Resonant and non-resonant tunneling in GaAsAlAs multi quantum well structures

Superlattices

383

Vol. 5, No. 3, 7989

and Microstructures,

RESONANT AND NON-RESONANT TUNNELING IN GaAs/AZAs MULTI QUANTUM WELL STRUCTURES HARALD

SCHNEIDER,

KLAUS

VON KLITZING,

AND KLAUS

Max Planck Institut fiir Festkiirperforschung, Heisenbergstr.

(Received

Transport

experiments

in an electric conditions,

structures

structures

on electrons

field perpendicular

sonant tunneling). time-of-flight

We further investigate

subbands

temperature tions.

and theoretically dependence

At low electric

band conduction a transverse

of the transport

field on the transport

Since the first proposal’ of growing heterojunction semiconductor superlattices (SL’s) to obtain synthetic materials with novel electronic properties a great deal of attention has been devoted to this area. There already exist in the literature many theoretical predictions and experimental manifestations of the fascinating features which arise from the quasitwodimensional band structure of the individual wells and the quantum mechanical coupling between them. For a general review of the electronic transport properties perpendicular to the SL layers, including the historical evolution of this field, we reference the review article by Capasso et al.‘. The systems under study in the present work are tight-binding multi quantum wells(MQW’s) (or weakly coupled SL’s) in an electric field perpendicular to the layers. The undoped MQW is embedded in a p-i-ndiode structure and the field is tuned with an external voltage. The MQW is depleted in the dark and the carrier density and distribution can be controlled by optical excitation at a suitable wavelength.

14 $02.00/O

and present

the transport

both experimentally

These

rates due to re-

photocurrent between

wells

times.

the conduction (photocurrent

We also discuss the

times and present

some model calcula-

effects in connection Finally,

characteristics

with mini-

we study the influence

of

and explain the results

point of view.

1. Introduction

0749-6036/89/030383+

are observed.

determine

spacings

fields we observe transport

mechanical

80, FRG

stationary

approximation).

and field induced localization.

magnetic

from a quantum

the transient

determined (t wo-band

Under

of the transport

data to the energetic

which are independently

spectroscopy)

photocurrent

which allow to directly

We relate these transport

multi quantum

in coupled GaAs/AZAs to the layers are reported.

with an enhancement

experiments

1, D-7000 Stuttgart

12 August 1988)

in the field dependent

are associated

PLOOG

Let us sketch the most important transport mechanisms occurring in such a system. If the barriers of the MQW are thick enough (uncoupled wells) and the electric field is not too high, then tunneling through the barriers can be neglected. Under these conditions, at high enough temperatures, thermionic emission processes are observed combined with a reduction of the effective barrier height due to Fowler-Nordheim tunneling. This situation was studied in a previous work314. Studies of thermionic emission across single Al,Gal_,As barriers have been reported by Solomon et al.s. We will focus here on tunneling processes which become important for thinner barriers. The tunneling probability is resonantly enhanced if the lowest subband of the well at one side of such a barrier has approximately the same energy as an excited subband of the well at the other side. This situation only occurs at certain electric fields and is hereafter referred to as resonant tunneling(RT). After subsequent relaxation into the lowest subband, the RT process is repeated (sequential RT) if the electric field is homogeneous enough. The first experimental study of sequential RT was carried out by Capasso et a1.6 in the

0 1989 Academic Press Limited

384

Superlattices

AlInAsjGaInAs system. Tunneling processes occurring without this resonance condition fulfilled will be referred to as non-resonant tunneling(NRT). We note that, in the context of tunneling through double barriers, the expression RT is commonly used in a different sense. One of our main results presented here is the direct determination of transit times from electrical time-offlight(TOF) experiments’. A consistent and reliable interpretation of such experiments demands the application of a wider scope of experimental techniques. Therefore we also present static photocurrent-voltage(I-V) measurements as well as photocurrent spectra. Further interesting topics will be the miniband conduction observed at low electric fields and the transport characteristics at high magnetic fields. Our samples are described in the following section. The behaviour of the photocurrent under static conditions is discussed in section 3. Some TOF experiments are presented in section 4. One more section contains the temperature dependence of the transport processes and a discussion of the role of impurities at low temperatures. The subband spacings are obtained via photocurrent spectroscopy in section 6 and theoretically in section 7. Subsequently, we discuss the observed transport times from a theoretical point of view. Finally, we focus on the transport characteristics at low electric (section 9) and high magnetic (section 10) fields and summarize our results.

and Microstructures,

Vol. 5, No. 3, 1989

grown by molecular beam epitaxy on (lOO)-oriented n+-substrates, etched into mesas of 0.04mm2 area and supplied with ohmic CT/Au-contacts at the top and substrate sides. The quoted parameters of the barrier and well widths of the MQW’s are determined from Xray refraction spectras. Capacitance-voltage measurements indicate a residual doping of less than 10’5cm-3. Relevant details of our experimental setups will be given in the corresponding sections. 3.

Static

Photocurrent: Transport Recombination

and

If recombination and avalanche multiplication effects can be neglected, then the photocurrent of a reverse-biased p-i-SL-n-diode is governed by the generation rate and there is no structure in the I-V curves since all the excited carriers are collected. As soon as the recombination time is of the same order or less than the transport time, the situation changes: In this case, a non-monotonic field-dependent behaviour of the transport times (as expected for sequential resonant tunneling) gives rise to structures in the fielddependence of the photocurrent.

