Local noise analysis of a Schottky contact: Combined thermionic-emission–diffusion theory

A theoretical model for the noise properties of Schottky barrier diodes in the framework of the thermionic-emission–diffusion theory is presented. The theory incorporates both the noise induced by the diffusion of carriers through the semiconductor and the noise induced by the thermionic emission of carriers across the metal–semiconductor interface. Closed analytical formulas are derived for the junction resistance, series resistance, and contributions to the net noise localized in different space regions of the diode, all valid in the whole range of applied biases. An additional contribution to the voltage-noise spectral density is identified, whose origin may be traced back to the cross correlation between the voltage-noise sources associated with the junction resistance and those for the series resistance. It is argued that an inclusion of the cross-correlation term as a new element in the existing equivalent circuit models of Schottky diodes could explain the discrepancies between these models and experimental measurements or Monte Carlo simulations. © 1998 American Institute of Physics. @S0021-8979 ~98!01705-8#


I. INTRODUCTION
Schottky barrier diodes are being extensively used in different applications, such as mixers and detectors, due to their good high-frequency behavior. 16][7] Under this approach the diode is characterized by means of the corresponding equivalent circuit.Hence, by associating each resistive element of the circuit with different noise sources ͑shot noise for the junction resistance, thermal noise for the series resistance, etc.͒ the total noise of the device is evaluated. 9Despite its wide use in electrical engineering, it should be noted, however, that this phenomenological procedure does not offer a relation between the equivalent circuit elements and the kinetic and material parameters of a device ͑barrier height, mobility, doping, surface recombination velocity, etc.͒.Moreover, it implicitly assumes that only a given type of noise can be assigned to each equivalent-circuit element and that the different sources of noise are uncorrelated.While the first assumption seems to hold in most of the cases, the second assumption might not, due to the existence of spatial correlation between different active regions of the device. 10o overcome this situation, and to avoid any a priori assumption regarding the different sources of noise, more recently microscopic Monte Carlo ͑MC͒ simulation has been performed. 8Although the MC technique describes the carrier transport on the microscopic level and takes into account a large variety of kinetic processes, there are some difficulties in its application to the study of Schottky barrier diodes.In particular, when the barrier height is sufficiently large (Ͼ5k B T/q) a depletion of carriers of several orders of magnitude occurs near the metal-semiconductor interface at low voltage biases.In such a case, the MC simulation becomes very time consuming because of poor statistics of carriers in the depletion layer, 8 which, in turn, controls the current and noise properties under these conditions.To overcome the problem the special algorithm of particle multiplication has been realized to calculate the I-V characteristics of the diode with a high barrier. 11Whether this algorithm is suitable for the noise calculation remains unclear.As a consequence, only the small-barrier diodes ͑near the flat-band conditions͒ have been simulated by the MC method. 8Therefore, the development of other methods able to provide theoretical analysis of the local noise properties of the Schottky contacts with arbitrary ͑low and high͒ barriers in the wide range of applied voltage biases seems to be highly desirable.
The purpose of the present article is to present precisely a simple analytical analysis of the different contributions to the net noise of a Schottky barrier diode.To this end we propose a combined model for the noise properties of these devices which takes into account in a common framework the two main sources of noise present in Schottky diodes, namely, the noise due to the thermionic emission of carriers across the metal-semiconductor interface and the noise due to the diffusion of carriers inside the semiconductor.By means of an analytical method recently developed 10 we have obtained the explicit expressions for the local noise and impedance distributions, which allow us to identify the regimes of the system parameters, where one or another noise source dominates.
Furthermore, we show that the equivalent noise circuit previously reported for these contacts [5][6][7] can be obtained as a limit of our model in some particular limiting cases.The equivalent-circuit parameters, such as the junction resistance, the series resistance, etc., acquire closed analytical formulas valid in the whole range of voltages including the crossover between the shot noise and thermal noise limits.In addition, our results suggest that in order to adequately describe the noise behavior of the Schottky diode under intermediate applied biases the traditionally used equivalent noise circuit model should be corrected by adding a new circuit element corresponding to the cross-correlation between the noise source associated with the junction resistance, and the noise source associated with the series resistance.This suggestion is based on the analytical expression for the cross-correlation term, and supported by our calculations.
The organization of the article is as follows: In Sec.II we review the thermionic-emission-diffusion theory which reflects the basic transport physics in an ideal n-type Schottky barrier diode.In Sec.III we propose the corresponding fluctuating thermionic-emission-diffusion model, which incorporates both the fluctuations due to the diffusion of carriers inside the semiconductor and the fluctuations induced by the thermionic emission of carriers across the metal-semiconductor interface.Then, the analytical solution of the model is derived, and the different contributions to the total noise and local noise distributions are identified.In Sec.IV the steady-state spatial profiles for the electric potential, electric field and carrier density are numerically obtained from thermionic-emission-diffusion equations, which will be used as input data to evaluate the noise characteristics.In Sec.V the equilibrium expressions for the analytical formulas found in Sec.III are derived and their compatibility with the Nyquist theorem shown.In Sec.VI the corresponding nonequilibrium results are presented and analyzed.Finally, in Sec.VII we sum up the main contributions of the article.

