Temperature dependence of the tunable luminescence, absorption and gain spectra of nipi doping superlattices — Theory and comparison with experiment

Superlattices and Microstructures, Vol. 6, No. 3, 1989

351

TEMPERA'K/RE DEPENDENCE OF THE "IUNABLE LUMINESCENCE, A B S O R P I I O N A N D GAIN SPECTRA OF NIPI DOPING SUPERLATTICES -THEORY A N D COMPARISON W I T H E X P E R I M E N T H. J. Beyer, C. Metzner, J. Heitzer, and G. H. D~Shler Institut f'dr Technische Physik, Universitiit Erlangen-NLirnberg E r w i n - R o m m e l - S t r a B e i, D-8520 Erlangen, F.R.G. (Received 15 December 1988)

Luminescence, absorption and optical gain spectra as a function of temperature and excitation level have been calculated for various design parameters of nipi structures. The relevant transition probabilities were numerically evaluated using self-consistently calculated wavefunctions, including those of the lower quasi-continuum subbands. In this way, the competition between reduced population and enhanced overlap of the excited o n e - p a r t i c l e states could be studied in detail, The resulting spectra were compared with experimental data, focussing interest on the transition from full to zero tunability with increasing t e m p e r a t u r e and on optical gain at room temperature.

1. Introduction The doping superlattices (DSLs) considered here consist of uniformly doped nand p-layers (npnp-structure). Due to the space charge induced separation of electrons and holes, the recombination lifetimes of e x c e s s c a r r i e r s in DSLs are enhanced and deviations from thermal equilibrium are quasi-stable, As photoexcited or electrically injected carriers partly compensate the impurity space charge, the subband structure and all associated optical and electrical properties become tunable t. The rea]-space energy band diagrams including electron and hole subbands in the ground state and an excited state are shown in Fig. 1. Radiative transitions contributing to luminescence in DSLs can roughly be divided into two major channels (indicated by arrows in Fig. 1), dominating the luminescence in different t e m p e r a t u r e regimes: The tunneling recombination across the spatially indirect gap, which leads to an energetic shift of the s p e c t r u m with the excitation level at lower temperatures, and the thermally activated vertical transitions with energies close to the bulk band gap, which dominate at higher temperatures. Some time ago the critical temperature, at which the luminescence changes from essentially full to essentially zero tunability, has been predicted by one of us, using a simple analytical estimate 2. Although experimental results clearly confirmed these predictions, a more refined theory of luminescence in DSLs should be based on detailed values for the thermal occupation and the mutual overlap of all the states involved, including those beyond the classical continuum threshold.

0 7 4 9 - 6 0 3 6 / 8 9 / 0 7 0 3 5 1 + 0 6 $02.00/0

In this paper we report the first direct numerical simulation of optical absorption, luminescence and gain spectra for finite temperature. The calculated luminescence spectra are compared with experimental results.

,- E

~

h~:,'. '~

~ t,~o~,:~ I

E;. . . . . . -- .. 2VeKcj.,-...... -

:

~

'

~

DIRECTION OF P E R I O D I C I T Y , z Fig. I. Schematic real-space band diagram of a nJpi doping super/attice for the ground state and an excited state.

© 1989 Academic Press Limited

Superlattices and Microstructures, Vol. 6, No. 3, 1989

352 2. Theory At first, the wavefunctions and energies of all relevant states have been computed self-consistently in the Hartree approximation. As usual, the calculations were done in the framework of the envelope function scheme, where the total wavefunction is written as a product of the bulk Bloch-function at zero waveveetor u~(r) (band index b=el,lh,hh for electrons, light and heavy holes, respectively), a superlattice Bloch function in z-direction, ~ . . ( z ) (subband index i.t=0,1,2...), and a plane wave parallel to the layers. ~b

Irk

(r) : ub(t)'(~b~ (z)'elk,,r. 0 ~'~z

(I)