2. Experimental A schematic band diagram of our samples is shown in Fig. 1. The undoped MQW’s consist of 50 periods of 12.3nm GaAs/a.lnm AlAs (sample 1) and 14.&m AZAs (sample 2), respectively, and are GaAs/3.4nm sandwiched between AZo,5Gao.sAs window layers. The A~,-,.~GcQ.~Aslayers are n-doped on the substrate side and p-doped on the top side, respectively, both to about 5*10’7cm-3. The A6 content at the edges of the window layers is continuously varied on a length scale of 70nm (graded regions in Fig. 1) in order to avoid additional depletion zones. The whole structures are

p* II (AlGalAs1 I I

I I n’ I I

I

IIA~GoIAsI

MQW

II

I

II

n'-GaAs

CB

3

substrate

pm_________

.‘..

J-L________---_

-9 Figure gram).

1:

Sample

structure

r(schematic

band

dia-

-25

-20

-15

-10

Applied

Voltage

( VI

-5

0

Figure 2: Voltage dependence of the photocurrent obtained from sample 2 under static 670nm illumination at different temperatures.

To illustrate this we plot some I-V-curves obtained from sample 2 at different temperatures (Fig. 2) using a HP4145 parameter analyzer. A similar measurement can be found in ref. 9. The optical excitation is chosen well above the band gap so that the field dependence of the absorption’OJ1 can be neglected. The peaks observed at an applied voltage of U = -2.6V, -8V and -15V (the negative sign of U denoting reverse direction) are associated with RT between the lowest (el) and the first three excited (ez, es, e4) electronic subbands, respectively. We now discuss the relevant recombination times. Due to the reduction of the overlap between the electron and hole ground states, the recombination times become larger with increasing electric field. At T =lOK, field dependent photoluminescence lifetimes

Superlattices

and Microstructure.%

from about 0.3 to 1OOns were reported” for quantum, wells with well widths similar to our case. If the electric field is large enough, then the decay of the photoluminescence is dominated by transport processes resulting in a reduction of the observed photoluminescence lifetimes. At increased temperatures, the recombination lifetime is influenced by the dissociation of excitons and by non-radiative recombination processes13. The transport times will be studied in the following section. The temperature dependence of the I-V-curves in Fig. 2 between a bias of U = -16V and +1.5V is as follows: From T =lOK to T =130K there is a decrease of the photocurrent for increasing temperature. Since the transport times, as shown later, are only weakly temperature dependent in this regime, we conclude that this behaviour is caused by the above mentioned decrease of the recombination lifetime due to non-radiative processes. Simultaneously, the RT-peaks become more and more clearly pronounced. For 130K< T <290K the photocurrent increases with temperature, especially for the field regions where no RT is expected. We note that some additional structures evolve: The peak at about U = -4.lV (data for T =220K) which can be observed for T Ll5OK is associated with RT from the thermally populated second subband ez into the third subband es. Similarly, the shoulder-like structure at U w -9.5V is due to RT from ez into ed. Further evidence that thermal population of ez is important in this temperature regime will be given later. The increase of the photocurrent at very high electric fields (V < -16V) is caused by avalanche multiplication’4. The probability for avalanche muItiplication decreases for increasing temperature. The electric field for avalanche break-through is also influenced by this effect although the dark current (
-

X = 632.6 nm 10-e

-25

-aI

-15 Applmd

~t 6ul -10

Voltage

385

Vol. 5, No. 3, 7989

-! IV)

Figure 3: Photocurrent-voltage curves (sample 2, static 632.8nm illumination, T =lOK) for different illumination intensities between lmW/cm’ and 5W/cm*.

Let us discuss the intensity dependence occurring at lower fields. Under extremely weak illumination the curves are nearly featureless, and we conclude that the recombination lifetime at T =lOK exceeds the transport times (for U 5 -2V). Thus the efficiency for carrier collection at the contacts is of the order of 1. Increasing the illuminationintensity results in an increase of the recombination rates due to the higher carrier density. Hence, RT gives rise to peaks in the I-V curves. The behaviour of the luminescence in this regime, as studied by Mssumoto et al.l5 and Tarucha et al.“~‘, shows that local maxima in the field dependence of the photocurrent actually correspond to local minima in the luminescence intensity. At high illumination intensity, the RT structures are broadened due to field inhomogeneities arising from space charge e&&s. For very high carrier densities, the local electric field is given in a self-consistent manner by the space charges and the resonant nature of the tunneling process. These effects lead to a formation of several domains along the SL, each of which corresponding to an electric field characteristic for RT. In the data of Fig. 3 we find peaks in the I-V-curves at certain applied voltages where just one domain exists as well as some additional iine structure (e. g., between V = -0.5V and V =+lV) which arises from the motion of the domain boundaries by integer multiples of the SL period. The latter effect was tist observed by Esaki et al.” in an n-doped SL. More recently, such samples were used as detectors for infrared radiation in the range of the energetic subband spacings”. We note that the RT peaks at higher temperatures (e. g., T=80K) are already observed at arbitrarily low illumination intensities. The resonances are less pronounced for samples with thinner barriers since, due to the shorter transport times, the recombination effects play a less important role. 4. Time-of-Flight

Experiment

While static photoconduction measurements only allow one to draw crude conclusions about the actual carrier drift velocities we will now directly study the transport times via TOF experiments. We sketch here the wavelength dependence of the shape of the photocurrent transients under pulsed illumination. In the ideal case, the penetration depth of the exciting light is so small that only a sheet of carriers is generated close to the top contact. While the holes immediately reach the contact, the electrons cross the whole intrinsic region giving rise to a constant photocurrent if their drift velocity is constant. Due to a diffusion-like spatial broadening of the electron population arriving at the opposite contact, a Gaussian transport model predicts an Error-function like decay of the signal’ (TOF edge). In reality there is always a finite penetration depth of the exciting light. Choosing a wavelength which is too short causes additional current components to be created since almost all the light is absorbed in the

Superlattices and Microstructures, Vol. 5, No. 3, 1989

386

\ ’ 10-4; -., -

‘d 5

,,/,#I.,

lo-5’5...___

20

\

.A__\\ “0 ‘.\_

-.