II. THERMIONIC-EMISSION-DIFFUSION THEORY
For ideal Schottky barrier diodes the thermionicemission-diffusion theory of Crowell and Sze 12 has long been recognized as an appropriate theory to describe the transport properties of these devices.The theory assumes that the two main current-limiting processes occurring in a Schottky barrier diode are the diffusion and drift of carriers through the semiconductor part of the contact and the thermionic emission across the metal-semiconductor interface.These two processes are effectively in series, and the current is determined predominantly by the process which causes the larger impediment to the carrier flow or, in a general case, by a combined action of both of them if they are comparable.The latter was pointed out as the most realistic situation. 3he first process, electron transport through the semiconductor, is described by means of the drift-diffusion model, that is an equation for the current coupled self-consistently to the Poisson equation Here, n(x,t) and E(x,t) are the electron density and the electric field, respectively, and A is the cross-sectional area.
The drift velocity v(E) and the diffusion coefficient D(E) depend, in a general case, on the local electric field.In this model the current across any section of the device is the sum of the conducting current J(x,t) ͑composed of the drift and diffusion components͒ and the displacement current.The current is conserved and equal to that in the external circuit I(t).
The second current-limiting process, the thermionic emission, is described by means of a boundary condition for the conducting current at the top of the potential barrier, which is expressed in terms of an effective recombination velocity v r as 12,13 J͑0,t ͒ϭq͓n 0 ͑ t ͒Ϫn 0 eq ͔v r A. ͑3͒ The term with n 0 v r represents the electron flux from semiconductor to metal, with n 0 being the electron density at the top of the barrier.The n 0 eq v r represents the opposite flux from metal to semiconductor, where the quasiequilibrium electron density n 0 eq ϭN c e Ϫq b /k B T does not depend on the bias, N c is the semiconductor effective density of states, and b the contact barrier height as seen from the metal side.The velocity v r is given by 12 v r ϭA*T 2 /qN c , where A* is the effective Richardson constant and T is the lattice temperature.
In this article we are interested in modeling the lowfrequency plateau of the noise spectrum ͑above the 1/f -noise region͒, corresponding to the time scale much longer than the dielectric relaxation time.Due to the frequency range chosen, the displacement current can be neglected.Therefore, by eliminating n, the system ͑1͒, ͑2͒ yields a nonlinear second-order differential equation for the electric-field profile, Moreover, by taking into account that in the frequency range of interest J(0,t)ϭI, the boundary condition at xϭ0 for Eq.͑4͒ following from Eqs. ͑3͒ and ͑2͒ becomes

͑5͒
On the opposite side of the semiconductor the flat-band boundary condition is assumed, that is where the sample length L is chosen to be large enough, i.e., much larger than the Debye screening length L D ϭ(⑀k B T/q 2 N D ) 1/2 to guarantee the quasineutrality condition n(L)ϭN D .͑This condition is also equivalent to putting the Ohmic contact at xϭL.)The results for a short Schottky barrier diode (L of the order of L D ) will be published elsewhere. 14o distinguish different regimes in the thermionicemission-diffusion theory, it is useful to introduce some dimensionless parameters as follows: Since the width of the depletion layer forming the potential barrier is of the order of several Debye lengths L D , the typical absolute value of the maximum of the electric field at the top of the barrier can be measured in units of E th ϭk B T/(qL D ).Accordingly, the characteristic diffusion velocity v D appeared in the theory 12 is of the order of E th , with ϭvЈ(E)͉ Eϭ0 being the lowfield mobility.Therefore, the parameter ␤ϭE th /v r can be used to estimate the importance of different regimes limiting the current.We will see in Sec.VI when we compute the impedance of the diode, that for ␤Ӷ1 the transport is diffusion limited while for ␤ӷ1 it is thermionic-emission limited ͑for ␤ϳ1 both processes contribute similarly͒.
Notice, that the parameter ␤ only enters into the boundary condition ͑5͒, where the term with the current is proportional to ␤.Therefore, in the diffusive limit when ␤→0, this term is small in respect to others, and the electron density n 0 remains approximately at the equilibrium value n 0 eq and is not altered by the applied voltage ͑current͒.This is equivalent to assuming that at the interface the quasi-Fermi level in the semiconductor coincides with the Fermi level in the metal. 2 In this case the decrease of the quasi-Fermi level occurs through the depletion region.In the opposite thermionicemission limit when ␤ is large and the boundary condition at the interface depends strongly on the current, the quasi-Fermi level for electrons remains flat throughout the depletion region ͑the mobility is large͒ and drops down inside the metal.Therefore, the boundary condition ͑5͒ mirrors the behavior of the quasi-Fermi level nearby the metalsemiconductor contact.

