Solid State Communications, Vol. 75, No. 10, pp. 825-829, 1990. Printed in Great Britain.

0038-1098/90 $3.00 + .00 Pergamon Press plc

STARK L O C A L I Z A T I O N IN GaAs-GaA1As S U P E R L A T T I C E S U N D E R A LOW ELECTRIC FIELD

P. Tronc and C. Cabanel

Laboratoire d 'Optique Physique, Ecole Supdrieure de Physique et Chimie Industrielles, 10, Rue Vauquelin, 75231 Paris Cedex 05, France

J.F. Palmier

Centre National d 'Etudes des Tdl6communications, 196, Avenue Henri Ravera, 92220 Bagneux, France

and

B. Etienne

Laboratoire de Microstructures et de Microdlectronique, Centre National de la Recherche Scientifique, 196, Avenue Henri Ravera, 92220 Bagneaux, France

(Received 12 March 1990 by M. Balkanski)

We show, from the p hotoluminescence o fa GaAs-Ga0.,sAl035As super- lattice of period 70 A, that the localization of the electrons takes place at low field ( ~ 1 x 104 Vcm ~). Before the localization, the energy of the photoluminescence line does not vary appreciably with the electric field. When the localization begins, the binding energy of the heavy- hole exciton is approximately the same whether the electron wave- function is centered on the same well as the hole or on an adjacent well.

1. I N T R O D U C T I O N

ELECTRIC field induced localization has been pre- dicted by theoretical studies in semiconductors super- lattices [1] and experimentally demonstrated by several authors using photoluminescence [2, 3], photo- conduction [2, 3], electro-absorption [4] or electro- reflectance [5] techniques. Due to the lack of resonant coupling between adjacent quantum wells when the electric field is applied along the growth axis, the spectrum converges towards the evenly spaced "Stark ladder". The peak corresponding to the transition between a hole and an electron with a wavefunction whose maximum is n periods away from the hole state is labelled n or - n , the sign indicating that its energy is respectively greater or smaller than the energy of the peak corresponding to an electron and a hole localized in the same well (peak labelled 0). The energies of the different peaks were plotted versus the electric field F 12~r values of F equal to or greater than 10kVcm [2 4]. D being the period of the superlattice, one expects, in the strong localization regime, i.e. for F sufficiently high, the energy spectrum to be E0 _+ m'FD where k2]. is the eigenenergy of a single quantum well [1]. Two corrections have to be added. Firstly the binding energy of the direct exciton (corresponding to the peak labelled 0) increases when passing from the supcrlattice regime (F = 0) to the high field regime: it varies from 5 meV [6] in a 35-35 A superlattice with

Ga0.~,5 A10.35 As barriers to 13 meV [7] for a heavy-hole exciton in an isolated 35-A quantum well. Moreover one may assume the existence of an indirect exciton with an electron and a hole located in adjacent wells. Secondly the electric field slightly reduces the eigenenergies of electrons and holes once they are localized in an isolated well [8]. The induced reduction of the energy of the transition is less than 1 meV [8] for F equal to a few tens of kV cm

At low electric field the localization is weak for electrons and light holes due to their effective masses. Transitions occur from the bot tom of the electron miniband. The variation in energy of the heavy-hole 0 peak results from two opposing effects: the optical width of the electron miniband decreases with increasing electric field whereas the binding energy of the exciton increases. The photon energies of the laser lines used in photoluminescence [2, 3] and photoconduction [2, 3] experiments were in the range 1.60-1.76eV. These energies were less than the bandgap of the barriers and moreover did not allow electrons to be excited into the second electron miniband. The slope of the 1 heavy-hole peak plot versus the electric field [2, 3] was in reasonable agreement with its expected value, if one assumes that the whole e(v - %.i.) potential was applied to the undoped part (width w) of the device (v and v~,.i, are respectively the external voltage and the

825

826 S U P E R L A T T I C E S U N D E R A LOW ELECTRIC F I E L D Vol. 75, No. 10

i i ( i i I I

=

- 2 - J ) -1 0 +0 .6V

iii - 2 . 2 V

- 2 . 5 V

- 3 V

-3.9V

- 5 V I L I l 1 [

1,65 1.C'7 1.73 1.69 1.71

ENERGY (ev)

Fig. 1. Photoluminescence (PL) spectra, recorded with a H e - N e laser, of a 35-35A GaAs-Ga0.65A10.35As superlattice for various external voltages V applied between the electrodes of the p + - n ÷ junction. The laser beam power density is 0.42 W cm 2.

built-in voltage). The electric field is then F = e ( v - v , ~ . , . ) / w .