‘.\

-.._

30

time ins) Figure 4: Time dependence of the photocurrent (sample 1, pulsed 53Onm illumination, 2’ =lOOK) for different applied voltages.

3v

av : -1ov .\\\

.\

-..-. -. ‘.+ .---.qJ,

Sample 2

\ --\

\

\

:

10 d-

:

\ ‘: \ ‘,, \\ ,, \,,i

/

quantum dfects.

For Vz&- 3.6V the transients

become

shorter with increasing field, for -3.SV&U&-SV they become longer. At even higher fields the transport time decreases again. The local minimum of the transport time at -3.6V is associated with RT from el to e2. For a plot of the field dependence of the transport times we refer to Fig. 13 in section 9. The second resonance (el+e2) occurring at U =llV is also visible in Fig. 13. Fig. 5a shows TOF experiments performed with sample 2. Since there is some interaction between the transport and the recombination times (giving rise to the RT peaks in Fig. 2 as mentioned above), the pho-

! i

10-G

time (set)

,““““‘,““““‘I”“““‘j

/‘\

--l4V ---___,5V _______,

contact layers and not at the beginning of the SL. Illumination near the SL band edge, on the other hand, results in a quasi-homogeneous excitation giving rise to enhanced recombination effects and less effective suppression of the hole current. Consequently our measurements are mostly carried out using a wavelength depth” of of 530nm corresponding to a penetration 125nm in the SL and 300nm in the Alo.sGw.sAs. The light pulses are generated by a nitrogen-pumped dye laser system (500 ps pulse width, 386-QQOnm wavelength, 5Hc repetition rate). The time-resolved photocurrent is detected by a fast transient recorder (500MRz bandwidth). We avoid space charge effects by working with low mean carrier densities (< 10’4cm-3 per pulse). A selection of photocurrent transients obtained from sample 1 at T =lOOK are shown in Fig. 4. As long as the transport times are not too short the observed shapes of the transients are well described by the Gaussian transport model mentioned at the beginning of thir section (e. g., data for -2V and -8V in Fig.4). Here the observed diffusion broadening of the electron sheet can be directly traced back to the transition rates for the carrier motion from one well to the next. Such a model, including computer simulations, has been presented elsewhere3v4 in a different context (thermionic emission). The transport times are strongly influenced by

_

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-----y?:F;7-.c.‘y .....

10-7

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ki 0” z r a

0

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I

T= 100K h=530nm

+jV

------li3V

0 (b)

Time

10 ( ns1

20

30

Figure Sa,b: Time dependence of the photocurrent (sample 2) (a) at T=lOOK and X=530nm, (b) at T=lOK and X=670nm, respectively, for different applied voltages.

tocurrent decays by up to one order of magnitude before the TOF edge is reached. Hence, the shape of the transients is more complicated than predicted by a Gaussian transport model. Nevertheless we clearly observe the non monotonic field dependent behaviour of the transport times as expected from Fig. 2. At U = -8V the transport time is about half as long as at U = -lOV and also decreases when changing the bias from -4V to -3V. The expected increase of the transport time at electric fields below the first resonance (U > -2.8V) could not be observed because of the mentioned recombination effects. The resonance between el and e4 is not so pronounced because the energy of e4 is already far above the conduction band edge of the GaAs but it still gives rise to a local minimum in the transport time at U X-15V as shown in Fig. 5b. This measurement was carried out at 10K in order to see the effect as clearly as possible. The additional slow component of the photocurrent occurring at low temperatures is probably related to impurities (see also section 5).

Superlattices

and Microstructures,

5. Temperature

Dependence Times

of the Transport

Strongly temperature dependent transport mechanisms become important at higher temperatures. Thermionic emission processes should not be observable due to the large conduction band discontinuity at the r-point. Thermal population of higher subbands, however, can lead to additional transport channels via tunneling. The temperature dependence of the transport time for the case of Sample 2 is shown in Fig. 6. The transport times show a saturation behaviour towards lower temperatures which is associated with nearly temperature independent tunneling processes out of the lowest subband. If the temperature dependence at higher temperatures is due to a thermally activated transport channel, then the transport time 7 must be described in terms of an effective tunneling time 7tun and an activation energy EA: 1 -=T

1 Ttun

387

Vol. 5, No. 3, 1989

0

10

20

Limeins/

+ a exp( $9.

The full lines in Fig. 6 are obtained from a numerical fit of eq. (1) to the measured data. The time constants ~~~~ determined in this way can approximately be read off at the right hand side in Fig. 6. They show a clearly non monotonic field dependence associated with RT. These anomalies are smeared out for higher temperatures due to the non-resonant behaviour of the temperature dependent component. We find activation energies Ea between ‘72meV and 86meV in the range from U = -3V to U = -lOV and 63meV for U = -13V. The energetic subband spacing between el and ez for these bias values varies between 63meV and 70meV (see section 6) and we therefore conclude that this thermally activated transport process is based on a thermal population of the second subband e2.

i----

/

0005 l/Temperature

I

001 (l/K)

Figure 6: Temperature dependence of the transport times (sample 2, pulsed 530nm illumination) for different applied voltages as indicated. The full lines are results of a model which is described in the text.