A. General formulation
According to the thermionic-emission-diffusion theory described in the previous section two current limiting processes are assumed to exist in Schottky barrier diodes: the diffusion of carriers through the semiconductor and the thermionic emission of carriers across the metal-semiconductor interface.Consequently, by associating, as usual, a noise source with each current-limiting process, two different sources of noise should be present in the system: a diffusion noise source and a thermionic-emission one. 15While the former accounts for the different random scattering processes occurring inside the semiconductor, the latter describes the random nature of the thermionic-emission processes across the metal-semiconductor interface.Although the existence of both noise sources seems to be out of discussion, none of the existing carrier-transport models incorporates them in a unified way.Concerning the equivalent circuit model, it does not incorporate them in a proper way, as will be shown below.
Here we present a simple model in which both sources of noise are taken into account in a common framework.The diffusion noise is modeled, as usual, as a bulk current noise source which accounts for the velocity fluctuations 16 K͑x ͒ϭ4q 2 n͑x ͒D͑ x ͒, ͑7͒ where the nonlocal heating of carriers is neglected.
On the other hand, we model the source of noise due to the thermionic emission of carriers across the metalsemiconductor interface as a surface noise source associated with a stochastic boundary condition.Its spectral density, as shown in Appendix A, is given by with I c ϭqn 0 eq v r A. This approximation is based on the assumption that the thermionic-emission processes occur inside a layer of the order of the electron mean free path p which is assumed to be a small fraction of the barrier width.The two sources of noise, as well as two local noise sources of Eq. ͑7͒ corresponding to different slices x and xЈ are assumed to be uncorrelated, since the characteristic scale is such that ͉xϪxЈ͉ӷ p .
As pointed out by Van Vliet, 17,18 in the case that surface as well as bulk noise sources are present in a given system the most convenient method to analyze the fluctuations and noise properties of the system is the Langevin response formulation.Essentially, this formulation consists in adding to carrier transport equations the corresponding stochastic noise sources and then, by solving these equations and computing the appropriate averages, to obtain the statistical properties of the different magnitudes of interest.For the case under consideration, this means that the fluctuation of the electric field ␦E x at slice x should satisfy the linearized version of Eq. ͑4͒ with a source noise term for the current ␦I x with the operator L ˆgiven by

͑10͒
Here, ␦I x represents the stochastic current induced by the random scattering of carriers inside the semiconductor.It has zero mean and ␦-type correlation function Eq. ͑7͒ and ⌬ f the frequency bandwidth.Furthermore, at the metal-semiconductor interface the fluctuations satisfy the linearized version of Eq. ͑5͒ with a surface Langevin-like term ␦I s present.Explicitly, Here, ␦I s describes the random nature of the thermionic emission of carriers across the metal-semiconductor interface.It has zero mean and the dispersion ͗␦I s 2 ͘ϭS Ic ⌬ f with S Ic given by Eq. ͑8͒.
Due to the requirement on the length of the sample LӷL D and the flat boundary condition at xϭL, for the fluctuations of the field at this boundary we have with the meaning that the contribution to the noise at the contact from distances much larger than the Debye length is screened out.Equations ͑7͒-͑12͒ constitute a complete set of equations to analyze the noise properties of Schottky barrier diodes in the thermionic-emission-diffusion approximation.

B. Analytical solution
Note that L ˆis a linear second-order differential operator with space-dependent coefficients.To find the solution in an analytical form we use the method recently applied for the n ϩ n semiconductor junction. 10The general solution for the electric-field fluctuation is given by

͑13͒
where W(x)ϭ(x)uЈ(x)Ϫu(x)Ј(x) is the Wronskian, and (x) and u(x) are auxiliary functions, satisfying the equations L ˆ(x)ϭ0, L ˆu(x)ϭ0, i.e., they are the solutions of the homogeneous equation corresponding to Eq. ͑9͒.The expressions for the auxiliary functions are as follows: 10 ͑x ͒ϭdE/dx, ͑14a͒ The value of the integration constant W(0) is not actually important, since it will be canceled after substituting Eq. ͑14b͒ into ͑13͒, so we can assume W(0)ϭ1.The constant C in Eq. ͑14b͒ can be defined by the appropriate choice for the boundary condition of the function u(x).Since it does not influence on the final results, we take uЈ(0)ϭ0 which corresponds to the homogeneous boundary conditions for the Green functions of the operator L ˆand provides the most compact intermediate expressions.The integration constants C 1 and C 2 in Eq. ͑13͒ are determined by the boundary conditions at xϭ0 and xϭL.The condition ͑12͒ implies (L)ϭ0 and, hence, C 2 ϭ0.By differentiating Eq. ͑13͒ and using Eq.͑11͒ we obtain the stochastic value for the constant C 1 as Hereafter, the prime stands for a derivative, and the subindex 0 refers to the value of the corresponding function at xϭ0.
The final expression for ␦E x then reads,

͑16͒
where we have denoted ()ϭ⑀AD()W().Equation ͑16͒, together with Eq. ͑14͒, constitutes the analytical solution to the problem posed by Eqs.͑9͒-͑12͒.It expresses the electric-field fluctuations at point x in terms of the noise sources ␦I x , ␦I s and the stationary electric field profile which appears through (x), u(x), and (x).
It is worth noting that Eq. ͑16͒, apart from allowing us to derive explicit expressions for the impedance field and voltage and current fluctuations ͑see below͒, it could also be used to evaluate the spatial correlations of the electric-field fluctuations ͗␦E x ␦E x Ј ͘ between two different points ͑re- gions͒ under nonhomogeneous conditions.These correlations are active over several characteristic screening lengths of the system. 10ow, we are in a position to evaluate the fluctuation in the voltage at the terminal ends by integrating the fluctuation of the electric field given through Eq. ͑16͒ along the system,