Valence band mixing was neglected as well as impurity induced potential fluctuations. With the assumption of a strictly parabolic in-plane dispersion the energy of the states in (1) reads:

EVk) : ,~.(k~) : 1 ~: . 2 k~

(+

are expected to become delocalized already at relatively low carrier concentrations as a consequence of the efficient electronic screening of the potential fluctuations and the low effective mass (large Bohr radius) of the electrons in GaAs 4, Therefore, we have treated these states as ordinary CB-subband states. Inside the p-layers, however, the heavy holes were allowed to occupy a "band" of strongly localized aceeptor states, that was crudely modeled by a 8-shaped density of states, located energetically 28 meV (=binding energy E ace of Be acceptors in GaAs) above the local valence band edge. Thus, the acceptor charge density depends on the position z within the p-layers according to the local value of the Fermi-Dirac distribution function. Therefore, we write for the impurity space charge density pimp(z)

nD(z) =

n D for z * n-layer 0 otherwise

hA(Z) =

nA for z c p-layer 0 otherwise

The SL Bloch functions ~ , . _ ( z ) resulted from the iterative solution of the Schr~'b~inger equation

-2m

0z 2 + v ' ° ( z ) + E

b-,b.(k

where E b denotes the bulk band together with Poisson's equation

~2V,C(z ) = ~Z 2

~k (z) : 0, (3) edge

energies,

e E prree(z) + pimp(z) 3 "

(4)

~OEr

The contributions of the free electrons, light and heavy holes to the space charge in (4) can be written as

p-ooCz) : ~ .Y. % Y.n~)(':,'b,")

• I®b.(z)l 2 ,

(~)

b

where N : number of superlattice periods, q~o=quasiFermi level (~pel=~pn, ~ l h = ( P h h - ' C p p ) , and qel=-e, qlh:qhh=+e. The sheet concentration of type-b carriers in the g.-th subband is given by

4~)(~.T)= ~"~ k'r in[-I+ exp (

-~)3 .

C6)

In order to reduce computer time, the Schri:idinger equation was only solved for the edges (k =0,~ ~) of the minibands, using the exact boundary con(]itions of one-dimensional Bloch functions. Any kz-dependence of the equations has been eliminated through the introduction of "average" energies and squared wave functions:

~b,

=

l tb I Sb. (kz= ~, (k=0) + ~-

2 ..~zOO ' '

~-)

2, .,~z°~(z),

(7)

(8)

Of course, this approximation becomes critical if it is applied to the strongly broadened energy-bands above the classical continuum threshold. For the calculation of the impurity space charge in (4} all donors in the n-layers were considered as ionized, because the states of the donor impurity band

e{ nD(z) - nA(z) . f(EV+V'C+E . . . . q~p)} ,(9)

with the Fermi function f and with the doping profile

for

electr°ns I (2) for holes )

=

(11)

The quasi-Fermi level q~n in (6) has been adjusted numerically at every cycle of the selfconsistency loop to reproduce a given total electron concentration:

~.:

~n o,.. (2) (~ ., T) = no(2) Z,

(12)

I1

The value of ~ is then determined by the condition P , . of macroscopic charge neutral,ty over each period: d ~Pp: ypfre¢(z,Cpn,~p,T)+ plmP(z,cpp,T) dz = 0 (131 0

We have continued the Hartree procedure, until the relative change of the total space charge distribution was less than 1~ at every position z within the period

[0,d]. In the next step, the number of photons with wave vector Q (energy l'i~OQ = I~cIQI) and polarisation • emitted spontaneously per second through intersubband-recombinations has been calculated with the expression

Rio,°) - ~. ll2 • ~L"~(Qz)pL(~o). (,,) pv

h=lh,hh

The transition-strength of the subband pair (It,v)

-CL(%) = ~ l<®~.kz_Qzl exp(-iQ z)l®Utkz>l 2 has been approximated by an overlap for processes with Qz~0 _ ~.%:0.

