Measurements of non-Gaussian noise in quantum wells

A. Ben Simon,1,2 Y. Paltiel,1 G. Jung,2 V. Berger,3 and H. Schneider4

1Solid State Physics Group, Electro-Optics Division, Soreq NRC, Yavne 81800, Israel2Department of Physics, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

3Matériaux et Phénomènes Quantiques, Université Paris 7, Case 7021, 2 Place Jussieu, 75251 Paris, France4Institute of Ion-Beam Physics and Materials Research, Forschungszentrum Dresden Rossendorf, P.O. Box 510119, D-01314 Dresden,

GermanyReceived 30 May 2007; revised manuscript received 5 August 2007; published 11 December 2007

Gaussian generation-recombination is accepted to be a dominant mechanism of current noise source inquantum well systems biased by electric field normal to the layers. We have found pronouncedly non-Gaussianexcess current noise in n-type and p-type multiple quantum wells. The non-Gaussian noise has been attributedto metastable spatial configurations of electric field. The metastability likely originates from negative differ-ential conductance caused by intervalley scattering in n-type wells and heavy and light holes tunneling inp-type wells. At a constant bias, the quantum well system randomly switches between a high resistivity statewith low current flow and low resistive state with high current flow. The non-Gaussianity of the noise is morepronounced in p-type wells where the time traces of current fluctuations resemble closely a two-level randomtelegraph signal, which has not been straightforwardly observed in n-type wells. The non-Gaussian character ofthe noise in n-type systems has been revealed by measurements of nonzero skewness of the amplitude distri-butions. The difference between noise properties of n- and p-type systems has been attributed to small captureprobability of electrons in n-type wells, as opposed to very high capture probability of holes in p-type wells.As a consequence, the noise of any p-type multiwell system is dominated by fluctuations of a single well, whilein the n-type the noise appears as a superposition of many fluctuators associated with individual wells.

DOI: 10.1103/PhysRevB.76.235308 PACS numbers: 73.21.Fg, 72.20.Jv, 72.80.Ey

I. INTRODUCTION

Noise is frequently regarded only as an annoying nuisancedisturbing the experiment. In reality, noise measurements canprovide a unique insight into the dynamics of the investi-gated physical system.1 Noise in quantum wells QWs and,in particular, in quantum well infrared photodetectorsQWIPs was extensively studied in the last years.2–5 Al-though the noise measurements were primarily aimed at apractical goal of optimization of the signal-to-noise ratio inQWIP devices, they have also significantly contributed to ourunderstanding of transport processes in QW systems. It isnow generally accepted that generation-recombination GRnoise constitutes the dominant source of current fluctuationsin QWs.6,7 In practice, QWs are always biased with a voltagesource. At very low voltages, the current noise in QWs arisesfrom trapping and detrapping of charge carriers from boundstates.8 At moderate and low bias voltages, the bias depen-dent GR noise prevails.9–11 At very high bias voltages, 1 / fnoise, associated with the action of electronic defects in QWsystems, appears and becomes most pronounced at low fre-quencies.

A simple relation based on considerations of statisticalfluctuations of the number of charge carriers due to theiremission and capture by the wells connects the dark currentand the power spectral density PSD of the GR noise,4

SiV = 4qIg1 −Pc

2 , 1

where g is the gain, defined as a probability that a charge

carrier reaches the collector, q is the electron charge, I is thedc current in the system, and Pc is the probability of captur-

ing a carrier from the continuum to a QW. In deriving Eq.1, each well was treated as a discrete independent source ofGR noise. Relation 1 was the subject of controversial dis-cussions over the years.2–5

The power spectral density of GR noise has a Lorentzianform. PSD is frequency independent at low frequencies, upto a cutoff frequency located in the GHz range. Above thecutoff, the PSD decays as 1 / f2.12

Recently, fast and slow noise components of the currentfluctuations, manifesting themselves as two plateaus in thePSD of the current noise, were found in quantum wells.13,14

The plateau at higher frequencies originates from a conven-tional GR mechanism, while the low frequency plateau isconsidered to be a signature of an excess noise mechanism.14

Alternatively, the time constant related to the recharging pro-cess of depleted QWIP wells has been claimed to be respon-sible for additional low frequency cutoff in the GR noisespectra.13 The time constant of the recharging process is con-trolled by the QWIP resistance R and capacitance C. In atypical QWIP, the time constant is of the order of 10−4 s.Since the low frequency plateau in the noise PSD appears atfrequencies coinciding with the typical operating frequencyrange of practical QWIPs, the excess noise may significantlydeteriorate the performance of QWIP devices. Therefore, un-derstanding of origins and mechanisms of excess noise be-comes an important issue also from a practical point of view.