0

10

20

30

40

time ins1 Figure 7a,b: Time dependence of the photocurrent (sample 1, pulsed 530nm illumination) at (a) T =300K and (b) T =lOK for different applied voltages.

The situation is completely different for sample 1. Fig. 7a shows photocurrent transients at 300K. The resonance effects are still very pronounced: The transport time increases by a factor of 2 when the bias is changed from -3.6V (resonance) to -7V. Comparing Fig. 7a with Fig. 4 we note that the transport times (in the field range for NRT) vary by only a factor of 2-2.5 between 1OOK and 300K. Concerning eq. (1) we conclude that the temperature dependent component has a value similar to the former case and does not play a significant role since 7tunis shorter by about two orders of magnitude. The transport characteristics (sample 1) at 10K are plotted in Fig. 7b. The signal consists partly of a fast component which differs from the data at 100K (Fig. 4) by only 20-30% and also shows the resonance at U = -3.6V. There also exists, however, an additional slow component which is probably related to impurities. The slow process becomes more and more dominant at high electric fields (U 5 -8V) while the photocharge transported in the fast component decreases although there is one more resonance (elves) at U x -1lV. The stationary photocurrent, on the other hand, shows the usual saturation behaviour.

388

Superlattices

and Microstructures,

I,

c’r/‘,

/

1.46

150

1.52

1.54

,

I

I,“,

VT

“32lw

146

,,,

Vol. 5, No. 3, 1989

I/,

32,h

I

ah

156

Energy leV1

l$&y-

Figure 8: Photoluminescence spectra obtained from sample 1 at T =5K, 647nm illumination (4pW power) and different applied voltages.

Samplel,C6153,

OU’ 15

’

16

’

Energy The impurities contained in the SL are studied via photoluminescence measurements. Some spectra at different bias values are plotted in Fig. 8. Field induced quenching of the SL luminescence allows one to correct the data for the substrate luminescence. We note that this quenching is accompanied by an increase of the photocurrent. The splitting of the excitonic luminescence (lines at 1.545eV and 1.548eV) is probably related to impurities”. Such a behaviour can also be caused by monolayer fluctuations21 in layer thicknesses, but photocurrent spectra (see Fig. 9) suggest that these fluctuations are not so important for sample 1. For the voltages applied in Fig. 8 the free exciton line (1.545eV) shifts by about 1meV (Stark shift). The spectra of Fig. 8 reveal two more lines which are associated with the recombination of free carriers with neutral impurities. The line at about 1.525eV is only weakly field dependent while the other one shifts strongly from 1.503eV to 1.482eV. A strongly field dependent shift of extrinsic luminescence lines was also observed for 3.5nm and 5.0nm wide quantum wells in refs. 22 and 23 and explained via impurities located at the interfaces between the wells and the barriers. Under this assumption the energies of the impurity-bound carriers are much more strongly field-dependent than the subband energies. The extrinsic luminescence in refs. 22 and 23 was explained by the recombination of free electrons with neutral carbon acceptors since there was a difference of about 2OmeV from the free exciton line. We will not discuss the types of impurities involved here but emphasize that the SL actually contains impurities with relatively high ionization energies. It is important to note that these impurities, once ionized, should be able to influence electronic transport proces-

ov

20 0 ’’ ’

T= 5 K

’’

17 (eV

’’

’

16

1

Figure 9: Photocurrent spectra (sample 1) at 5K and different applied voltages. The arrows indicate the excitonic transitions as described in the text. The broken lines serve as a guide for the eye.

ses. There are two possible reasons why these effects become dominant at high electric fields. Firstly, the probability for the capture of carriers increases since their probability density is located closer to the interfaces. Secondly, the binding energy of these impurity states apparently increases leading to an enhanced trapping of carriers, especially at low temperatures. 6.

Spectroscopic Determination Subband Spacings

of the

In order to relate these transport data to the band structure of our samples we have measured photocurrent spectra (Figs. 9 and lo), which allow one to directly read off the experimental electronic subband spacings under the electric fields applied in the TOF experiments. The spectra are corrected for the number of incident photons from the monochromator/halogen lamp and the (arbitrary) units are of the order of pA. On the basis of well-known features 24,25 of such spectra and the field dependence of the peak positions and oscillator strengths we assign the peaks to specific excitonic transitions. In Figs. 9 and 10, ijh stands for the excitonic transition between the i-th conduction subband ei and the j-th heavy-hole (jh) valence subband, whereas ijl corresponds to light-hole transitions. The spacing between ei and es is visible from the replicas of up to five hole states (labeled lh, 11, 2ha, 2hb and

Superlattices

and Microstructures,

8001 Sample 2

,

R6108

389

Vol. 5, No. 3, 1989

electric fields. We will not discuss these features here but restrict ourselves to extracting the relevant data concerning the conduction subbands. Neglecting any level dependence of the exciton binding energy 26--28 we find for the spacing Es El between the two lowest electron subbands es and ei 86flmeV (-3.6V external voltage) for sample 1, 64flmeV (-2.8V) and 71k2meV (-8V) for sample 2, respectively. Between es and et we get 205f2meV (-3.6V) for sample 1, 155Ifr2meV (-2.8V) and 167f2meV (-8V) for sample 2, respectively. From the SL period one can now calculate the actual fields necessary for RT and compare them with the expected fields, i. e., by dividing the built-in (o1.55eV) plus the applied potential by the SL width L. Capacitance measurements indicate that the obtained deviation of 15-20% is mainly due to a partial depletion There remains a difference of 5-100/o, of the contacts. however, which must be related to space charge or screening effects.