2. E X P E R I M E N T

The experiments reported here were performed on an undoped superlattice made by alternation of 100 35-A GaAs layers with 99 35-A Ga065A1035As films clad on the p+ side of the diode by 4300A of undoped GaAs. This heterostructure was in the space-charge region o f a p+n + junction doped to 2 x 101Scm 3 carriers on each side. The width of the undoped region is 1.13/tin and the built-in voltage is estimated to be 1.55 eV. Photoluminescence (PL) measurements were performed at ~ 10K, with the 6328A line o f a H e - N e laser and the 4880 A line of an Ar laser. Both lines have an energy greater than the bandgap of the GaA1As barriers.

3. RESULTS A N D DISCUSSION

The PL intensity decreases very rapidly with increasing external voltage V. Typical spectra are

1.690

A3 A 4

>

>" A o n,, 1.680 lU Z W

< U.I =.

1.67C

A 2 / . _.o. . . . . . .;%. .

~ °'- " - 0

I I

"-4 I J I •

1 1

2i5 F1 F'2 50 75

ELECTRIC FIELD F (kV/cm)

Fig. 2. Recombination energies for the transitions appearing in Fig. 1 vs the electric field defined from the external voltage v by F = 0.881v - 1.55VJ.

displayed in Fig. 1. The peaks are assigned to 0 and - 1 transitions of the heavy-hole excitons. The plot of peak energies versus the electric field defined by F = e(v - %. i . ) /w (Fig. 2) makes it clear that the separ- ation of the 0 and - I peaks takes place in a very narrow interval of Fvariat ion. This allows us to define three regimes with increasing electric field F: before the separation the energy of the unique peak remains constant to within a good approximation; in the separ- ation interval the energy of the 0 peak increases linearly with F; after the separation, the energy of the - 1 peak decreases linearly with F, whereas the energy of the 0 peak increases very slowly. Two particular values F~ and F, of the field can then be defined. They corre- spond to the intersection of the straight lines averaging the three regimes of the 0 peak (Fig. 2). The intersection point A ~ of the F = 0 axis with the straight line corresponding to the - 1 peak provides the recombi- nation energy Efnd of the indirect exciton when the shift in energy between two adjacent wells due to the electric field is removed. Figure 2 shows that E~nd is constant over the range of the field strengths which allow us to separate the 0 and - 1 peaks and is there- fore equal to its value when the electron is localized in one well. A 2 (Fig. 2) provides the recombination energy Ether of the direct exciton corresponding to the field F 2. (It is obviously necessary to specify the field because ED~r increases very slightly with F above F~.) The difference between both energies is of the order of 1 meV, i.e. small when compared to the 13 meV value [7] of the binding energy of an exciton in a single quantum well. An important result can be drawn from above concerning the direct exciton: the heavy hole being localized due to its large effective mass, we can indeed conclude that, for fields around F2, the binding energy of the direct exciton is approximately the same whether the electron is spread over the well of the hole and the two adjacent wells or tightly localized in the

Vol. 75, No. 10 S U P E R L A T T I C E S U N D E R A L O W E L E C T R I C F I E L D 827

Fig. 3. Square of the moduli o f the electron and heavy hole p roposed eigenfunctions centered on the same well at electric field close to F].

same well as the hole. This result allows us to propose the meaning o f the Fj and F2 values of the electric field F. When F < F~, the decrease in the width o f the first electron min iband is compensa ted by the increase in the binding energies of the direct and indirect excitons. At F ~ F~ the electron wavefunct ion is spread over the central well and the two adjacent wells (Fig. 3). When F varies f rom F~ to F2, RD~ remains approxi- mately constant whereas the electron miniband optical width is still decreasing and becomes zero at F = ~ .