Important information about the physical nature of theexcess noise was obtained in our first experiments when wehave determined that in p-type quantum wells the excessnoise has a non-Gaussian character.14 Consistent with thecentral limit theorem, the classical generation-recombinationnoise originating from an action of many elementary fluctua-tors should be characterized by a Gaussian amplitude

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distribution.4,12 Proper characterization of non-Gaussiannoise requires measurements and analysis of higher momentsbeyond standard two-point correlations. Nevertheless, justthe mere appearance of non-Gaussian fluctuations alreadyproves that the excess noise cannot be produced by a com-bined action of many fluctuators.15 The noise generated byan assembly of fluctuators should be Gaussian, even if theelementary fluctuations are not Gaussian. In mesoscopic sys-tems, the non-Gaussian character of the noise is most com-monly related to a limited small number of active fluctuatorsin the system. In larger samples, the non-Gaussianity of thenoise is a signature of an action of a single or just a handfulof elementary macroscopic fluctuators influencing systemproperties on a length scale comparable with the systemsize.15,16 The non-Gausssian behavior of the excess noise in-dicates therefore that a new physical mechanism, beyond thewell known GR noise, dominates current noise in quantumwells.

The work was motivated by our first experiments inwhich non-Gaussian noise in p-type wells was initiallyobserved.14 In this paper, we characterize in detail noiseproperties of both p- and n-type quantum wells and find thepronounced non-Gaussian current at moderate bias voltagesin both types of QW. We interpret the results in terms ofswitching between metastable electric field distributions inthe investigated systems. Even if the non-Gaussian currentnoise was found to be always much more pronounced inp-type wells, our results prove that additional noise sourcebeyond the standard GR noise dominates noise properties ofboth n- and p-type QWIPs at moderate bias.

II. EXPERIMENT

Mesa QWs of n and p type were grown by metal-organicchemical-vapor deposition on 100 semi-insulating GaAssubstrates. Both types of structures have a diameter of200 m and consist of five periods of 4.6 nm GaAs wellsdoped at 51017 cm−3 separated by 50 nm undoped AlGaAs30% Al barriers. The areas under 500 nm width top and bot-tom Ohmic contacts are doped at 21018 cm−3 and are sepa-rated by a 100 nm spacer of undoped GaAs. The GaAs caplayer was grown at 650 °C, while all the other layers wereshown at 750 °C. Vacuum evaporated Ge /Au contacts for ntype and Zn /Au contact for p type were alloyed at 430 °C.

All measurements reported in this paper were performedexclusively at liquid nitrogen temperature, with the samplesimmersed directly in a liquid nitrogen bath. Special care wastaken to eliminate parasitic noise contributions from ambientelectromagnetic fields by extensive shielding and groundingarrangements. The sample was dc voltage biased by a highcapacity battery. The resulting current was delivered to theinput circuit of a homemade transimpedance current ampli-fier placed at the top of the cryostat at room temperature. Theamplified current signal was analyzed in time and frequencydomain by a computer assisted dynamic signal analyzer. Theperformance of the entire measuring system was checked byreplacing a QWIP with a dummy resistor having the sameresistance as the QWIP at a given temperature. The resistorand system noise were subtracted from the total noise foreach voltage and temperature.

III. RESULTS

The measurements reported in this paper were performedwith samples maintained in liquid nitrogen at 77 K. In thiscontext, the term “dark conditions” will mean that thesample is exposed to the radiation of thermal surroundings of77 K. Alternatively, the samples were exposed to the externalblackbody radiation of 300 and 1000 K.

A. Noise spectra

Power spectral density of the dark current noise in p-typeQWs recorded at various positive voltages is shown in Fig.1. The noise spectrum is frequency independent up to aclearly marked cutoff frequency. At frequencies above thecutoff, the PSD of the current noise decays to yet anotherhigh frequency plateau. At intermediate voltages, where theexcess noise is most pronounced, the decay above the cutoffis approximately proportional to f−2.

