Flicker noise by random walk of electrons at the interfacein nonideal Schottky diodes

J.I. Lee*, J. Brini, A. Chovet, C.A. Dimitriadis 1

LPCS, UMR 5531 CNRS, ENSERG, 23 Rue des Martyrs, BP 257, 38016 Grenoble, France

Received 15 August 1998; received in revised form 11 June 1999; accepted 7 July 1999

Abstract

An explanation of low frequency 1/f noise in nonideal Schottky barrier diodes is presented where the current

¯uctuation is attributed to the random walk of electrons at the metal-semiconductor interface via modulation of thebarrier height. The experimental results on TiN/n-Si Schottky diodes have been successfully analysed to give usefulinformation on the interface states. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction

Flicker noise, or 1/f noise, in Schottky diodes was

®rst studied by Hsu [1,2] with a trapping and tunneling

model where the ¯uctuations in the occupancy of bulk

traps in the depletion region modulate the barrier

height to result in a current noise intensity pro-

portional to the square of the forward current. Hsu's

models were criticized by Kleinpenning [3] on account

of several reasons, for instance, Hsu's assumption on a

uniform distribution of the trap states in both energy

and space is not realistic, and the trap densities

obtained are too large. This can be actually self-

explained if we consider the trap states to be localized

in a certain energy region of interest over several

values of kT and near the interface in some cases [4].

However Kleinpenning established a new model con-

sidering mobility and di�usivity ¯uctuation with a

Hooge parameter, which gives a noise intensity pro-

portional to the current. Later Luo et al. [5] corrected

Kleinpenning's model by considering the local, instead

of the terminal, ®eld. Introducing a position dependent

weighting factor in integration over the distance from

the neutral bulk semiconductor, Luo et al. [5] obtained

a similar expression which results in a lower value of

the Hooge parameter by about two orders of magni-

tude for the same value of the noise intensity.

However the above models did not consider the contri-

bution from the interface states which is important in

nonideal Schottky diodes. The normal ¯uctuation in

the occupancy of interface states, such as in the ther-

mal activation model [1], gives generation-recombina-

tion noise and not ¯icker noise.

However, random walk of electrons at the interface

via interface states has been known to generate ¯icker

noise in semiconductor devices [6,7]. Jaentsch [7]

applied the random walk model to account for the 1/f

noise in Schottky barrier diodes, where he considered

the ¯uctuation of the surface generation-recombination

current, which is directly related to the ¯uctuation in

the occupancy of the interface states according to the

Solid-State Electronics 43 (1999) 2185±2189

0038-1101/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0038-1101(99 )00187-2

* Corresponding author. Permanent address: Photonics

Reseach Center, Korea Institute of Science and Technology,

Cheongryang, P.O. Box 131, Seoul 130-650, South Korea.

Tel.: +82-2-958-5786; fax: +82-2-958-5709.

E-mail address: jil@kistmail.kist.re.kr (J.I. Lee)1 Permanent address: Department of Physics, University of

Thessaloniki, 54006 Thessaloniki, Greece.

generation-recombination theory of diodes. However,Jaentsch [7] did not consider the e�ect of random walk

on the volume current via the modulation of the bar-rier height, although he discussed the 1/f noise in ther-mionic tubes caused by random walk of electrons at

the surface of the cathode, via the ¯uctuation of thework function, which corresponds to the barrier heightin Schottky barrier diodes. His results on the noise

intensity for Schottky diodes give, however, manyorders of magnitude lower values in normal conditionsand, sometimes it is di�cult to separate the volume

current and the surface current in Schottky diodes.The model proposed by Jaentsch [7] might be usefulonly when the current is dominated by the surface gen-eration-recombination current. Here the e�ect of ran-

dom walk of electrons at the interface on themodulation of the barrier height is considered and theresulting current noise intensity is derived. In addition

experimental results on TiN/n-Si Schottky diodes arepresented and analyzed with the model.

2. Theory

From the thermionic emission model includingimage force correction, the current ¯uctuation DIF,due to the modulation of the electric ®eld DF, can beobtained as follows,

DIF � qIF

4kT

�q

peF0

�1=2

DF, �1�

where q is the electronic charge, kT is the thermalenergy at the absolute temperature T, e is the permit-tivity of the semiconductor and F0 is the average elec-

tric ®eld intensity at the metal semiconductor interface,or

F0 � qNdW

e, �2�

where Nd is the doping concentration of the n-typesemiconductor, and W is the width of the depletionregion given by

W �����������������������������������������������2eqNd

�VD ÿ VF ÿ kT

q

�s, �3�

where VD is the built-in potential and VF is the for-ward bias voltage. The modulation of the ®eld inten-sity due to the ¯uctuation in the occupancy of the

interface trap states is,

DF � qDNt

eA, �4�

where A is the area of the diode and DNt is the ¯uctu-

ation in the number of the trapped states. Using Eqs.(1)±(4), we obtain the noise current intensity as fol-

lows,

SI� f� ��

q2IF

4kTeA

�21

pNdWDNt

2�tm

tn

g�t� 4t

1� �2pft�2 dt:

�5�The distribution of time constants t given by the

random walk model is such that,

g�t� � G=t: �6�The constant G has the value

G ������sp2pl

10:1, �7�

where s is the capture cross section and l is the smal-lest possible mean free path which is the atomic dis-tance [7]. For tn<<1/2pf<<tm, the integral in Eq. (5)reduces simply to G/f.