͑17͒
We then obtain, where is the impedance of the metal-semiconductor contact associated with the thermionic-emission processes, and ٌZ(x) the bulk impedance field associated with the drift-diffusion processes in the semiconductor.The latter consists of two terms It will be shown below, that the term ٌZ dep is the contribution coming from the depletion layer and it dominates at low currents, while ٌZ ser is determined by the doping level inside the semiconductor and is associated with the series resistance.The latter dominates at high currents when the depletion layer disappears ͑flat-band condition͒ and the voltage drops mainly on the bulk of the semiconductor.From Eq. ͑18͒ the total impedance Z of the Schottky contact, as well as the spectral density of the voltage fluctuations S V , may be easily computed.One obtains, where and they represent different contributions to the total voltage fluctuations.On the other hand, we shall use the function Equations ͑23͒-͑30͒, once Eqs.͑19͒-͑22͒ are used, give the exact expressions for the total diode impedance, voltage and current fluctuations, and the noise temperature as obtained form the thermionic-emission-diffusion theory.Furthermore, they also give information about the spatial distribution of the impedance throughout the device by means of the impedance field ٌZ(x), and the local contributions to the voltage fluctuations by virtue of the spatial profile s V (x).These local distributions may be very useful in interpreting the results as will be seen in what follows.It is worth noting that in order to compute Eqs.͑19͒-͑22͒, and hence Eqs.͑23͒-͑30͒, we only need to know the stationary electric field profile.
Recalling now the equivalent-circuit-noise formula for the current spectral density of the Schottky diode 5 with R s and R j the series and the junction resistances, respectively, and comparing it with our Eq.͑29͒, it is seen, that we arrived at a more general result.In particular, the junction resistance R j usually used as a phenomenological parameter to fit the experimental I -V or noise characteristics, now can be evaluated from R j ϭZ c ϩZ dep and is given analytically by Eqs.͑19͒ and ͑21͒.The series resistance R s is equal to Z ser in our model and can be obtained from Eq. ͑22͒.The noise associated with the junction resistance 2qIR j 2 is given in our model through S Vc ϩS Vdep , which are followed from Eqs. ͑25͒ and ͑26͒.Moreover, the noise term associated with the series resistance 4k B TR s can be evaluated from S Vser given by Eq. ͑27͒.Finally, an additional term represented by S Vcros absent in the equivalent circuit model has appeared in our theory, which is relevant, as will be shown below, in a certain range of voltages.It is interpreted as a cross-correlation between the voltage-noise sources associated with the junction resistance and those for the series resistance.More detailed discussion of all the terms will be found in the subsequent sections.

IV. STEADY-STATE SPATIAL PROFILES AND I -V PLOTS
In order to evaluate the different expressions found in the previous section, we need to compute the corresponding stationary profiles for the electric field.To find these profiles we solve numerically Eq. ͑4͒ with the boundary conditions ͑5͒ and ͑6͒ by making use of a finite difference scheme.For the sake of completeness, also the electron density and potential profiles will be reported.It can be shown that the diode behavior is governed by three dimensionless parameters ͓for given shapes of v(E) and D(E)]: ͑i͒ the coefficient ␤ϭE th /v r introduced in Sec.II characterizing the type of transport regime; ͑ii͒ the ratio ␣ϭn 0 eq /N D which characterizes the height of the contact barrier, and ͑iii͒ the current, for which we use the following units JϭI/I R , with I R ϭqE th N D A.
In the following we shall use three different values of the parameter ␤ for a comparative analysis of different transport regimes: ␤ϭ20 for the case of thermionic-emission ͑TE͒ dominated transport, ␤ϭ0.01 for the case of diffusion ͑D͒ dominated transport, and ␤ϭ1 for the intermediate thermionic-emission-diffusion ͑TE-D͒ regime.
The parameter ␣ describes basically the nature of the contact under consideration.In this sense, for a rectifying contact ͑high barrier͒ ␣ is very small, while for not rectifying contacts ͑low barrier͒ ␣տ1.In the following for our calculations we take ␣ϭ10 Ϫ7 .This value provides a sufficiently high potential barrier (Ϸ16k B T/q at equilibrium͒ and a ''deep'' depletion layer with the electron concentration at the barrier top n 0 ϳ10 Ϫ7 N D .For simplicity we consider only the case of constant mobility and the diffusion coefficient D, although explicit field dependencies characteristic for the hot carrier transport may be used without additional difficulties.To be concrete, we take v(E)ϭE, Dϭk B T/q.The length of the sample is taken to be Lϭ15L D to guarantee the quasineutrality conditions at xϭL.
Having found the electric-field profiles E(x) for a given current J, the distributions of the potential (x) and the electron density n(x) can be easily restored.Knowing the total voltage drop Vϭ͐ 0 L E(x)dxϩV bi , with the equilibrium built-in voltage V bi ϭ(k B T/q)ln(1/␣), the I -V curves can then be obtained.In our calculations in which ␣ϭ10 Ϫ7 , the built-in voltage V bi Ϸ16k B T/q.A comparison of the I -V characteristics for the structures simulated for different values of ␤ is shown in Fig. 1.In all cases the qualitative behavior is similar: for low biases, when the current is controlled by the depletion layer, there appears an exponential region, while for high biases the characteristics becomes linear displaying the presence of series resistance effects once the depletion layer has disappeared.Apart from a quantitative difference, the exponential behavior is seen to be more pronounced for the case TE.The fact that both the thermionic emission processes as well as the diffusion processes ͑or a combined action of both͒ may induce a similar behavior on the I -V characteristics has to be taken into account in analyzing the properties of these diodes.In particular, care should be taken in drawing conclusion from an analysis of the I -V characteristics in respect to the dominant transport mechanisms acting on a given system.In general, precise information on the values of the different contact parameters, i.e., recombination velocity, mobility, doping, etc., are necessary to arrive at a correct interpretation of the results.As we will see, the same statement holds for the analysis of the noise properties of these systems.
The steady-state spatial profiles for the case D and for different currents J are shown in Fig. 2. The profiles display the usual behavior: for low currents a pronounced depletion layer is observed near the metal-semiconductor interface, while at higher currents the depletion layer progressively disappears ͓Fig.2͑a͔͒.Along with a disappearance of the depletion layer, the potential barrier also disappears ͓Fig.2͑c͔͒.The spatial distributions for another limiting case, TE case, are presented in Fig. 3. Apart from quantitative differences arisen from different typical scale of the current, it is seen that for the TE case at high enough currents the depletion layer can be substituted by an accumulation layer.

