.From the linear I-V characteristics we obtain a resistance of 1.1ϫ 10 11 ⍀, which implies a thermal noise level S I th =4K B T / R = 1.5ϫ 10 −31 A 2 / s and, by assuming an electron mobility = 1.0ϫ 10 3 cm 2 / ͑V s͒, a resistivity of the material = 2.0ϫ 10 11 ⍀ / cm.
Figure 2 reports the noise-current characteristics of both samples and the theoretical results obtained using ␤ Ͻ 0.1 ͑i.e., a negligible value of the strength of the GR contribution, which is obtained for a Debye diffusion time D = L D 2 / D much longer than the lifetime of ionized traps t Ј͒ and ␣ as fitting parameters.We notice that for the samples considered here D = 0.18 s.In particular, we see that when ␣ = 1 the usual hyperbolic cotangent behavior is obtained for the transition between thermal and shot noise.The best fit, for both sets of experimental data, is obtained with ␣ = 60 000, which implies a lifetime of free carriers n Ј ͑esti- mated of 0.1 s͒ much shorter than that of ionized traps t Ј.
This means that, in the samples, Coulomb interaction is highly screened by ionized traps, giving rise to a strong anomaly in the transition between thermal and shot noise.A similar anomalous crossover between thermal and shot noise was already predicted and experimentally observed for the case of semi-insulating CdTe. 6,9 deeper understanding of the noise characteristic can be achieved by the analysis of the impedance field profiles, calculated as described before.In the plots of Figs.3-5 we report the normalized impedance field in the presence and in the absence of Coulomb interaction y on and y of f as defined in Eqs.͑51͒ and ͑52͒, as a function of the normalized position x / L. The dimensionless parameters used are defined as y 0 = L / L E ͑which controls the intensity of the applied electric field͒ and y 1 = L / L ˜D ͑which controls the importance of Coulomb correlations͒.
Figure 3 reports the profile of the impedance field for increasing values of the normalized electric field y 0 in the absence of Coulomb correlations ͑i.e., y 1 =0͒.In the presence of an electric field, the impedance field exhibits a monotonic increase which is maximum at the right contact and tends to vanish at the left contact.This asymmetric shape of the impedance field is associated with the presence of the diffusion current, which is responsible for damping the fluctuations in the number of carriers inside the sample to keep the homogeneous steady-state solution of the system.
Figures 4 and 5, report the profiles of the impedance field at low and high electric fields, respectively, in the presence of increasing intensities of the long-range Coulomb interaction, parametrized by increasing values of y 1 .The results show that, at a fixed electric field, the presence of Coulomb interaction tends to flatten the profile of the impedance field.Indeed, the Coulomb interaction is able to move the noise spectral density toward thermal equilibrium conditions, with an effect contrasting that of the applied voltage.This is the why, at the increasing Coulomb interaction, the crossover between thermal and shot noise becomes anomalous and occurs at higher voltages with respect to the standard condition V Ӎ 3k B T / q.In other words, the fluctuations in charge density are now damped by space charge more effectively than by the diffusion mechanism alone on a time scale which is that of dielectric relaxation.At high electric fields, the dynamic transit time becomes comparable to or shorter than the dielectric relaxation time, the Coulomb interaction loses importance, and the profile of the impedance field becomes again steeper, as shown in Fig. 5.

V. CONCLUSIONS
We have carried out a theoretical investigation of the impedance field and current noise in a macroscopic Ohmic resistor pertaining to semi-insulating semiconductor materials.We have shown that this system can display shot noise at the highest voltages, but that the shot noise is due entirely to the diffusion noise source.The GR contribution can be quadratic on current ͑voltage͒ or constant, but in no case proportional to the current.In other words, GR noise cannot be a source of shot noise.Independent results in the same direction obtained by means of numerical resolution of the noise equations have been reported for bipolar systems, 19 thus providing an indirect confirmation of our predictions.We have found that these systems can exhibit an anomalous transition from thermal to shot noise, which is confirmed by the experimental data.Moreover, this transition is physically explained by analyzing the impedance field profiles.We have found that from one side the increase of the applied voltage favors the transition from thermal to shot noise, while from another side the Coulomb screening due to space charge tends to restore the thermal equilibrium ͑Nyquist͒ value of the noise.Theory is applied to available experiments carried out in semi-insulating CdZnTe samples nominally of n type at room temperature.The agreement between theory and experiments is found to be quite satisfactory.On this ground, the shot noise measured at increasing applied voltages when the sample exhibits a linear I-V characteristic is attributed to the diffusion noise source.In this region of applied voltages we attest to the remarkable and nonintuitive fact that GR noise is of negligible importance apart from the role played by ionized traps in renormalizing the Debye screening length.This result is associated with the wide separation of the characteristic times that describe the relaxation of free carriers to traps, of ionized traps to their neutral state, and of Debye diffusion according to the trend n ЈӶ t Ӷ D .Present estimates are n Ј=10 −7 s, t =6ϫ 10 −3 s, and D = 0.2 s.The renormalization of the Debye screening length implies an anomalous crossover from thermal ͑Nyquist͒ to shot noise.Similar features were observed in semi-insulating CdTe ͑Refs.6 and 9͒ and offered evidence of shot noise in a macroscopic resistor.This evidence is here confirmed on a more general ground for the case of semi-insulating CdZnTe.