, T=lOK

600

7. Some

Theory

We check these results using the Schradinger tion in the two-band approximation2g--31: 0

-ML

0 145

15

2Aa

155

16

Energy

(eV)

equa-

1 v

165

17

Figure 10: Photocurrent spectra (sample 2) at 10K and different applied voltages. The arrows indicate the excitonic transitions as described in the text. The broken lines serve as a guide for the eye.

3h), between ei and es (as well as between es and es) from the 2ha, b and 3h replicas. For sample 2 (Fig. 10) some additional fine structure exists which is probably due to monolayer fluctuations of the well width. These fluctuations also lead to certain anomalies in field-dependent static photoluminescence measurementsg. The broadening of this fine structure for increasing electric field leads to some uncertainty concerning the subband spacings at high fields. The 12ha and 12hb transitions were previously observed by Collins et. a1.24 who related these lines to a mixing between the 12h-exciton and an exciton state deduced from the 2s-state of the ill-exciton. In our spectra we can clearly see, to the best of our knowledge for the first time, that the same feature also exists for the excitonic transitions associated with the second electron-subband es. The field-dependence of the ratio between the oscillator strengths of the 22ha and 22hb transitions is similar to the behaviour of the eireplicas. As in ref. 24, the 2ha hole-subband is dominant at low electric fields, whereas transitions related to 2hb become more and more important for increasing

Non-parabolicities are included here via the k: pinteraction which is described by a matrix element A,, between the conduction band and an effective valence band. These have wave functions \E, and g,,, respectively. The potentials V, and V, of the conduction and the valence bands are spatially constant in a bulk semiconductor and differ by the band gap Es. Inserting plane waves (9,, Qj,, N exp(ikz)) into eq. (2) we obtain the momentum dispersion relation in the bulk material E+,(~~Ji+~~2~2).

(3)

An expansion in k2 yields a relation the effective mass m’ 1

between

K,, and

4 7rZ”

z=3Es and the non-parabolicities are given by the higherorder terms in k2. x,” is determined from the effective mass of GaAs (m&,,. = O.O67ms), and, using eq. (4), we obtain m;l[,, to be 0.1367mc from the AlAs bandgap3’ of 3.05eV. Since this is in close agreement with the experimental value32 0.13ms we assume w,, to be material independent. We note that, projecting eq. (2) onto the conduction band, one gets

390

Superlattices

22

.zn .16

P

4

Vol. 5, No. 3, 1989

Finally we note that a two-band formalism yields a significantly better description of the observed subband structure than the usual one-band Hamiltonian. This becomes clear when one considers the analytically solvable case of a particle in a box with infinitely high barriers. Making use of eq. (4) we get the relation

26

.a

5‘ -5

and Microstructures,

.16 .I,

S, =

.1e

Ei

i = 1,2, . . .

(6)

.m

between-the “two-band” eneryg E, and the “one-band” energy E, of the i-th subband34. Clearly these two models deviate increasingly for higher subband energies.

.06

.wi .04

.o*

8.

0. 0

50

Electric

loo

Field

150

(kV/cm)

Figure 11: Calculated electronic subband energies for a system of four 11&m GaAs-wells separated by three 1.6nm AlAs-barriers as a function of the electric field (two-band model). The energy is measured from the GoAs-conduction band edge and the spatial origin is in the middle of the second well. The arrows indicate the most important resonances.

2 h2 d 4 xc” ---2 dr 3 E - V,(z)

d+

Discussion

of the

Transport

Processes

200

(5)

which can be interpreted in terms of a one-band Schrodinger equation with an energy-dependent effective mass. Eq. (2) is numerically solved with the parameters of our SL’s. We model this by a system of four wells separated by three barriers in an electric field F. The whole system is confined in a “big” well. Using the band gaps of the host materials and a 65/35 rule for the band edge discontinuities our calculations reproduce the observed subband spacings to within less than 4%. An example of the numerical calculations is shown in Fig. 11. Here slightly different parameters (11.8nm wells, 1.6nm barriers) were taken in order to show the anti-crossing behaviour occurring at the resonances. We briefly discuss here the most important features. Three groups, each of four delocalized states, corresponding to the first three conduction subbands, are visible at the electric field F =O. The four states of each subband become localized with increasing electric fields and suffer shifts proportional to the field (Stark ladder formation). This effect was recently observed experimentally by Mendez et al.33 in a 30A/35AGaAsfAlo,35Gao,asAs-SL. The anti-crossing behaviour occurs at higher fields when different subbands of adjacent wells are at about the same energy and is closely related to the RT process since the corresponding wave functions become delocalized over these two wells. The field dependence of the subband spacings arises due to the Stark shift of the subband energies in the individual wells.