The ¯uctuation of the number of electrons is equalto the ¯uctuation in the occupancy of the interfacestates, or,

DN2t � A

�P�1ÿ P �DitdE1AkTDit�EF�, �8�

where P is the Fermi±Dirac distribution function, EF

is the fermi energy, and the interface states density Dit,is assumed to be constant or vary slowly with theenergy. Utilizing Eqs. (5)±(8), the noise intensity due

to the random walk of electrons at the interface isobtained as a function of the interface states density asfollows,

SI� f� � G

f

�qIF

4e

�2 bqDit

pNdWA, �9�

where b � q=kT.

3. Experimental results and discussions

3.1. Device and measurements

The low frequency noise measurements were per-

formed on a set of TiN/n-Si Schottky barrier diodeswith 1-mm diameter at room temperature. The TiN®lm was deposited via reactive magnetron sputtering

[8] under substrate bias voltage of ÿ40 V, depositiontemperature of 4008C and nitrogen ¯ow rate of 2.2sccm, on a phosphorous doped n-Si substrate of dop-

ing density Nd=2 � 1015 cmÿ3, 0.38 mm thickness andarea R8 � 8 mm. The electrical characterization wasdone by the analysis of the I±V curves of the diodes

J.I. Lee et al. / Solid-State Electronics 43 (1999) 2185±21892186

utilizing novel `slope and intercept' methods [9] toobtain device parameters, such as the ideality factor n,

the series resistance Rs, the reverse saturation currentIs. For noise measurements, the diode was connectedto a battery in series with a 10 kO resistor to provide a

constant current through the diode. The power spectraldensity of the voltage ¯uctuation across the diode wasthen measured utilizing HP 35665A Dynamic Signal

Analyzer with Ultra Low Noise Preampli®er 5004,EG&G in the frequency range between 0.1 Hz and 100kHz. The forward current level was varied up to 2 mA

by changing the battery voltage. The voltage noise atzero current was subtracted from the measured voltagenoise at each current level to account for the back-ground noise such as the noise from the measurement

set-up, the shot noise of the device, etc. The measuredvoltage noise intensity ®rst increased and then tendedto saturate as we increased the current in most of the

diodes.

3.2. Series resistance

The values of the series resistance were in the rangeof 10 to 20 O, which are somewhat bigger than the

contribution from the substrate which must be lessthan 10 O indicating existence of some extra series re-sistance. Since the spectral density of the ac open-cir-cuit voltage noise of the diode is [3],

SV � SI,jn

�dVF

dIF

ÿ Rs

�2

�SI,srR2s , �10�

where SI,jn is the current noise in the junction includ-

ing the depletion layer with the dynamic resistance�dVF=dIF ÿRs� � nkT=q�IF � Is� and SI,sr is the currentnoise in the series resistance increasing with I 2, thenoise due to series resistance will eventually dominate

with increasing forward current. However the satur-ation of the voltage noise intensity at higher currentsmeans that the noise from the series resistance is not

dominant within the current range of this measure-ment. In some diodes fabricated at a separate run, thevalues of the series resistance were exceptionally large

in the range of 60 to 80 O and the voltage noise inten-sity increased proportional to I 2 indicating that thenoise from the series resistance is dominant in thesame current range in these `bad' diodes. The excep-

tionally large series resistance could be from badohmic contact at the backside or multi-spot contact atthe TiN/n-Si interface due to some unintentional sur-

face treatment. Analysis of these `bad' diodes utilizing,e.g., multi-spot contact model [10] would be interestingbut remains beyond the scope of this work. Since the

noise from the TiN/n-Si junction is dominant in mostof the diodes with saturating voltage noise intensity,the power spectral density of the current noise SI, was

obtained by dividing SV by (dVF/dIFÿRs)2 or [nkT/

q(IF+Is)]2.

3.3. Current noise and comparison with other models

Figure 1 shows typical results for the frequency

Fig. 1. X-axis is the measurement frequency in Hz and y-axis

is the noise current power spectral density, SI, in A2 s, for

three di�erent forward current levels, 2, 1 and 0.7 mA, from

top to bottom for a TiN/n-Si diode.