͑A7͒
By substituting the linearized Poisson equation into the linearized current equation and by making use of the Einstein relation D = k B T / q, we finally arrive at the following equation for the carrier density fluctuations: ‫ץ‬ 2 ␦n͑x,t͒ where the Langevin force ͑x , t͒ is with the boundary conditions ␦n͑0,t͒ = ␦n͑L,t͒ = 0.

͑A10͒
The equation for the density fluctuations is a second-order linear differential stochastic equation with constant coefficients which can be solved in a closed analytical form. 13The solution reads ␦n͑x,t͒ = A͑x,t͒e Here, A L ͑t͒ and B L ͑t͒ are integration constants determined by the boundary conditions For the sake of completeness we write also the solution found in terms of the low-frequency Green's function of the problem G͑x , xЈ ;0͒.In this case it is

͑A28͒
The three contributions to the current fluctuation are uncorrelated.Indeed, since ␥͑x , t͒ is uncorrelated with J D ͑x , t͒, the second contribution is not correlated with the others.Concerning the first and the third contributions, although their origin is in both cases the diffusion noise, their correlation is proportional to

͑A29͒
where in the last equality we have used the homogeneous property and the fact that G ˜͑x͒ vanishes at the boundaries.Hence, these contributions are also not correlated.

e 2 L
− e 1 L e − 1 x − 2 e 2 L ͑e 1 L − 1͒ e 2 L − e 1 L e − 2 x ͪͬ.͑51͒ If, by imposing L / L D → 0, we switch off the Coulomb interaction, then the impedance field in the absence of space charge, ٌZ of f , becomes

FIG. 1 .
FIG. 1. ͑Color online͒ Low-frequency current spectral density normalized to the thermal Nyquist value as a function of the applied bias normalized to the thermal value for several values of the parameter ␤ for the cases ͑A͒ L / L ˜D = 50 and ͑B͒ L / L ˜D = 0.1.

FIG. 2 .
FIG.2.͑Color online͒ Current spectral density as a function of the current.Experimental data refer to CdZnTe at room temperature and theoretical curves to the fit carried out with the analytic formula for the reported parameters with ␤ Ͻ 0.1.͑The experiments at the lowest current are obtained from the Nyquist formula at the measured resistance.͒

͑e − 1 ͑e 1
xЈ − e − 2 xЈ ͒͑e 1 x e 2 L − e 2 x e 1 L ͒ ͑e 2 L − e 1 L ͒ ͑x−xЈ͒ − e 2 ͑x−xЈ͒ ͒͑x − xЈ͒, ͑A19͒ where ͑x − xЈ͒ is the step function.Green's function satisfies the equation for the fluctuations and the boundary conditions G͑0,xЈ;0͒ = G͑L,xЈ;0͒ = G͑x,0;0͒ = G͑x,L;0͒.͑A20͒Moreover, it is continuous at x = xЈ but its derivative is discontinuous at this point, as it should be.Now we are in the position to calculate the fluctuation of the total number of carriers,;0͒ ϵ L ˜D 2 G ˜͑x͒, ͑A23͒ G ˜͑x͒ = 1 − e 1 L ͑e 2 L − 1͒ ͑e 2 L − e 1 L ͒ e − 1 x + e 2 L ͑e 1 L − 1͒ ͑e 2 L − e 1 L ͒ e − 2 x .͑A24͒Notethat G ˜͑0͒ = G ˜͑L͒ = 0.By substituting the expression for the combined noise source, we finally arrive at ␦N͑t͒ = A ͵ the second contribution and by using the properties G ˜͑0͒ = G ˜͑L͒ = 0, we finally obtain␦N͑t͒ = A ͵ 0 L dxG ˜͑x͒␥͑x,t͒ − A ˜D͵ 0 L dxG ˜Ј͑x͒J D ͑x,t͒ ͑A26͒ ϵ␦N GR ͑t͒ + ␦N D ͑t͒.͑A27͒We note that there are two different contributions to the fluctuations of the total number of carriers, the first one associ-with GR processes and the second one associated with diffusion processes.The two contributions are related since the corresponding noise sources are uncorrelated.The equation of current fluctuation then reads ͑x,t͒ + ͩ I N ¯ͪ␦N GR ͑t͒ + ͩ I N ¯ͪ␦N D ͑t͒.