We have recorded PL spectra for various values of the power density P of the laser beam. Ft and F2 increase and the slope s o f the - 1 peak energy plot versus F decreases with increasing P. On the other hand the locations of the A0, A ~ and A3 points (Fig. 2) remain the same. To explain these results we suggest that ballistic carriers are generated because the photons energy of our laser beam is larger than the ba ndgap of the GaA1As barriers. These carriers accumula te at both ends of the superlatt ice and set up an electrical field ~ in a direction opposi te to that o f the built-in field. The - 1 peak plot remains straight (Fig. 2), suggesting that, for a fixed value of P, e is p ropor t iona l to F:

e. = - k ( P ) F k ( P ) > 0

The measured slope S o f the - 1 peak plot there- fore is S O [1 - k ( P ) ] , S ° = 0 .7meV ( k V c m ~) being the slope in the case of no ~ field.

Conversely F~ and F, are respectively equal to FI ~ and F~ ~ (their values when ~: is cancelled) divided by 1 - k(P) . The overall effect o f the r, field is therefore to per form an affinity t ransform, the axis of the trans- form being defined by F = 0 and the ratio equal to [1 - k(P)] ' ( F i g . 2).

To verify this model it has to be shown that the products S • F~ and S • F, remain constant when P is varied. The experimental data agree satisfactorily with the model (Table 1). F rom S ° = 0.7 meV (kV cm i ) ] one gets:

FI ~ = 9 . 4 k V c m i and F{ ~ = l l k g c m i.

Both values show that the localization of the electrons takes place at low actual field F ° applied on the super- lattice. The decoupling of the quan tum wells becomes complete when the potential d rop eF~D between two adjacent wells is 8 meV.

The knowledge of the actual field F ° in our experiments allows us to deduce an approx imate value of the electron miniband A from the model o f a super- lattice under an electric field by Bleuse et al. [I] and to check the electron wavefunct ion at F ° = ~ we have proposed above. In this model the superlatt ice eigen- function centered on the q~h well is:

It

where n is expended over all the N wells o f the super- lattice, D being the period and O(z) the envelope eigen- function of the ground state of an isolated quan tum well centered at the origin. When the total potential d rop in the superlattice N e F ° D is larger than the electron miniband, which is the case in our samples for U ~ > 0 . 5 k V c m ~ if A is assumed to be equal to 35 meV, C,,q is given by:

C,,q = J,, ~; (A/2eF°D )

where J,,(x) is the Bessel function of integer index m. We can therefore predict that the ratio p = I ~/Io of the intensities o f the peaks labelled respectively - i and 0 will be maximal at the zeros of Jo(x). Indeed, due to the strong localization of the heavy holes p may be taken as equal to [J l(x)/Jo(x)] 2 and the over lap between the electron and heavy hole wavefunct ions

Table 1. Values o f FI, F2, S, F~ • S, F2 • S.['or various lasers, lines and beam power densities P

Laser Line P F] F2 S (A) ( W c m 2) ( k V c m i) ( k V c m ') ( m e V ) ( k V / c m i) i

Fl" S F ~ . S ( m e V c m i) ( m e V c m t)

H e - N e 6328 0.42 30.8 34.5 0.22 H e - N e 6328 0.09 15.8 18.7 0.40 Ar 4880 1.1 18.8 22.8 0.35

6.77 7.59 6.31 7.50 6.57 7.97

828 S U P E R L A T T I C E S U N D E R A L O W E L E C T R I C F I E L D

o,5 A

i 0% i ~ ~ ~ 5 x

Fig. 4. Ampli tude o f the Bessel functions J,,,(x).

should then be very small for the direct exciton and large for the indirect exciton. Figure 4 shows that the first zero o f J0(x), located at x = 2.4 is close to the maximum o f J~(x), located at x - - 1.8. With the 35meV value o f A, x = 2.4 corresponds to F ° = 10 .4kVcm ~. The second zero o f Jo(x) is then at x = 5.52; this value corresponds to F ° = 3.9 k V c m a field too low to allow separation o f the - 1 and 0 peaks in our experiments. F rom our results (Fig. 2, curve v = - 2 . 2 V which corresponds to F ° = 10 .4kVcm ,) p is approximately equal to 1.2 at x = 2.4. This value o f p is obviously not as large as one would expect f rom the Bleuse's model (Fig. 5). One can wonder whether the 35 meV value assumed for A is correct in view o f the possible discrepancies o f the well and barrier widths and of the p ropor t ion o f aluminium in the barriers. For example, calculations show that A would be 42meV for well and barrier widths equal to 33 A and 26 meV for 37 A [9]. I f the actual value o f A is larger than 35 meV p would pass through a maximum when F ° increases above 10 .4kVcm ~ (Fig. 4). It is obviously not the case (Fig. 1). On the other hand, if a value o f lower than 35meV is proposed, then according to the Bleuse's