Fig. 2. Current dependence of the current noise power spec-

tral density, SI at 10 Hz for the same diode as in Fig. 1. The

result of the random walk model with interface states density

of 2.7 � 1011/eV cm2 is indicated as `RWM' (solid line) and

the prediction of the mobility and di�usivity ¯uctuation

model by Luo et al. [5] with a Hooge parameter of

1.7 � 10ÿ9, is indicated as `Luo et al.' (dashed line).

J.I. Lee et al. / Solid-State Electronics 43 (1999) 2185±2189 2187

dependence of SI for di�erent current levels in the fre-quency range from 0.5 to 50 Hz for a diode with

n= 1.18, Rs=16 O and Is=373 mA. The noise spec-trum has in general a 1/f behavior, except at very lowcurrents (below 20 mA) where the frequency depen-

dence becomes 1/f 2 at frequencies below 10 Hz (notshown here). The 1/f dependence was extended tohigher frequencies until it became ¯at in white noise

frequency region (not shown here). The current depen-dence of SI measured at 10 Hz is presented in Fig. 2for the diode of Fig. 1. For the forward current higher

than 0.2 mA the slope (indicated as RWM in solidline) is close to 2 and the noise intensity can beexplained by our random walk model (Eq. (9)) with aninterface states density of Dit � 2:7� 1011=eV cm2. For

the current lower than 0.2 mA, the current noise inten-sity has a linear dependence on the current (indicatedas Luo et al. in dashed line) and can be explained by

the mobility and di�usivity ¯uctuation model [5] witha Hooge parameter of 1.7 � 10ÿ9. The value of theHooge parameter is in good agreement with the theor-

etical value of 3.4 � 10ÿ9 calculated by Kousik et al.[11] and the experimental value of 4.2 � 10ÿ9 obtainedby Luo et al. [5] for a 2.4 � 1016 cmÿ3 n-type Si diode.

The value for the interface states density obtainedfrom above can be somewhat reduced if we considerthe contribution from the bulk traps as in Hsu's model[1]. To compare our random walk model (Eq. 9) to

Hsu's trapping model [1], we calculate the ratio g ofthese two expressions,

g � SIrandomwalk

SIHsu0s trapping� 4Gb�VD ÿ VF� Dit

DTW, �11�

where DT is the density of bulk trap states and DTWrepresents the e�ective areal density of bulk states inthe depletion region. One can see from Eq. (11) that

the noise due to interface traps can dominate the noisedue to bulk traps for DitrDTW. To explain our datafor VDÿVF=0.4 V by Hsu's trapping model only, we

need DT=4 � 1016/eV cm3 which is rather too high tobe justi®ed. In addition, in the study of the depositiontemperature dependence of the characteristics of TiN/

n-Si diodes fabricated via reactive magnetron sputter-ing [12], we observed that both the ideality factor andthe current noise intensity at higher currents increasedwhereas Hooge parameter remained almost constant as

the deposition temperature was reduced from 4008Cdown to rt. Since Hooge parameter is a bulk propertyin the depletion region of Schottky diodes and the ide-

ality factor is related to the interface, one can concludethat the increase in the current noise is due to theincrease in the interface states density via the random

walk model. The deviation of the slope at the higherend (in Fig. 2) can be explained by the energy distri-bution of the interface states density. Since those inter-

face trap states half occupied are most important, weare probing the interface trap states at around the

Fermi level. At higher currents, i.e. as the Fermi levelapproaches toward the conduction band, the interfacestates density increases, which is a well documented

fact.

4. Conclusion

We have presented a random walk model for the

low frequency 1/f noise in Schottky barrier diodes. Therandom walk of electrons via the interface statesmodulates the barrier height and in turn, results in thecurrent ¯uctuation. The model explains well the exper-

imental results on TiN/n-Si Schottky diodes. At highercurrent region where a quadratic current dependenceof the current noise spectral density is observed, the

interface states density is determined utilizing our ran-dom walk model and the variation of the exponent ofthe current is related to the energy distribution of the

interface states density. At lower current region, thenoise current intensity is linear to the current and canbe explained by the mobility and di�usivity ¯uctuation

model [5] with a Hooge parameter in good agreementwith others [5,11]. It is noted that the noise due to therandom walk via interface states can dominate thenoise due to the trapping at bulk traps and the noise

due to the mobility and di�usivity ¯uctuation, in noni-deal diodes with a signi®cant density of interfacestates.

Acknowledgements

J.I.L. appreciated helpful discussions with GerardGhibaudo, technical assistances from Sylvie Mercier,Luc Bouro and Adrian Ionescu and supports from the

Ministry of Science and technology and KOSEF,Korea and CGE, France during his stays at LPCS.C.A.D. is grateful to the Greek General Secreteriat of

Science and Technology for ®nancial support.

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