The bias dependence of dark current noise in n-type QWsis illustrated in Fig. 2. The fastest decay of the PSD is seen at−2.2 V, where PSD decays above the cutoff frequency asf−1.5. In general, the decay rates and differences between thehigh and low frequency plateaus in n-type QWs are signifi-cantly smaller than in p-type QWs.

102 103 104

10-8

10-6

10-4

1 V

1/f2

4 V3.5 V

2.5 V2 V

3V

PSD(nA2 /Hz)

Frequency (Hz)FIG. 1. Power spectral density of current noise in a p-type QW

system under dark conditions for different bias voltages.

102 103 104

10-8

10-7

PSD(nA2 /Hz)

Frequency (Hz)

-3.8V

-1.6V

-2.2V-3V

-3.4V1/f1.5

FIG. 2. Power spectral density of current noise in n-type QWsystem at 77 K.

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B. I-V characteristics

P-type QWIPs are characterized by very high captureprobability and, consequently, by very low gain. In then-type QWIPs, the capture probability is lower than in thep-type devices and, as a result, tunneling is less crucial.Since the n-type QWs have higher gain, practical devices areusually based on n-doped systems.

An example of typical I-V characteristics of a p-type QWsystem under dark conditions i.e., exposed to the back-ground radiation of 77 K is shown in Fig. 3. At high volt-ages, the I-V curves can be well fitted to the exponentialgrowth law,

I = I0 expV , 2

with exponent =2.5 V−1 at the positive voltage branch and=1.8 V−1 at the negative one. At low voltages, the curve isasymmetric and deviates from a purely exponential behavior.Moreover, a clear bump is seen in negative voltages around−1.2 V.

Typical I-V characteristics of an n-type well system underdark conditions and under illumination by 300 K backgroundand by 1000 K blackbody radiation are shown in Fig. 4.Similar to p-type QWs, current in n-type QWs increases ex-ponentially with increasing voltage. Under dark conditions,

the exponential fit to Eq. 2 =1.2 V−1 for negative volt-ages V3.5 V. In the bias range 2 V V3.2 V, the ex-ponent is approximately constant and takes a value of 0.7 V−1. At intermediate voltages, I-V characteristics ofn-type QWs also deviate from a purely exponential behavior.Moreover, in n-type QWs, a pronounced current plateau ap-pears in the I-V curves at intermediate voltages under1000 K blackbody illumination. Deviations from the expo-nential behavior of the I-V characteristics in n-type quantumwells have been previously associated with gain changes,tunneling effects, and appearance of negative differentialresistance.17,18

C. Statistical analysis of current fluctuations in time domain

Already, our first experiments have shown that the lowfrequency excess current noise has a non-Gaussiancharacter.14 For the Gaussian noise, all higher order timecorrelation functions and any of their Fourier relatives arefully determined by two-point correlation and correspondingPSD functions. Therefore, all available information about theprocess is obtainable from the PSD. A proper analysis ofnon-Gaussian fluctuations requires a determination of higherorder statistics. Accordingly, a correct proof of the non-Gaussian character of the noise should come from appropri-ate statistical tests involving measurements of higher mo-ments. However, the non-Gaussian character of the excessnoise in p-type QWs is so pronounced that it can be asserteddirectly from the time records of the fluctuating current, evenwithout performing proper statistical tests.

1. Random telegraph noise

It follows from Figs. 1 and 2 that with increasing bias, thelow frequency noise increases above the expected GR highfrequency plateau level. Nevertheless, at very low voltages,the excess noise in QWs is still very weak and the distribu-tion of the time domain dark current fluctuations appears asGaussian. In p-type wells at intermediate voltages, at whichthe I-V curves strongly deviate from the exponential behav-ior, the distribution of the current fluctuations becomes pro-nouncedly non-Gaussian.14 In this bias range time traces ofthe p-type current noise resemble closely the two-level ran-dom telegraph noise RTN.19 The experimental amplitudedistribution of the current noise can be fitted with two Gauss-ian distributions, each centered at the corresponding RTNlevel, as shown in Fig. 5. Here, the RTN up state correspondsto a high current level, while the down RTN state to a lowcurrent level. Since the sample is biased with a constant volt-age, transitions from high to low current state are, in fact,transitions from low to high resistance state of the QW sys-tem. With further bias increase, the relative difference be-tween the excess noise level and the GR plateau level in themeasured PSDs diminishes and the non-Gaussian characterof the noise becomes less visible. Eventually, at high biasvoltages the noise amplitude distribution returns to be againundistinguishable from the Gaussian one.