V. EQUILIBRIUM NOISE AND THE NYQUIST THEOREM
The relative significance of the different terms contributing to the impedance and noise of the Schottky diode can be ascertained by considering the equilibrium case Iϭ0, for which very transparent expressions can be obtained.For this case the equation for the balance of the current nEϩ(k B T/q)dn/dxϭ0 can be integrated, giving n͑x ͒ϭn 0 eq exp͓q͑x ͒/k B T͔, ͑32͒ where we have used the boundary condition for the electric potential (0)ϭ0.Furthermore, all the expressions are simplified, giving E(L)ϭ0; W(x)ϭexp͓q(x)/k B T͔; E(x)W(x) ϭL D0 2 (d/dx), where L D0 ϭ(⑀k B T/q 2 n 0 eq ) 1/2 is the screening length corresponding to the electron density at the interface.Substituting these relations into Eqs.͑20͒-͑22͒ we obtain ٌZ eq ͑ x ͒ϭR͑ x ͒ϵ showing that the bulk impedance field is equal to the local ͑per unit length͒ resistance.Figure 4 shows the spatial profiles for ٌZ eq (x) and its constituents ٌZ dep eq (x) and ٌZ ser eq (x) calculated by using the equilibrium distribution n(x) shown in Figs. 2 or 3 ͑at equilibrium they are identical for all three transport regimes͒.The term ٌZ dep eq (x) is seen to give the main contribution near the interface xϽ5L D , since it represents the depletion layer contribution, in which the carrier density is very small.Whereas for xϾ7L D the impedance is determined mainly by the series resistance term ٌZ ser eq which is constant over almost the whole structure and is determined by the donor density.
The contact impedance given by Eq. ͑19͒ is estimated by taking into account the relation 0 ЈϭE 0 /L D0 2 valid at equilibrium.One gets where R c is the well-known contact resistance, 4 used to characterize the nature of the contact.Thus, the total equilibrium impedance ͑resistance͒ of the diode is given by Z eq ϭR c ϩ͐ 0 L R(x)dxϭR c ϩR dep ϩR ser .After integration along the structure the depletion-layer resistance is much greater than the series resistance, because R dep /R ser ϳ1/␣ӷ1, as it should be for a rectifying contact.The total equilibrium diode resistance is then governed by the sum R c ϩR dep .Introducing the characteristic series resistance as R R ϭE th L D /I R ϭL D /(qN D A) the contact resistance can be evaluated by R c ϭR R (␤/␣), whereas the depletion resistance R dep ϷR R /␣.Therefore, R c /R dep Ϸ␤, and one or another resistance dominates (R c or R dep ) depending on the regime considered ͑TE or D͒.
A similar spatial analysis is performed for the voltage fluctuations.For the local noise contributions in Eqs.͑26͒-͑28͒ one gets Hence, for the net local noise s V eq (x)ϭs Vdep eq (x) ϩs Vcros eq (x)ϩs Vser eq (x)ϭ4k B TR(x), which is the Nyquist theorem, since R(x) is the local resistance.It should be pointed out that in recovering the Nyquist theorem all contributions, including the cross-correlation term, have been necessary.
The local noise distributions Eqs.͑37͒-͑39͒ calculated by using the equilibrium profile n(x) are shown in Fig. 5.As a consequence of the Nyquist theorem the thick solid lines in Figs. 4 and 5 coincide.Note that near the interface the main contribution to s V eq (x) comes from the depletion-layer term s Vdep eq (x).The series-noise term dominates in the quasineutral FIG. 4. Equilibrium bulk impedance field ٌZ eq (x) and its components ٌZ dep eq (x) and ٌZ ser eq (x), all normalized by (qAN D ) Ϫ1 .
region, and the cross-correlation term is active in the crossover from the depletion layer to the quasineutral part of the diode.
For the contact contribution to the voltage noise we get by means of Eqs.͑8͒, ͑25͒, and ͑36͒, which again consists of the Nyquist theorem.The total voltage fluctuations are then given by S V ϭ4k B T(R ser ϩR dep ϩR c ).Similarly, we obtain for the current fluctuations S I ϭ4k B T/(R ser ϩR dep ϩR c ).These results show that the Nyquist theorem is satisfied both locally and globally, as should be.