We have the following quantum mechanical picture of the RT process. At fields where the above mentioned anti-crossing between different subbands of adjacent wells occurs there are two states with slightly different energies delocalized over the two wells. An electron in this energy range which is localized in the first well must be described as a linear combination of these two states. It is clear from the time-dependent Schrijdinger equation that their energy difference (level repulsion) AE leads to a difference in the time evolution of the phase factors for these two states. In the most extreme case the wave function oscillates between the two wells with the frequency AE/2xh. The transport of carriers is then due to an energetic relaxation into the lower subbands of one of the wells from which tunneling backwards is forbidden. If there is also a resonance between this well and the following one, then the RT process is repeated (sequential RT). Assuming a barrier height of 1eV at the Iminimum, the anti-crossing causes a level repulsion of about 1.2meV (sample 1) and O.lmeV (sample 2) for the resonance between el and ez. These values are so small that RT across one of the barriers should only occur within very narrow field ranges. Hence, only a part of the MQW can be in resonance because of slight inhomogeneities of the electric field along the SL axis due to background impurities. The width of the peaks corresponding to RT observed in the experiment (see Figs. 2 and 13), can therefore be explained assuming a residual doping of zz 10’5cm-3. We note that approximately the same values were obtained from capacitancevoltage measurements34 which were performed under forward bias. Another mechanism for smearing out the field dependent RT features involves energy and momentum relaxing scattering processes2,6x35. The full width AF of the field depenat half-maximum(FWHM) dent photocurrent is approximately given by35 AF x h/edTl, where d is the period of the SL and 711 the relaxation time of the momentum parallel to the layers. Hence, our data could also be explained, although not conclusively, by assuming 711 ZlOOfs. A similar value was obtained from RT in the AlInAsjGaInAs system’.

Superlattices

and Microstructures,

391

Vol. 5, No, 3, 1989

From this discussion it is clear that the observed structures in the field dependent transport characteristics arise from a combination of RT and the nonresonant tunneling(NRT) process. Charge transport due to sequential RT is presumed to happen relatively fast with the quantum mechanical limit given by xh/AE which, in our case, amounts to about 1.7~s (sample 1) and 20~s (sample 2) per SL period, respectively. These values are already of the same order as the limiting time constant of 5-10~s for inter-subband Charge transport due to NRT should occur much slower than for the RT case since it is forbidden in the sense that the carriers have to tunnel into one of the energy gaps between the subbands of the adjacent well. The transmission probability !I’of a particle of energy E through a barrier of the potential shape V(z) can be approximated by the WKB-expression37

-1 y=

E~T~ljc?Xp(-&)

PO)

3

(

The coupling between the two individual wells can now be described using the fact that a particle in the jth state has probability rj to be found in the right-hand well. Hence, our model yields the “transport probability” for the motion towards the right Zl II,

= C p,rj j

= TkBT C TjJj Up(-&). 3

(11)

B

In a real MQW there is also a finite probability for carrier transport against the field direction. This effect is taken into account by assuming that sn electron is located in the right-hand well. Similarly we define the transport probability II, and the normalization factor 7 by replacing Jj by Tj (and vice versa) in eqs. (9)-( 11). The total transport probability II is then given by II = II--II+

= (7-i)ksTC~jJjexP(-~).

(12)

One possible method to estimatess the transmission time relies on dividing the phase velocity ‘up= &ZJG of the particle by twice the well width tu:.

’

Multiplying this “attempt’‘-frequency v by the transmission probability T we actually obtain reasonable NRT-times. The regime between the first (er--+ez) and second (e,-+es) resonances of our samples is correctly described within a factor of 5. This agreement should not be taken too seriously, however, since the above approximations are somewhat questionable. Neglecting the effect of the time evolution of the quantum mechanical phase, we would also expect increasing transport rates since delocalization of the wave functions leads to enhanced transition probabilities. Thus one can describe RT and NRT simultaneously. To consider this in detail we choose a double well (confined in a box) at an electric field F. We first assume a thermal occupation (Boltzmann statistics) of the carrier states in the left-hand well in order to model an electron localized there. The occupation probability p, of the j-th state in the double well is then giveh by

p, =

Tz, -exp(-&)dE

J E,

= +&exp(-&).

Eq. (12) was evaluated with parameters for sample 2 n&g’a one-band Hamiltoman. Our results are shown in Fig. 12. The peaks of II are related to RT. The peak values for RT out of thermally occupied, higher subbands ei (i > 1) of the wells are, to first approximation, thermally activated, with the activation energies being given by the energy spacing between e, and er. We note that, from Fig. 12, a near-resonant alignment between electron states already gives rise to an enhancement of II and that the calculated RT peaks are much narrower (FWHM xO.lkV/cm) than observed in the experiment since we did not account for the inhomogeneities of the electric field.

(9)

Here 1, denotes the integral of the probability density (j-th state) in the left-hand well. The integral rJ in the right-hand well then satisfies the relation rj x 1 - Jj (neglecting the occupation of the barrier region). In eq. (9), the integration takes account of the continuous two-dimensional density of states of each subband. From the condition that pr + pz + . . . = 1, the normalization factor 7 is

-8 250

‘____‘__ am

I,, _, __~_,__. 150

loo

Electric Field (kV/cm)

so

tt-

a00

0

,e

$

r;”

Figure 12: Temperature and field dependence of the transport probability II (see text) using the parameters of sample 2. The resonance ei+eb is indicated by ik.

392

Superlattices

and Microstructures,

We do not attempt to specify a relaxation time constant which, combined with calculated values for II, would allow one to estimate the transport times. Firstly, there are uncertainties related to the occurrence of extremely low occupation probabilities in combination with low energies. The numerical problems aside, it is certainly not realistic to assume thermalisation into “forbidden” states before transport occurs. Secondly, we did not fully take into account the inelastic tunneling processes (phonon emission and absorption). The effect of acoustical and optical phonon emission was studied (under certain approximations) by Tsu and Dijhle?’ and by Palmier et al.4°--42, respectively. When this is accounted for, the transport times will be reduced but the qualitative behaviour should be the same. Finally, transport might also be affected by hot electron effects, lattice defects and monolayer fluctuations”. Consequently, we view Fig. 12 as a qualitative picture of RT and NRT. The zero-field anomaly is related to the auti-crossing between states deduced from the lowest subband el and will be described in detail in the following section.

