Table Of Contenthttp://www.terahertz.tn.tudelft.nl/ http://www.kinsler.org/physics/
Towards quantum well hot hole lasers
∗
P. Kinsler and W. Th. Wenckebach
Department of Applied Physics,
Faculty of Applied Sciences, T.U. Delft,
Lorentzweg 1, 2628 CJ Delft, The Netherlands.
It should be possible to improve hot-hole laser performance by moving from bulk materials to
a quantum well structure. The extra design parameters enable us to alter the band structure by
changingthecrystal orientation of thegrowth direction; tousethewell width toshift thesubband
offsets,enablingtheeffectoftheLOphononscatteringcut-offtobecontrolled;andtousemodulation
doping to ensure a high hole concentration to increase the gain without the dopants being present
in the gain region. We present the first simulations of THz quantum well hot-hole lasers that can
produceinversion and optical gain.
2
0
0
I. INTRODUCTION
2
n
a Hot hole lasers [1, 2] emit in the THz (far-infrared)
J with an unusually broad gain spectrum, allowing ampli-
2 fication and generation of laser pulses on a picosecond
2 time scale. The THz band has important potential ap-
plications in (e.g.) medical imaging and office communi-
1 cations. Bulk hot-hole lasers have been realised in p-Ge,
v producing gains of∼0.25cm−1 around4THz. Ofthe III-
6
V materials, both GaAs and InSb are suitable for hot
9
3 hole lasers;althoughtheir performance is not as goodas
1 in Ge [3]. Investigation of these is the most useful for
0 industrial applications because of the existing ability to
2 grow high quality III-V structures.
0
We present a discussion of likely modes of operation
/
t of quantum well hot-hole lasers in GaAs, together with
a FIG. 1: Band structure of a quantum well hot-hole laser.
m predictions from Monte Carlo simulations for one de- Uppergraphs: 100˚Awellwitha[101]growthdirection;lower
- sign. The simulations use an infinite-well k.p bandstruc- graphs: a 100˚A well with a [001] growth direction. The kx
d ture,andincludeopticalphonons,acousticphonons,and variation is shown on the left-hand graphs, and ky on the
n piezoelectricphonons[4,5,6]. Asyettheydonotinclude right. The vertical scale is 0–100meV, and the horizontal 0–
o ionisedimpurity scattering,but this is a smalleffect and 1×109m−1.
c shouldnotaffectthecharacteroftheresultssignificantly.
:
v
i
X subbands of two simple cases schematically. Note that
II. QUANTUM WELLS the [101] well is not symmetric in x and y; and that for
r
a the [001] well the HH dispersions are no longer even ap-
proximatelyparabolic,withtheHH2havinganoticeable
Bulk hot-hole lasers can be described using the heavy
local maximum at the origin.
andlight hole valence bands: anelectric field accelerates
the heavy holes in a streaming motion to high energies The [101] well has two heavy-hole subbands (HH1,
E > E ; from where (ideally) they scatter into light HH2) that need to be considered. For wells over about
LO
hole cyclotron orbits formed by the magnetic field; and 50˚Awide,the twolowestenergy,andhencemostheavily
thenemitaphotonandreturntolowenergyintheheavy populated subbands will be HH1 and HH2. This means
holebandtorepeatthecycle. Inaquantumwelleachva- that in contrast to the bulk case we do not need a mag-
lence band breaks up into a set of subbands: heavy hole netic field to confine the light holes in cyclotron orbits.
subbands HH1, HH2, HH3, ...; and similarly for light The LH1 is well above the HH2 for a 100˚A well, and so
holes LH1, LH2, etc. The non parabolicity of the bulk has minimal effect on hot-hole laser operation. Along
bandstructureleadstoavarietyofpossiblequantumwell one direction (y) both HH subbands are very flattened
bandstructures,dependingontheorientationofthecrys- at the base, but with a LH-like curvature for larger ky;
tal axes in the well material. Figure 1 shows the lower whereas along the other (x) HH2 sits between HH1 and
LH1. This means that the heavy-hole distributions will
beroughlyrectangularinoutline;andanincreasingelec-
tric field will shift the distributions along its direction.