The two-level fluctuations of the dark current differ fromthe canonical RTN noise by the fact that in the observed timetraces, the time of transition between the levels t

-3 -2 -1 0 1 2 310-1

101

103

105

Current(nA)

Voltage (V)

FIG. 3. Dark current of a p-type QW system as a function of thebias voltage. The dashed line represents the best fit to Eq. 2.

-6 -3 0 3 610-2

100

102

104

Voltage (V)

(a)

1000K300K77K

Current(nA)

FIG. 4. I-V characteristics of n-type QWs at 77 K illuminatedby 77, 300, and 1000 K radiation.

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0.05 ms is not negligibly short with respect to the lifetimeof the system in each level. Nevertheless, as in the classicalRTN signal, the lifetimes at both levels of the current noisewere found to be Poisson distributed, as shown in Fig. 5b.The average lifetime in each state can be obtained by deter-mining the constant of the exponential decay of the experi-mental time distributions yielding the same result as by av-eraging over the measured lifetimes. For example, thusdetermined average lifetime of the up state at −2.5 V is up=0.28 ms, which is longer than the average lifetime of thedown state dn=0.16 ms. It means that at −2.5 V, the systemstays predominantly in the low resistance up state. The sym-metry of the RTN-like signal changes strongly with bias. Atlow voltages, the system stays predominantly in the highresistance down state while at high voltages, predominantlyin the low resistance up state. The excess noise contributionto the total noise reaches the maximum at voltages for whichup=dn, consistent with the properties of the classicalRTN.19

2. Skewness

The asymmetry of a distribution is characterized by thenormalized skewness i=1

N xi−3 /N3, i.e., the third mo-ment i=1

N xi−3 /N normalized to the third power of thestandard deviation 3. If the distribution is skewed posi-tively, its mean will be larger than its median. The opposite istrue for negative skewness. Gaussian noise has a zero thirdmoment since the existence of the third moment is related tothe breaking of time reversal symmetry. In this sense, mea-

surements of the nonzero skewness are proper indications ofthe non-Gaussian character of the noise.

The bias dependence of the normalized skewness of thecurrent noise in p-type QW is shown in Fig. 6. The changefrom a positive to a negative skewness around V=−2.5 Voccurs at the same bias at which updn and excess noisereaches its peak value.

In n-type QWs, the appearance of a low frequency plateauin the noise PSD has been observed at all illumination levels.However, the non-Gaussian character of the noise could havebeen clearly revealed only under illumination by 1000 Kblackbody radiation. Moreover, in a marked difference to thenoise seen in p-type QWs, no clear RTN-like wave formsappear in the current noise time traces even under 1000 Killumination. The non-Gaussian character of the photocurrentnoise in n-type wells was therefore verified by measurementsof the nonzero skewness shown in Fig. 7. Notice that thestrength of the non-Gaussian character of the noise changeswith changing bias. Similar to the case of p-type QWs, thebias dependence of the noise skewness in n-type QWs ex-hibits one negative and one positive maximum. The zeroskewness at V=−2.8 V in between the peaks corresponds tothe maximum of excess current noise, as it was in p-typeQWs.

-0,5

0,0

0,5

1 2

V= − 2.5V

(a)

Amplitude(nA)

0 0 750 1500Time (ms)

Time (ms)

(b)

Counts

0,0 0,51

10

100

(c)

τdn = 0.16ms τup = 0.28ms

Counts

0 1

(d)

FIG. 5. a Real time records of dark current fluctuations at V=−2.5 V for a p-type QW system. b Amplitude distribution in therecord shown in a. The lines represent the best fit to two overlap-ping Gaussian distributions. Lifetime distribution of the c downstate and d up state of the RTN-like signal from panel a. Thedashed lines are fits to the Poisson distribution.

-4 -3 -2 -1-0,1

0,0

0,1

Volage (V)

Skewness

FIG. 6. Skewness of the amplitude distribution of dark currentnoise in a p-type QWIP as a function of bias voltage.

-5 -4 -3 -2 -1 0

-0,01

0,00

0,01

Volage (V)

Skewness

FIG. 7. Normalized skewness of an n-type QW system illumi-nated by 1000 K blackbody radiation as a function of the bias volt-age. Note that at low and high voltages, the skewness approacheszero, as deviations from the Gaussian character of the noise becomenegligible.