The mobility decreases at higher fields (eFd > 2A) because the carriers become completely localized and the NRT-process becomes dominant. These localization effects were first studied by Tsu et aL4& in GaAs/Al,Gal_,As-SL’s. Under static illumination, regions of negative differential photoconductivity were observed. Such behaviour also occurs in other materials, e. g., in the AlInAs/GalnAs-system46. The above mentioned experiments only allow one to determine the qualitative behaviour of the transport dynamics since they depend on the interplay between the transport and recombination times. Usual TOF experiments, as described above, are not suitable for studying the dynamical azpccts of these transport processes since the recombination rates are too high. Within the framework of a Gaussian transport model’, however, the photocurrent amplitude I,,, is proportional to the drift velocity VD. Therefore, neglecting contributions from hole transport, we can determine vg (and the transport time 7 = L/vD) if we know the drift length L and the charge Q. of the generated electrons: VjJ

=

Vol. 5, No. 3, 1989

I L”“2.

(15)

80 9.

Miniband

Conduction

At vanishing electric fields, there is a quantummechanical coupling between all the states of the SL, i. e., there are no more subbands but minibands of quasiperiodic (Bloch-)states completely delocalized across the whole SLa. In reality, however, these states tend to become localized within at most a few SL periods due to external perturbations like phonon scattering and SL defects2s43. This gives rise to a coherence length which also depends on the actual value of the energetic width 2A of the miniband (see the discussion in ref. 2). Recent experiments performed by Mendez et al.33 suggest a lower limit of about 5 SL periods for this length in their sample. When one applies an electric field across an (ideal) SL, the wave functions become localized since the translational symmetry of the SL of period d is destroyed. At low fields (eFd < 2A), the states are still delocalized over several SL periods (for large enough coherence lengths). One simple model to describe the transport processes in this regime relies on an analogy with the bulk cas&2*44. The mobility pI for miniband conduction depends on the relaxation time 71 in the following manner

The measured photocurrent amplitude does not depend on the recombination lifetime if the latter appreciably exceeds the total rise time of the experimental setup. This condition is satisfied in our case since the recombination time isl3 >2ns at 80K and our experimental rise time only 700;s. The results of such an experiment7 are shown in Fig. 13. We note that the integrated photocurrent (integrated over the first 50ns after the laser pulse) shows a plateau of Q0=0.5pC between -3V and -18V applied voltage, followed by an increase due to avalanche

APPLIED

where the effective mass m*I at the bottom niband is approximately given by

*

Ii2

mL=zi’

of the mi-

(14)

VOLTAGE

(VI

Figure 13: Peak photocurrent (log. scale) for sample 1 plotted versus applied voltage at 80K. The intensity of the 530nm illumination corresponds to 0.5pC excited photocharge per pulse. The inset shows the peak photocurrent on a linear scale for voltages near the built-in potential.

Superlattices and Microstructures,

Vol. 5, NO. 3, 1989

multiplication at higher fields. Hence, Qs is the generated photocharge. Making use of eq. (15), we find good agreement between the amplitude measurements (Fig. 13) and the TOF results (Fig. 4). We note that the local maximum of the peak photocurrent around U = -1lV is caused by the second resonance (ei-res). The plateaulike behaviour at high fields is due to the fact that the transport time becomes too short for the experimental time resolution. In this regime, the measurement of the amplitude is qualitatively equivalent to the stationary experiment. At lower electric fields, we observe a decrease of the peak photocurrent with increasing electric fields (range between U =+1.3V and U =OV) caused by fieldinduced localization. Around F =O, the amplitude is almost linearly field dependent (between U =1.65V and U =1.45V, inset of Fig. 13). We find a low-field mobility (p = vo/F) of about 1.6cm2/Vs. This value is about 20 times as large as in the NRT regime and 3 times as high as at the resonance ei+ez. The width 2A of the miniband is estimated from the width of the mobility regime39~45 and amounts to 2-3meV which is in a reasonable agreement with our calculated value (xl.rlmeV). We note that the relaxation time approximation (eqs. (13) and (14)) breaks down due to the low experimental mobility since the relaxation time would give rise to a large collision broadening*. This is partly due to the fact that hsT >> 2A which means that not only the states near the bottom of the miniband are involved. The other reason is due to the finite coherence length. Consequently it seems that the observed mobility cannot be deduced from the miniband effective mass. The observed localization phenomena can qualitatively be understood in terms of resonantly enhanced transition probabilities, as already contained in the two-well model (Fig. 11). As in the RT case, inelastic tunneling processes should also be taken into account in order to achieve quantitative agreement. Hence, at least from the experimental point of view, it looks as if there are no fundamental differences between RT and miniband conduction. Carrier mobilities, as calculated for LO-phonon scattering, can be found in refs. 40-42. A direct comparison with these values was not possible, however, since these calculations refer to A10,3G~.7As barriers. A more recent theory of miniband conductivity takes account of impurity scattering47.

10.

RT in a Transverse

Magnetic

Field

A magnetic field B=(B,O,O) parallel to the transport direction z has relatively little influence on the carrier motion. This is due to the fact that tunneling between states belonging to different Landau levels is forbidden since the momentum parallel to the layers is conserved.