"average"

Using the definition

h

¢~¢I

squared

(IS)

i'l e®h~.kz°O , ~ t,.k~:o > 2 +

~ N l~- I. I

(is)

2

Superlattices and Microstructures, VoL 6, No. 3, 1989

353

with mred -1 = m e t -1 + m h -1 ,

[

8.0 the effective combined density of states phv((aQ) in (14) in the case of spontaneous emission can be written as I.tv •

T=77K

7,0

'I

i

6.0

,.

i

In(2)=8 •i012crn-2 i ',

=

t Q~

m 5.0

-

f(~

•

Ath (taQ)-¢pp)],(I8

n(Z)=6•ii~I[Ii'F0ZZcrn-/I~ Ir¢=

~ 4.0

where O(x) denotes the Heaviside stepfunction. In the case of absorption/stimulated emmision formulas (14)-(16) remain valid, whereas P~vh"p°nt~Q"I has to be replaced by ".v°h'abs"'t"n'~'Q'l ~ O ( ~ % - %--¢1 _--h % ))'

.~0~3.0 2.0

n[Z)=4.1OlZcrn-2 ~1 ,

1.0

(19)

t?.O 0.5

•{ f(E-'~ + A¢h ((aQ)-(pp) - f(Tet+ A¢;l((.aQ)-¢pn)} .

0.7

0.9

1.1

1.3

1.5

1.7

Energy (eV)

It should be noted, that in our present model the aeeeptor impurity band affects the calculation of the optical spectra only during the Hartree procedure in such a way that it influences the space charge distribution and reduces the density of holes within the VB-subbands. In reality, there is additionally a direct contribution to the spectra through optical transitions between the localized acceptor states and states of the CB-subbands or the CB-hand tail, respectively. At room-temperature, however, a large fraction of the holes occupies extended subband states, which should dominate the spectra because of their enhanced overlap with electronic wavefunctions.

Fig• 2. Calculated luminescence spect.ra at T=77K for three different excitation densities n(2)or Fermi-level differences 4pn p (dashed lines). The multiple small peaks within each spectrum correspond to specific CB-VB-subband-transitions. ' 11.0

The GaAs DSL used for the experiments consists of 10 n- and 11 p-layers, with the following parameters: nD=4.10Zaem -3 , nA=l.10Zgcm -3., d =25nm i dp =35nm. • This npnp-structure has cladding ~ayers of undoped AIo.zGao.TAs. It was grown by molecular beam epitaxy using a silicon shadow mask to form grown-in selective contacts to the n- and p-layers 5"6. The electroluminescence measurements were carried out at room temperature (300K)• Voltages of the range from 0.9V to 6V were applied to the DSL. A 0.22m Spex double monochromator combined with a North Coast Ge-detector and a Stanford Research Lock-ln amplifier was used to detect the luminescence signal. The spectra were corrected to the spectral response of the optical system. 4. R e s u l t l and D i s c u s s i o n 4.1. T e m p e r a t u r e D e p e n d e n c e of L u m i n e s c e n c e

Calculated luminescence spectra at T=77K are plotted logarithmic al,ly in Fig. 2 for three different excitation densities n (2) or Fermi-level-differences A¢pn. (dotted lines), resp. They are related to a nipi-struerure with dn=30nm, d_=35nm, dis0, nD=nA=4.10tacm-3. The multiple small peaks within each spectrum correspond to specific CB-VB-subband-transitions, whereas the high-energy flank simply reflects the Boltzmann tail of the thermal occupation probability.

T 300K

,"

/=

9.0 8.0

I'

n(2)= 8 110t2 c m - i ~ " ~

10.0

"~

3. E x p e r i m e n t a l

1

'

' ~

\

n(21=6 • I OlZcm -2

d 7.0

t

,

t[0 = 3.05'04"0n[2)= 4" 1 0 n c m ' ~

0,0 0.5

0.7

0.9

1.1

1,3

i 1.5

1.7

Energy (eV)

Fig. 3. Calculatcd luminescence spcctra at 7"--3001(, for three different excitation densities n(2)or Fermi-level differences dq~np (dashed lines).