VI. NONEQUILIBRIUM FLUCTUATIONS IN A SCHOTTKY DIODE
In this section we will discuss the results obtained from our theory for the nonequilibrium case.Note that to evaluate the different expressions derived in Sec.III B we only need to use the stationary electric-field profiles obtained in Sec.IV.
In Fig. 6 we have plotted the bulk impedance field ٌZ(x) and its components ٌZ dep (x), ٌZ ser (x) for different values of the current.The term ٌZ dep (x) is seen to be mostly localized in the depletion layer and it plays the main role at low currents whenever the depletion layer is appreciable.On the other hand, the spatial profile for ٌZ ser (x) is flat over almost the whole structure with the magnitude corresponding to the donor density.It is seen that its shape depends weakly on the bias.This clearly explains why this term represents the series resistance of the sample.
Integrating along the structure the impedance-field components, the depletion impedance Z dep , and the series impedance Z ser are computed.By adding them to the contact impedance Z c , the total impedance Z is evaluated.Figure 7 shows the results as functions of the applied bias in the different transport regimes.It is seen, that for low biases the main contribution comes from either Z dep ͑case D͒ or Z c ͑case TE͒ ͑or both at TE-D regime͒, while at high voltages the series impedance Z ser always dominates.Note that the series impedance Z ser is almost constant in the whole range of voltages considered.
A similar analysis is performed for the voltage fluctuations.As before, the meaning of the different terms can be  justified by means of the local noise analysis of the diode.In Figs.8͑a͒ and 8͑b͒ the local noise distributions s V (x)ϭs Vdep (x)ϩs Vcros (x)ϩs Vser (x) are depicted for two different transport regimes ͑D and TE͒ and for two different values of the current.It is seen that the term s Vdep is mainly localized in the depletion region, the term s Vser dominates in the quasineutral part of the sample, and s Vcros is important in the crossover between the depletion and the quasineutral regions and represents their cross-correlation.At low currents the depletion contribution dominates, while at high currents, when the depletion layer has disappeared, all the components are comparable in magnitude.The main difference between the D and TE cases is in the position of the local noise maxima at low currents.For the case TE it lies at the interface xϭ0 where the electron density is lowest, while for the case D it is displaced towards the depletion region, since for this case the carrier density at xϭ0 is fixed by the boundary condition and does not fluctuate significantly.
Integrating along the device the spatial profiles for all the components, the spectral densities of voltage fluctuations S Vdep , S Vcros , and S Vser are found.To get the total voltage noise they should be added to the contact noise term S Vc , which is localized at the metal-semiconductor interface.The relative significance of different components for different applied biases is shown in Fig. 9.For low voltages the main contribution comes from either S Vdep ͑case D͒ or S Vc ͑case TE͒ ͑or S Vc ϩS Vdep for TE-D case͒, while at high voltages S Vser dominates.At intermediate biases S Vcros is seen to contribute appreciably.
Having found the total impedance Z and the voltagenoise spectral density S V , the spectral density of current fluctuations S I is computed from Eq. ͑29͒.The results for different transport regimes are shown in Fig. 10.A first important point to be noted is that in all cases the current fluctuations display a similar behavior: for very low current levels (JՇI c /I R ϭ␣/␤) S I is constant, since it gives the thermal equilibrium noise with different diode resistances according to different ␤.Whereas at higher currents (␣/␤ՇJՇ10 Ϫ2 ) all the curves exhibit a 2qI shot-noise-like behavior. 19hese two different behaviors can in fact be predicted explicitly from our analytical results.Indeed, by neglecting at low biases (VՇV bi ) the series-resistance and cross contributions, one gets which in the depletion approximation, valid under this lowvoltage regime, yields ͑see Appendix B͒ S I ϭ2q͓Iϩ2I sat ͔, ͑42͒ Ϸ2qI, for IӷI sat ,