∗Electronicaddress: [email protected] InthexdirectionthetwoHHsubbands(nearly)crossat
1
2
The [001]wellHH2 subbandhasa distinctlocalmaxi-
(a)
mum at k =0; and also LH1 is rather close to HH2, and
shouldnotbe neglected. Ourcode doesnotyetallowfor
these more complicated dispersions, but we can see that
a lasing cycle might be as follows: A HH1 of moderate
positive k is acceleratedby an electric field +F , where
x x
itmightscatterintoHH2neartheanti-crossing. Thenit
willscatterbackontotheinvertedpartofHH2,eitherby
acousticphononscatteringorLOphononemission. Here
it will be accelerated backwards past k = 0 to the point
ofinflection(−k )wherethe HH2 effective massisin-
LHi
finite;andsubsequentlyemitaphotonanddropdownto
HH1, from where it will eventually scatter to moderate
positivek andwillrepeatthecycle. NotethatHH1also
x
has a local maxima, but it is smaller and has less effect;
(b) and its point of inflection where holes can collect is off-
set from the HH2 one, and so should not interfere with
population inversion too much. In contrast to the [101]
well,weexpecttheemissionspectrumofthis[001]wellto
be peaked and centered at EHH2(kLHi)−EHH1(kLHi),
where the HH2’s will accumulate.
Quantum well hot hole lasers also benefit from modu-
lation doping. One significant source of unhelpful scat-
teringisthatduetoholesscatteringoffeitherotherholes
orimpurities. Wecanhalvethiscontributiontothescat-
teringsimplybymovingimpuritiesusedtoaddtheholes
to the device to outside the active region. Also, we can
vary the well widths to vary the inter-subband separa-
tions to greater or less than E to enhance or reduce
LO
FIG.2: [101] 100˚A quantumwell hot-holelaser, withelectric optical phonon scattering as required; or even use the
fieldFx =250V/cm(a)Contourplotshowingtheregionofin- well’s crystal orientationto adjust the densities of states
version;(b)spectrumofthegaincross-section fory polarised
(DOS) of the subbands. The [101] well has HH1, HH2
light. Both graphs are affected by statistical noise from the
withflatE(k)inthek direction,leadingtoanenhanced
simulations. y
density of states, hence making those regions into rela-
tively preferred destinations for scatterings. This differs
apointabout15meVabovethebottomofHH1,allowing markedly from the DOS effects in bulk material [3].
for fast inter-subband scattering. Holes that get further
up HH1 to an energy equal to E above the bottom
LO
of HH2 will quickly emit an LO phonon and return to III. CONCLUSIONS
populate either HH2 or HH1. Note that HH2 is flat to
larger k values than HH1, making it relatively easy to
y Wehaveshownthepotentialforquantumwellhot-hole
get inversion where HH2 is flat but HH1 is not.
lasers by an investigation of the possible bandstructure
Fig. 2(a)showsthe difference inthe distributionfunc-
in combination with computer simulation. Although we
tions between HH2 and HH1, for an electric field of
have not yet explored the full parameter space of possi-
250V/cm along the x direction. We see regions of in-
ble designs, our first attempt (the 100˚A [101] quantum
version for k ≈ 0 and k ≈ 5.00×108m−1. There is
x y well) produced simulations which showed inversion over
a small displacement in the direction of the field, and
arangeofelectricfieldstrengthsanddirections,withthe
the amount of inversion decreases with increasing field
optimum being for a 250V/cmfield in the in-plane x di-
strength. Ifthe electricfield is appliedalongthey direc-
rection. Further, the 100˚A[001]wellhas a bandstructure
tion,weseeonlyoneinversionpeak,becausetheotheris
which promises a good inversion. We aim to continue to
wiped out as the HH1 distribution is shifted underneath
test different designs and field combinations, including
it; and a field applied along x=y has a similar, but not
theadditionofmagneticfields,inordertodeterminethe
somarked,effect. Figure2(b)showstheopticalgaindue
most practical designs for experimental investigation.
to the inversionobtained andshownin (a). We see opti-
calgainoccurringoverthe rangeofenergiesbetweenthe Acknowledgements: This work is funded by the Eu-
small splitting at the HH1-HH2 anti-crossing (≈ 0meV) ropean Commission via the program for Training and
and the separation of the subband minima (≈15meV). Mobility of Researchers.
3
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