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3. Standard deviation

The possibility of fitting noise amplitude distributions inp-type QWs to two overlapping Gaussian distributions en-ables one to easily determine the standard deviation of thenoise at each RTN-like level. The experimentally determined increases exponentially with increasing bias following thelaw

= 0 expV . 3

The fit of the experimental data to Eq. 3 is shown in Fig. 8.For negative voltages, the fit to the down level noise yields=1.2 V−1. The standard deviation of the noise around theup level grows exponentially with =0.6 V−1 for voltagesbelow −2.5 V and with =1.3 V−1 for negative voltagesabove −2.5 V.

According to Eq. 1, the variance of GR noise is propor-tional to the current. The standard deviation =SiVfshould be therefore proportional to the square root of thecurrent. In the bias range of exponential I-V curves, the valueof is expected to be close to half of the value of . This istrue for low and high voltages. Different values obtained atthe measured voltages confirm that an additional noisemechanism, beyond the standard GR noise, contributes sig-nificantly to the dark current fluctuations in the system

Figure 9 illustrates the bias dependence of the standarddeviation of current noise in the n-type QW system in darkconditions and under illumination by a blackbody radiationof 300 and 1000 K. The standard deviation of the dark cur-rent noise also increases exponentially with increasing bias.For voltages below 1.4 V and above 3.5 V, the dark currentnoise fits Eq. 3 with =0.56 V−1. Consistent with thepredictions for the GR noise, this is close to half of thevalue of the exponent derived from fitting I-V characteris-tics to Eq. 2. However, at intermediate voltages, =0.25 V−1, which is only about one-third of the parameter.Again, the fact that is smaller than the expected GR valueis a clear indication of a deviation from a pure GR Gaussiannoise in the bias range where the excess noise appears. Thisproves that even in dark conditions the excess noise, differ-ent from the GR fluctuations, is present in the system even if

its non-Gaussian character cannot be straightforwardly de-tected by the means employed in our experiments.

Under illumination of the blackbody radiation, the stan-dard deviation of the current noise in n-type QWs increaseswith increasing illumination level and deviates even strongerfrom the predictions of the GR model. For low negative volt-ages of 0.7 V V1.2 V, the under illumination fits Eq.3 with =1.35 V−1. This is about half of the value of=2.77 V−1 derived from fitting the I-V curves measuredunder illumination to Eq. 2. For intermediate voltages cor-responding to the bias range of the current plateau in the I-Vcharacteristics, deviations from the expected GR noise be-havior became evident, and a smaller exponent =0.7 V−1 isobtained.

D. Hysteretic behavior of I-V characteristics

If the appearance of the non-Gaussian excess currentnoise is associated with transitions between metastable resis-tance states in the system, then one should expect an appear-ance of a hysteretic behavior in I-V curves in the bias rangeat which the excess noise is clearly visible. The hysteresisshould be seen for sufficiently fast bias changes, which pre-vent the system from decaying to the equilibrium state. In-deed, such behavior was experimentally observed in n-typeQWs for both positive and negative biases. Figure 10 showsI-V curves recorded under 1000 K blackbody radiation forincreasing and decreasing negative biases. At intermediatevoltages, within the plateau range, the I-V curves show aclear hysteretic behavior which, as expected, disappears athigher and lower voltages.

The hysteretic I-V behavior shown in Fig. 10 is consistentwith a simple model of two metastable conductivity states.At high voltages, the QW system prefers to stay mainly inthe up state, while at low voltages the system is mainly in thedown state. When the sweep starts at high voltages and goestoward lower voltages, the system remains in nonequillib-rium low resistivity, high current up state. Conversely, for theincreasing voltage, the system remains in the high resistivity,

1 2 3 4 510-2

10-1

100

101

Standarddeviation(nA)

Voltage (V)

UpDown

FIG. 8. Standard deviation of the current noise in a p-typeQWIP at the up full symbols and down level empty symbols asa function of voltage. The dashed and dotted lines show the best fitof the data to an exponential growth in Eq. 3.

-4 -3 -2 -1

2

4

6

1000K300K77KS

tandarddeviation(nA)

Voltage (V)FIG. 9. Standard deviation of the current noise in n-type QWs

measured at three illumination levels: 77 dark conditions, 300,and 1000 K. The dashed lines represent the best fits of the darkcurrent to Eq. 3.