͑43͒
where I sat is the saturation current of the diode which includes both the effects of thermionic-emission and diffusion processes, and it is given by Eq. ͑B13͒.Note, that in general, I sat is a voltage dependent function due to the diffusion effects.It is worth noting that the validity of Eq. ͑42͒ extends over not only the ohmic regime but the hot-electron regime as well ͑see Appendix B͒.For the case of ohmic transport, however, the I -V characteristics takes on a simple form 4 IϭI sat ͓exp͑ qV/k B T͒Ϫ1͔, ͑44͒ and Eq.͑42͒ can be rewritten as 9 S I ϭ2qI coth͑qV/2k B T ͒, ͑45͒ which does not include explicitly any system dependent parameter.It can be seen that the data plotted in Fig. 10 closely follow this law independently on the transport regime considered.
At high biases (VϾV bi ) the current fluctuations S I deviate from the 2qI law and tend to saturate.In this limit we may approximate ZϷZ ser and S V ϷS Vser ϭ4k B TZ ser , which are independent on the bias ͑see Figs.7 and 9͒.Consequently, S I ϭ4k B T/Z ser is independent of the current, that explains the saturation behavior at high voltages.
The effective noise temperature T n as a function of bias in different transport regimes is depicted in Fig. 11.One can see, that starting from T n ϭT at zero bias ͑thermal equilibrium noise͒ it drops sharply towards T/2 for low voltages that is a consequence of the 2qI law and is typical for the shot noise behavior. 5,9At higher biases T n increases again approaching the ambient temperature T, corresponding to the diffusion noise of the series resistance.Note that T n may attain T or may not, depending on the length of the Schottky diode L ͑the magnitude of the series resistance͒. 14The noise temperature T n can be used as a figure of merit in order to evaluate the effects of the different sources of noise on the device performance.In particular, it can be seen that the more the transport is thermionic-emission dominated, the more T n approaches the value T/2.In this sense, one can conclude that the TE case is less noisy than the D case.
The importance of the cross-correlation term can be made more explicit by plotting S I and T n with and without including this term.Figure 12 clearly shows that both functions deviate significantly starting from the voltage VϷ10k B T/q (JϷ10 Ϫ2 ) when the series effects starts to appear.A similar behavior has been observed in some experiments, 6 where the noise temperature was found to deviate from the equivalent circuit results ͑without the cross term͒ precisely as shown in Fig. 12.Therefore, a part of the excess noise observed could be assigned to the crosscorrelations between the different noise sources.This statement is further confirmed by the recent MC simulation, 8 which also found out the discrepancies with the equivalent circuit results ͑compare Fig. 12 of the present article with Fig. 3 of Ref. 8͒.

VII. DISCUSSION
In this article we have presented an analytical analysis of the noise properties of an n-type Schottky barrier diode under forward bias conditions.Our study has been based upon a noise model which incorporates in a unified way both the noise due to the diffusion of carriers through the semiconductor and the noise due to the thermionic-emission of car- riers across the metal-semiconductor interface.By means of the analytical solution of the model, the different contributions to the net noise of the diode have been identified, and with the help of a local analysis, assigned to different spatial regions of the device.Furthermore, the relevance of the different contributions depending on the characteristic parameters and the applied voltage range has been analyzed.In particular, it has been shown that at low biases (VՇV bi ) the noise properties are dominated either by the diffusion of carriers in the depletion layer near the interface (␤Ӷ1) or by the thermionic emission of carriers across the metalsemiconductor interface (␤ӷ1).Here, ␤ is the coefficient between the diffusion velocity and the contact recombination velocity.At high biases (VտV bi ), the noise properties are shown to be dominated in all cases by the diffusion of carriers through the quasineutral region of the device.Finally, starting from the applied voltage VϳV bi at which the depletion layer becomes small, an additional contribution has been identified.Due to its spatial localization in the crossover between the depletion layer and quasineutral region of the device, this additional term has been argued to represent the cross-correlation between the voltage noise sources originated in these two parts of the device.
A detailed comparison between the results predicted by means of our model and those predicted by means of the conventional equivalent circuit noise technique has been performed.In particular it has been shown that: ͑i͒ The so-called junction resistance R j , whose origin is not clearly established in circuit analysis, consists in general of two contributions: Z c and Z dep , for which we give explicit expressions.͑ii͒ The series resistance R s is given through Z ser , whose value can also be obtained from the corresponding expression.͑iii͒ The 2qIR j 2 behavior associated with R j at low current levels is in fact given through S Vc ϩS Vdep , which reproduces this type of behavior at low voltages, but holds in the whole range of biases.Note, that both the thermionic emission of carriers through the metal-semiconductor interface as well as the diffusion of carriers in the depletion region contribute to this behavior.͑iv͒ The noise due to the series resistance is given through S Vser , which reproduces correctly the well known thermal noise behavior.͑v͒ There is an additional contribution S Vcros to the voltage fluctuations not present in the conventional equivalent circuit analysis.This contribution is due to the cross-correlation between the fluctuations generated in the depletion layer and those in the quasi-neutral region of the semiconductor.It starts to become relevant in the transition from the 2qI behavior towards the series-resistance noise ͑near the flat-band conditions͒.It has been argued that this additional term could explain some discrepancies observed between experimental measurements or MC simulations and the results of the equivalent circuit model precisely under near flat-band conditions.It should be pointed out that mm-wave and sub-mmwave mixers operate precisely at this regime, 7 so that our theory can be useful for the local-noise analysis of these devices in order to improve their performance.
It is worth remarking that our theory provides closed analytical expressions to evaluate all the terms appearing in the impedance as well as in the voltage and current fluctuations under the full range of applied biases.Hence, it can be used to improve the existing equivalent circuit model.These expressions can in fact be easily inserted into the developed numerical programs which calculate the steady I -V characteristics and electric field profiles for Schottky diodes, thus adding also information about the local and global noise properties of these systems.In addition, the method presented in this article, besides its fruitful application for the Schottky barrier diodes, it can be incorporated into any device model described by means of the drift-diffusion approach and its modifications. 21Our work then offers new perspectives on what concerns the spatial analysis of the noise properties of space-charge-limited devices, such as n ϩ -n -n ϩ diodes, field-effect transistors, etc.Furthermore, for these values of the applied bias the depletion-layer approximation holds for which the electron concentration inside the depletion layer is neglected, since n(x)ӶN D . 2 For this case (x)ϭdE(x)/dx becomes ͑x ͒Ϸ ͭ 0 , 0ϽxϽw, 0, wϽxϽL, ͑B2͒ where 0 ϭqN D /⑀ and w is the length of the depletion layer.Under these conditions the electric field is approximated as 2 E͑x ͒Ϸ ͭ E 0 ͑ 1Ϫx/w ͒, 0ϽxϽw, 0, wϽxϽL.͑B3͒ As a result for the depletion-layer impedance by using Eq.͑21͒ we have where C E ϭ(E 0 ϪE L )/(⑀A 0 Ј), and we have defined the dif- fusion velocity v D by 22 1 The expressions for the contact impedance and voltage spectral density follows from Eqs. ͑19͒ and ͑25͒, 4q 2 An 0 eq v r ͬ .͑B7͒ Now, the only unknown term in Eq. ͑B1͒ is the depletion voltage-noise spectral density S Vdep which can be found from Eq. ͑26͒ To proceed further we integrate Eq. ͑1͒ with the help of the integrating factor 1/W(x) finding the following relation: Substituting this relation into Eq.͑B8͒ and using the identity ͑can be proved by integrating by parts͒ with f (x)ϭ1/͓D(x)W(x)͔, one arrives at