The spectra in Fig. 3 are calculated for the same sample parameters at room temperature. Now the occupation of high excited states leads to a smoothing of the fines,rue,use and to a considerable broadening of the spectra. The "shoulder" at energies near the bulk band gap, which grows with increasing excitation level, is due to thermally activated transitions between deloealized electron states and confined hole states• In the spirit of simple analytical estimates, which predict a roughly exponential relation between the electron-hole lifetime and the photon energy 2, we have calculated average inverse lifetimes through energy integration over each of the spectra.

354

Superlattices and Microstructures, Vol. 6, No. 3, 1989

i

/ T--Z25K

T=300K

i

~., /

f

/

,1.o

i .....

5DO

8no

1909

.

-oo .... i : 1200

A~p/meV

In Fig. 4 these values are plotted logarithmically versus /~p p which at T=77K is approximately equal to the peak photon energy . The exponential behaviour is conserved also in this refined model. It is interesting to note, that the slope of log I vs. Atpnp does not change within the considered t e m p e r a t u r e range. Obviously, higher t e m p e r a t u r e increases the intensity by a large factor, which is however not dependent on the excitation level. of L u m i n e s c e n c e

At room t e m p e r a t u r e the tunability tends to be reduced by the thermally activated transitions, but it can be optimized through a suitable choice of the design parameters. So we have carried out e l e c t r o l u m i n e s c e n c e m e a s u r e m e n t s for a strongly doped sample, described in section 2. As shown in Fig. 5, the peak m a x i m u m can be shifted by about 0.6eV even at this temperature. The curves, however, can not be directly compared with theory, because the lateral injection of carriers through the selective contacts leads to position dependent excitation levels Acpn~,(x), producing a complicated superposition of local s p e c t r a in the far field. This behaviour is demonstrated in Fig. 6, where the intensity at wavelength X=lpm is plotted versus position x. Therefore, we have also measured near-field spectra to get distinct values of ~¢Pn-' The dotted line in Fig. 7 belongs to the "nipi-pipi =e j u n c t i o n "5' 6 at m a x i m u m applied bias Unn, whereas the other spectra are located near t ~ e "nini-nipi junction" for different voltages U p n . " The theorettcal spectra in Fig. 8, calculated for the s a m e design parameters, show good agreement with the overall shape and the amount of tunability of the spectra. Our simplified theoretical model, however, neglecting potential fluctuations, predicts an abrupt drop down of the luminescence intensity at the effective band gap energy, whereas the experimental curves extend to lower energies due to band tailing effects.

......:

,.

~

I,

- 1.o I I

1400

Fig. 4. Calculated relation between integrated intensity {average inverse electron-hole lifetime) and Fermi-level difference.

4.2. Tuaability

..... .

o7

' 0.8

0.9

I

1,1

1.2

1.3

1.4

1.5

1.6

I7

1

E n e r g y (eV)

Fig. 5. Elcctro/umincsccncc spectra, measured in the far field (spatially unresolved) at room temperature. The peak shift was achieved by a variable bias Upn between the n- and p-layers in the range of Upn=O. 7V to 6.0V.

10

T=300K

i

Up~=2V

i

76! m -

I nini

nlpi

,' l i

pipi

I ' I,

i

i

l 2

r

: 19.0

19.2 X-Direction

19.4

19.6

ml~

TZ

nini

i

,

o,zo

nipi pipi GaAs SUBSTRATE

Fig. 6. Spatial distribution of the eleetroluminescence intensity at wavelength 2=lgm, measured locally across the surface of the nipi probe. The lower picture schematically depicts the geometrical structure of the sample, as it results from the shadow masking technique.