400

~200

0 100 t'r' I.IJ Z i,, 0

-100 i ~ ,

0 200 400 600 800 900

z(A)

Fig. 5. Square o f the modulus o f the electron eigen- function at actual electric field F ° = 10 .4kVcm after Bleuse's model [I], with an electron miniband width A equal to 35 meV.

Vol. 75, No. 10

400

300,

A

>~ 2 0 C

(3 10C n- uJ Z " ' C

-100 0

i i i

200 400 600 800 900

z(A)

Fig. 6. Square o f the modulus o f the electron eigen- function at actual field F~ = 9 . 4 k V c m t after Bleuse's model [1], with an electron miniband width A equal to 22 meV.

model, the 1.2 value o f p at F ° = 1 0 . 4 k V c m ' results in A = 22meV. J0 would then be zero at F ° = 6 . 5 k V c m -~. It can be noticed that the photolumi- nescence curve v = - 2 V (Fig. 1), which corresponds to 9.9 kV cm ' exhibits a value o f p which larger than 1.2, making it clear that A is less than 35 meV. Unfor- tunately, it is not possible f rom our measurements to determine directly at which value o f the actual electric field p is maximal for the 0 and - I peaks cannot be separated at F ° less than l0 k V c m ~ approximately. Nevertheless, if one admits the 22 meV value o f A, the Bleuse's theory provides at F ° = F~ = 9 . 4 k V c m J an electron wavefunct ion which is spread over three wells i.e. in good agreement with that we have pro- posed above in Fig. 3 (Fig. 6).

4. C O N C L U S I O N

We have investigated Stark localization in G a A s - GaAIAs superlattices in the low electric field regime. The heavy-hole exciton photoluminescence peak does not undergo appreciable variation in energy before splitting. Fur thermore the splitting takes place in a very narrow interval o f electric field variation. When the localization begins the binding energy of the indirect heavy-hole exciton is very close to that o f the direct exciton. The model o f eigenfunctions proposed by Bleuse et al. [1] for superlattices under electric field fits our results satisfactorily but it has not been possible to deduce a precise value o f the electron miniband width because the necessary electric field strengths are too low to allow separation for the direct and indirect optical transition peaks. A reduction of the electric field applied to the superlattice has been obtained by using a laser beam with photon energies greater than the bandgap width o f the barriers. This effect provides accurate measurements o f the low field localization.

Vol. 75, No. 10 SUPERLATTICES UNDER A LOW ELECTRIC FIELD 829

Acknowledgement - We thank P. Voisin, J.Y. Marzin, H. Le Person and B. Sermage for helpful discussions and E. Umdenstock for technical assis- tance.

REFERENCES

1. J. Bleuse, G. Bastard & P. Voisin, Phys. Rev. Lett. 6, 220 (1988).

2. E.E. Mendez, F. Agullo-Rueda & J.M. Hong, Phys. Rev. Lett. 60, 2426 (1988).

3. F. Agullo-Rueda, E.E. Mendez & J.M. Hong, Phys. Rev. B40, 1357 (1989).

4. J. Bleuse, P. Voisin, M. Allovon & M. Quillec, Appl. Phys. Lett. 53, 2632 (1988).

5. P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert & A. Regreny, Phys. Rev. Lett. 61, 1639 (1988).

6. A. Chomette, B. Lambert, B. Deveaud, F. Clerot, A. Regreny & G. Bastard, Europhys. Lett. 4, 461 (1987).

7. E.S. Koteles & J.Y. Chi, Phys. Rev. B37, 6332 (1988).

8. G. Bastard, E.E. Mendez, L.L. Chang & L. Esaki, Phys. Rev. B28, 3241 (1983).

9. P. Voisin (private communication).