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low current down state. With increase of the illuminationlevel, the hysteresis becomes more pronounced. In this case,the metastability and associated non-Gaussian effects be-come stronger.

IV. DISCUSSION

Our experiments show that excess non-Gaussian currentnoise appears in both n- and p-type QWs and that spectralproperties of the excess noise in both types of QWs are simi-lar. In both cases, the PSD plateau at high frequencies rep-resents the true GR noise level, while the low frequencyplateau appears due to additional excess noise processes.This conclusion is supported by the fact that the level of thehigh frequency PSD plateau increases with increasing bias.We have verified that at voltages at which the high frequencyplateau is fully developed within our experimental frequencybandwidth, the level of the GR plateau increases exponen-tially with increasing bias, following SIV=S0 expV,with being close to the value of the exponent obtainedfrom fitting the I-V curves to Eq. 2, exactly as expected forthe GR noise.

I-V measurements have revealed that strong non-Gaussiannoise shows out at voltages at which I-V characteristics de-viate from a purely exponential behavior. Moreover, in thesame voltage range, the value of the experimentally deter-mined exponent in exponential bias dependence of standarddeviation is smaller than the one expected for the GR noisemechanism in both types of QW. We conclude that the addi-tional noise mechanism is responsible for excess noise inboth p- and n-type quantum wells.

A. Metastability of electric field distribution as a sourceof non-Gaussian noise in quantum wells

For both types of wells, the appearance of metastable dis-tributions of electric field seems to be a general source ofexcess non-Gaussian noise. At certain voltages, more than

one potential distribution may possibly appear in the QWsystem. Such multiple potential distributions are responsiblefor the existence of metastable current states in the system.The nonlinear behavior due to tunneling or impact ionizationinduces changes in the gain. Two or more field distributionsmay appear for the same potential difference between thecollector and emitter the same bias, as illustrated schemati-cally in Fig. 11a.

In n-type wells exposed to external radiation, the addi-tional noise can be associated with the processes of interval-ley scattering resulting in two or more possible solutions tothe current continuity equation

-3 -2300

320

340

360

-2.8 -2.4

0.315

0.320

Decreasing voltageIncreasing voltage

Current(nA)

Voltage (V)FIG. 10. Current plateau in the I-V curve of an n-type QW

system measured for negative bias under 1000 K blackbody illumi-nation. The hollow symbols represent data obtained with increasingbias while the filled symbols represent data recorded at decreasingbias. The inset shows a zoom of the I-V range marked with a dashedsquare.

b)

Collector

IinI w

I in

d

IwE2

E1

d

1 QW

E2

"up" state"dn" state

Iw I in I w

With NDC

a)

X Position

QW 1QW 2

Emitter

c)

d)

Iin (1-Pc)

Pc

Single

E2

I in IwIw

No NDC

State

Pc

Pc <<1

Electric field

FIG. 11. a Example of two possible potential distributions in aQW system at the same bias voltage. Random switching betweenthem induces non-Gaussian current noise. X labels the position,QW1 and QW2 indicate two QW barriers, and I is the currentcrossing the two-well system. b Physical meaning of the quantitiesused in the continuity equation Eq. 4 in a one-well system. Elabels the electric fields of the two barriers, I the currents, Pc thewell capture probability, and d the barrier thickness. c In the sys-tem in which current grows monotonically with increasing bias andno NDC regime, there is only a single solution to the continuityequation. d Negative differential conductance can lead to multiplesolutions to the continuity equation in a single quantum well. I-Vwith the negative conductance regime is therefore a sufficient con-dition of appearance of excess non-Gaussian current noise.

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Iin = 1 − PcIin + Iw, 4

where the physical nature of currents listed in the continuityequation is illustrated in Fig. 11b and Pc Pc1 is thewell capture probability. Each solution is linked with a dif-ferent potential distribution in the system, corresponding to adifferent resistance of the system and, consequently, to a dif-ferent current flow in a device. The discussed solutions aremetastable, and random switching between them is the mostprobable mechanism of the observed non-Gaussian excesscurrent noise.