4.3. Absorption and Optical Gain We expect our simulation model to be equally well suited for quantitative predictions about the tunable

Superlattices and Microstructures. VoL 6, No. 3, 1989

355

0 T=300K 4,o

r.

r

10 s

3

T=300K 10 4

@

n{ZI=B.1012cm -2

10 3

3.0 -4

f~) 10 2 "~ o

2.0

_~

1.0

101 \ •2

1

o,o

I

i

I

1.3

1.35

1.4

Photon -

1.o ° 7 .

i

0.8

0,9

1

1.1

1.2

1,3

1.4

1.5

1.6

1.7

I0 s E n e r g y (eV)

i

Energy i

r

I

1.45

1.5

I

1.55

eV

in

)

i

r

T=3OOK

Fig. 7. Spatially resolved electroluminescence spectra

"

at T=3OOK. The dotted curve was measured near the nipi-pipi junction (compare Fig. 6) at m a x i m u m applied bias. The other spectra belong to the nini-nipi junction for different applied voltages Upn.

g

,

,

,

,

"~

10 3

o U

10 2

'~ 0

i0 l

.~

,

12.0

10 4

n 2 =9 1012cm -2

1

T=300K. 11.0

n(2)= 1015m-2- - - 5" 1016m'2

....

/._

~

o

I

I

I

<

1.3

1.35

1.4

Photon

9.0

Energy

I

I

1.45 in

J

1.5

r

1.55

eV

Fig. 9. Absolute value of the absorption/gain coefficient in cm -1 versus photon energy in eV, calculated for two different excitation levels at room-temperature. The absorption coefficient is negative (gain) for photon energies below the difference dq~n~ of the quasi-Fermi levels (vertical lines) and posiIlve above.

~7.o ~6.o _c 5.0 o~4.o 3.0 2,o

/

1.o 0.0 0.7

0.8

0,9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

E n e r g y (eV)

Fig. 8. Theoretical luminescence spectra for different excitation dcnsitics n (2), calculated for the same temperature and design parameters as the experimental data in Fig. 7.

optical gain in DSLs. Figure 9 represents logarithmic plots of the calculated room-temperature coefficient of absorption (ho>ACpnJ and gain (l~(o
carrier absorption. This reduction, however, is less than 100 cm-" for any reasonable carrier density. The spectra clearly show, that net gain is obtained within a wide spectral range, if the excitation level is sufficiently high. The required values of A¢ p are thoroughly realistic, as the corresponding ere~ctron densities indicate. Acknowledgement We would like to thank G. Hasnain, D. Mars and J. N. Miller for providing the sample with grown=in contacts to us.

References

nD=8']0 1 8 cm- 3 and nA=2'10 19 cm- 3 . In order 1oP obtain

the spectra in absolute units, we have identified the calculated absorption at energies significantly above the bulk band gap (riOre~l.SeV) with the absorption coefficient of pure GaAs. In practice, the net optical gain at low photon energies is reduced due to free

I. For recent reviews see: G. H. Critical Reviews in Solid State Sciences 13, 97 (1987); G. H. Journal of Quantum Electronics (1986).

DShler, CRC and Material D~hler, IEEE QE-22, 1682

356

Superlattices and Microstructures, Vol. 6, No. 3, 1989

2. G. H. DShler, J. Vac. Sei. Technol. B 1, 278 (1983); K. Kbhler, G. H. Dtihler, J. N. Miller, and K. Ploo8, Superlattices and Mierostructures 2, 339 (1986). 3. G. H. DShler, G. Pasol, T. S. Low, and J. N. Miller, Solid State Commun. $7, 563 (1986); K. KiJhler, G. H. D~ihler, J. N. Miller, and K. Ploo8, Solid State Commun. 58, 769 (1986).

4. P. Ruden and G. H. DShler, Phys. Rev. B 27, 3538 (1983). 5. G. H. DShler, G. Hasnain, and J. N. Miller, Appl. Phys. Lett. 4.9, 704 (1986). 6. (3. Hasnain, G. H. D~ihler, J. R. Winnery, J. N. Miller, and A. Dienes, Appl. Phys. Lett. 49, 1357 (1986).