͑B11͒
Now we can use the boundary condition for n(0) given by Eq. ͑3͒ and substitute all the terms into Eq.͑B1͒.One gets S I ϭ2q ͫ Iϩ2 qAn 0 eq v r v D v r ϩv D ͬ ϵ2q͓Iϩ2I sat ͔, ͑B12͒ where is the saturation current of the diode including both the effects of thermionic-emission and diffusion processes.Note the universality of Eq. ͑B12͒ which holds for different transport regimes ͑D, TE, and TE-D͒ and conduction mechanisms ͑ohmic, hot-electrons͒.For the D case under ohmic conditions a similar derivation of the formula ͑B12͒ has been performed by van der Ziel. 23

FIG. 1 .
FIG. 1.I -V characteristics for a Schottky diode operating at different regimes: D, TE-D, TE.Current is in units of I R ϭqE th N D A.

FIG. 3 .
FIG. 3. Stationary distributions of the electron density n ͑a͒, the electric field E ͑b͒, and the electric potential ͑c͒ of a Schottky diode in the regime TE for different currents J.

FIG. 5 .
FIG. 5. Spatial profiles for the contributions s Vdepeq (x), s Vcros eq (x), and s Vser eq (x) to the net local equilibrium noise s V eq (x), all normalized by 4k B T⌬ f /(AN D ).

FIG. 7 .
FIG. 7. Impedance Z vs applied voltage V at different regimes: D ͑a͒, TE-D ͑b͒, and TE ͑c͒.The relative contributions from the contact Z c and the bulk Z ser , Z dep are compared.

FIG. 8 .
FIG. 8. Spatial profiles for the local noise s V (x) ͓normalized by 4k B T⌬ f /(AN D )] in regimes D ͑a͒ and TE ͑b͒.For each type of line, the curve on top corresponds to the lower current.

FIG. 10 .
FIG. 10.Current-noise spectral density S I vs current I at different regimes: D, TE-D, and TE.The low-current limit is shown to be nicely described by Eq. ͑45͒ ͑dashed line͒.S I and I are normalized by 4qI R and I R ϭqE th N D A, respectively.
also subdivided into the corresponding components s Vdep (x)ϩs Vcros (x)ϩs Vser (x), in order to characterize the local contribution of different space regions to the net noise measured between the terminals.The spectral density of the current fluctuations S I can be also computed by using Eqs.͑23͒ and ͑24͒ asSI ϭ S Vc ϩS Vdep ϩS Vcros ϩS Vser ͑ Z c ϩZ dep ϩZ ser ͒ 2 .͑29͒ Finally, the noise temperature 4k B T n ϭS V /Z takes on the form Vc ϩS Vdep ϩS Vcros ϩS Vser Z c ϩZ dep ϩZ ser .͑30͒