The appearance of the negative differential conductivityNDC regime in the QW I-V curves is a sufficient conditionfor the appearance of non-Gaussian noise in a single quan-tum well. The continuity equation Eq. 4 states that thecurrent injected to the well PIin should be equal to the cur-rent coming outside of the well Iw see Fig. 11b. The totalvoltage drop V across a system composed of a single welland two barriers, each having the same thickness d, will bethe sum of the voltage drop across the first and second bar-riers, respectively, V=dE1+dE2, where E1 and E2 are electricfields in the relevant barrier. Usually, the current in a QW isexponentially proportional to the electric field. For a fixedbias voltage V, the field E2 will grow with increasing Iw,while Iin will decrease because E1=V /d−E2. In the absenceof the NDC regime, the current Iw will increase monotoni-cally with increasing bias voltage and the continuity equationwill have only a single solution, as shown schematically inFig. 11c. In a marked difference, the appearance of theNDC leads to a nonmonotonic behavior of Iw or Iin, which,in turn, allows for two metastable solutions to Eq. 4, asshown schematically in Fig. 11d.

In bulk GaAs, NDC appears as a result of intervalleyscattering between , L, and X conduction energy bands. InQW systems, miniband tunneling20,21 and intervalley scatter-ing in barrier regions can significantly reduce current comingout from the wells at certain voltages. The voltage at whichNDC will appear depends on the bulk properties of the ma-terial from which wells and barriers are fabricated and on thespatial configuration of QWs.18,19 In this case, the source ofcurrent fluctuations can be a transition of the charge carriersto the lower mobility band.22–24 Additionally, under strongillumination, the intervalley scattering in n-type QWs maygenerate electric field domains associated with NDC, whichleads to further enhancement of the non-Gaussian componentof the current noise.25,26

Above a certain threshold impact, ionization27,28 could re-sult in additional two-step-like noise. The ionization processdepletes the wells, changing the voltage distribution to thedash distribution of Fig. 11a, which is the high currentmetastable state. Small fluctuation in the recharging processof one well can increase the charge in that well, changing thevoltage distribution to a state where the voltage is below theimpact ionization threshold. In this state, the current is lowand the voltage distribution follows Fig. 11a solid line.

B. Difference in noise behavior between p- and n-typequantum wells

Due to the high effective mass of charge carriers, the cap-ture probability in p-type QWs is close to unity. As a conse-

quence, fluctuations of current coming out from a single wellcan dominate the current noise of the entire system. Thenon-Gaussianity of the noise in p-type QWs is therefore dueto a dominating single elementary fluctuator associated witha single well, and RTN-like fluctuations appear in time tracesof the total current noise. Difference in the rates of tunnelingfor light and heavy holes may cause nonlinear behavior andappearance of metastable solutions to the continuity equationat intermediate voltages. The RTN noise is known to haveLorentzian PSD, which decays as f−2 above the cutoff fre-quency, determined by the average lifetimes in both RTNstates fc=1 /up+1 /dn. This type of spectral behavior can beseen in PSD of the current noise of p-type device at V=2.5 V in Fig. 1. Note that in the scenario of RTN-likejumps between high and low metastable current states, thechange of the electric field distribution is almost instanta-neous, while the current responds within the recharging RCtime constant, as indeed seen in the experiments.

N-type QWs have significantly higher gain than thep-type well, and all QWs of the device contribute to thecurrent noise. Possible non-Gaussian current fluctuations ofindividual wells are incoherently superimposed. As a result,measurable manifestations of non-Gaussian noise in n-typeQWs can be seen only in nonzero skewness of the noiseamplitude distributions recorded under illumination see Fig.7. Moreover, the metastable potential distributions in n-typeQWs can be associated with the NDC regime and strongnonlinearity of I-V curves, which are stronger under illumi-nation.

Skewness, or the normalized third moment, exhibits apositive peak at −2.2 V and a negative one at −3.4 V, and asecondary maximum at −1.8 V. The skewness reaches zeroat V=−2.8 V, but the kurtosis fourth moment of the distri-bution at this bias is markedly different from the Gaussianvalue. The zero skewness voltage coincides with the voltageof the maxima in dynamic resistivity peak and normalizedPSD. Voltage shift at high illumination levels can be attrib-uted to changes in the electric field distribution induced bythe illumination. With increasing illumination temperature,the electric field in the barriers close to the emitter becomeshigher, while that in the barriers closer to the collectordecreases.25 Thus, an extra voltage is needed for the wellscloser to the collector to contribute to the fluctuations.