393

I -25

-20

-10

-1s

Appled

Voltage

-5

0

IV)

Figure 14: Photocurrent-voltage curves (sample 2, static 632.8nm illumination, T =45K) for different magnetic fields. Each plot is progressively shifted vertically by log(2).

We will study here the effect of a magnetic field B’ parallel to the layers, i. e., perpendicular to the electric field, on the RT process. To this end, a quartz fiber was used to shine light from a HeNe laser onto the sample which was mounted in a magnet. Static I-V curves for different magnetic fields are plotted in Fig. 14. The relevant observations are as follows: For increasing B, the resonances shift towards higher electric fields. This effect is approximately proportional to B’. An additional struc_ture becomes visible at lower electric fields. Its B dependence is similar as for the other peaks. All the structures become sing magnetic field.

broader

with increa-

Further experiments reveal that, at constant B, these structures also become broadened with increasing temperature. We give here a theoretical explanation for these results. The Hamiltonian is well known48--50 in the Landau gauge

Here we have defined the center coordinate to be z0 = hk,/eB and the cyclotron frequency to be w, = eB/m*. Eq. (16) is yielded by the following assumed form for the wave function @(z,y,z) = 9!(z)exp(-iL,y -,ik,t). The wave number k, enters into eq. (16) via z. while kz gives a trivial free dispersion. For vanishing electric field, the energy spectrum of a SL is periodic in z. (or k,), with the period of zo given by the SL period d. We note that the value k, = 0 is not a special one. However, it is often convenient to define the z-origin such that k, = 0 yields the lowest possible energy. Numerical calculations of these energy states

394

Superlattices

AE =

($+ +d + zo,)*

(19)

AE

2kBT -

-=2B ed

Figure 15: Total potential for B = 0 (left) and B > 0 (right). The situation corresponds to RT between (a) er+es and (b) er+ei, respectively, for B > 0.

were performed for MQW’S~~~“,~~ and for single wells4s. Other experimental techniques to study these features include cyclotron resonance4s,52,53, reflectance54, photoluminescences and excitation spectroscopysl. The observed shift of the resonances towards higher electric fields is easily understood from eq. (16): The magnetic potential m*wz(z - rs)r/2 acts in shifting towards higher energies the unoccupied states of the adjacent wells, as seen by a carrier at the ground state of a well. Classically, the shift is due to the Lorents force which transforms part of the electrostatic energy into kinetic energy. Hence, in order to be at resonance, the electric field at B > 0 must be higher than at B = 0 (Fig. 15a). The relaxation process now occurs not only between subbands but also by shifting zs towards the energetic minimum (momentum relaxation). A perturbative treatment of eq. (16) yields the shift AF of the resonances:

(17)

This value agrees with our data within the experimental uncertainty of about 20-30%. For the situation of Fig. 15b, we expect an additional resonance ei+ei at the electric field given by eq. (17). The position of the above-mentioned additional peak actually agrees with the expected position if we take into account the field dependence of the nonresonant current and a built-in voltage of zz 1.5V. Finally we relate the broadening of the RT peaks to the distribution of the carriers into the individual states. Defining the ground state (with respect to k,) of a well by k, = 0, we estimate the region of occupied states by the condition ti2k:/2m* 5 ksT or, in terms of 20, by

(18) We then evaluate the magnetic potential at the next well (first order perturbation theory). In this way, the I, interval of eq. (18) corresponds to the energy interval (for zs,,, 5 d)

/-- m’

up to which a tunneling resonance is smeared out by the magnetic field. This value has the correct magnitude to explain the observed adening of the RT peaks. 11.

= $w;d.

+ d - zo,,$)

Note that, due to the definition of the ground state, the expectation value (2) of the position operator vanishes. This energy regime AE corresponds to the field interval

(al

(b)

AF

- $w:(z

= 2m*wZzOm.

4

%%

Vol. 5, No. 3, 1989

and Microstructures,

Summary

d-field

induced

bro-

and Conclusions

We have investigated resonant and non-resonant tunneling processes occurring in coupled GaAsIAIAs MQW systems in an electric field perpendicular to the layers. We have shown that the resulting drift velocities and mobilities can be accurately determined from electrical time-of-flight experiments. The measured field dependence of the transport times is consistent with the behaviour of the photocurrent under static conditions, when the collection efficiency of the contacts is controlled by the field dependent ratio between the transport time and the recombination time. The lowfield transport characteristics are related to miniband conduction and field-induced localization. The RT and NRT processes were also discussed from a theoretical point of view, including some model calculations. The energy spacings between the conduction subbands were determined from photocurrent spectra and agree well with numerical simulations. The subband spacings are consistent with the electric fields for which RT is observed. Applying a transverse magnetic field, the tunneling resonances shift towards higher electric fields. This shift and a few other experimental observations are explained quantum mechanically. Acknowledgements - The authors would like to thank S. Bending and S. Tarucha for valuable discussions, to M. Hauser for sample growth and to J. Collet for critical reading of the manuscript. Part of this work was sponsored by the Bundesministerium fiir Forschung und Technologie and by the Stiftung Volkswagenwerk. References l)L. Esaki and R. Tsu, IBM J. Res. Develop. 14, 61 (1970). 2)F. Capasso, K. Mohammed, and A. Y. Cho, IEEE J. Quantum Electron. 22, 1853 (1986). 3)H. Schneider, K. v. Klitzing, and K. Ploog, Journal de Physique 48, Colloque C5-431 (1987).

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