Modifications of I-V curves induced by external illumina-tion, illustrated previously in Fig. 4, can be better seen in theplots of voltage dependence of the differential resistance =dV /dI shown in Fig. 12a. The nonmonotonic behavior of seen at all illumination levels becomes significantly en-hanced with increasing illumination level. Moreover, volt-ages at which the maxima of differential resistance appear,change with changing illumination level. At 77 K, the maxi-mum is located at −2.2 V, while for 300 and 1000 K illumi-nation, the peak shifts up to −2.8 V.

Figure 12b shows the voltage dependence of the level ofthe low frequency plateau in current noise PSD normalizedto the square of the dc current, Si=SI / I2. The maximum ofthe noise coincides here with the center of the current plateauin I-V curves, clearly indicating that additional noise sourcesare active in the intermediate bias range. Bias dependence ofthe differential resistance provides yet another connection

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between the excess noise and the NDC regime. The voltageat which the normalized PSD reaches the maximum coin-cides with the voltage at which differential resistivity inn-type QW system goes through a maximum.

Current plateaus and the differential resistivity maxima,which can be related to intervalley scattering,20 are morepronounced at negative bias voltages. The asymmetry be-tween positive and negative voltages may be attributed todifferences between barriers of the emitter and collector or toasymmetric doping.21

The gain in n-type QWs calculated using Eq. 1 for darkconditions and for the 300 K illumination case is plotted inFig. 13 as a function of bias voltage. The character of thedependence is very similar to that of Fig. 12. One can there-fore relate low frequency noise to the nonlinear behavior ofthe gain, which is consistent with the intervalley scatteringscenario.20,21 Under illumination conditions, the shape of thebias dependence of the gain in Fig. 13 resembles the shape ofthe bias dependence of the standard deviation in Fig. 9. Thisadditionally supports the conclusion that intervalley scatter-ing is responsible for non-Gaussian noise in n-type QWs.

V. CONCLUSIONS

We have observed non-Gaussian noise components inboth n-type and p-type quantum wells. The non-Gaussiancharacter of the noise is significantly more pronounced inp-type wells where clear random telegraphlike fluctuationsappear in time domain records of current fluctuations. Inn-type wells, non-Gaussianity of the noise has been demon-

strated by measurements of the nonzero third moment. Weattribute different degrees of non-Gaussian character of thenoise to different capture probabilities in both types of wells.The noise can be related to a nonlinear behavior of the gain,resulting in metastable potential distributions. In the p type,the nonlinearity can be attributed to the difference in tunnel-ing rates for light and heavy holes. For n-type wells, theintervalley scattering seems to be the dominant reason for theappearance of the nonlinear gain. In both cases, additionalnon-Gaussian noise can originate from impact ionization.Our five-well sample creates several possible voltage distri-butions. In that way, in order to differentiate between thesuggested mechanisms, experiments on a one-well sampleshould be done.

The appearance of non-Gaussian noise is attributed to twosolutions of the continuity equation in the NDC regime, al-lowing for the existence of two metastable spatial voltagedistributions. Two distinct spatial voltage distributions undera constant external bias voltage correspond to two possiblestates of the system: a high resistivity state with low currentand a low resistivity state with high current. Each state ischaracterized by its specific bias dependent average lifetime.The finite time of transition between the metastable states,which is not negligible with respect to the average lifetimes,is determined by the charging time constant by the capaci-tance and resistance of the QW system.

For practical purposes, at the nonexponential regime ofthe current, where we have negative differential gain, excessnoise could appear. In those regimes, the noise will reducethe signal-to-noise ratio, reducing the device function ability.A change in the operating voltage moving out of the meta-stable area will improve the signal-to-noise ratio.

ACKNOWLEDGMENT

This work was supported by the Israeli Ministry of Sci-ence Tashtit Program 3444.

107

109

1011

-5 -4 -3 -2 -1 0

2x10-10

3x10-10

4x10-10 (b)

Voltage (V)

dV/dI(Ω)

1000K300K77K

S I/I2[1/Hz]

(a)

FIG. 12. a Differential resistance of n-type QWs illuminatedby 77, 300, and 1000 K radiation. b Bias dependence of the nor-malized PSD, Si=SI / I2 at 100 Hz for 77 K background radiation.

-4 -2 0

0,3

0,6

0,9

1,2

-4 -2 0

0,6

1,2

77K

Gain

Voltage (V)

300K

FIG. 13. Gain of n-type QWs in dark conditions and underillumination by 300 K background radiation.

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