ENERGY LOSS MECHANISM FOR HOT
ELECTRONS IN GaAs
D. Neilson, J. Szymanski, De-Xin Lu
To cite this version:
D. Neilson, J. Szymanski, De-Xin Lu. ENERGY LOSS MECHANISM FOR HOT ELECTRONS IN GaAs. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-263-C5-266.
<10.1051/jphyscol:1987556>. <jpa-00226760>
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Submitted on 1 Jan 1987
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JOURNAL D E PHYSIQUE
Colloque C5, suppl6ment au null, Tome 48, novembre 1987
ENERGY LOSS MECHANISM FOR HOT ELECTRONS IN GaAs
D. NEILSON, J.
SZYMANSKI(~)
and D E - X I N L U ' ~ '
School of Physics, University of New South Wales, P.O. BOX 1,
Kensington NSW 2033, Australia
RBsumB. Nous avons Ctudik systCmatiquement le mecanisme de la perte d'knergie des
Clectrons chauds inject& dans une couche mince dopCe de GaAs. Les Cnergies incidentes sont
gardkes en dessous du seuil de diffusion entre-val. On trouve que pour les densitks de porteurs
typiques de celles utiliskes dans les appareils de GaAs, les pertes d'Cnergie sont dominCes par les
processus d'excitations semblables A pure Clectrons.
Abstract. We have systematically investigated the energy loss mechanisms for hot electrons
injected into a thin doped GaAs layer. Incident energies are kept below the threshold for intervalley
scattering. We find for carrier densities typical of those used in many GaAs devices, the energy
losses are dominated by pure electron- like excitation processes.
1. I n t r o d u c t i o n
In this paper we investigate the importance of the various different scattering mechanisms
on the transport of hot injected electrons in GaAs. Let us consider a typical ballistic hot-electron
device, the Planar Doped Barrier (PDB) transistor.[l] Electrons are injected from the emitter into
the narrow base layer.
We deal here with the limitations to the performance of such a device due to the scattering
of the electrons in the base region leading to non-baliistic transport. To keep the RC delay time
associated with the base emitter capacitance to less than 1 ps the base region width needs to be
> 1000A, and the dopant concentration 10'~cm-~.[2]
A hot electron injected into the base region can (i) excite a coupled optical-phonon / plasmon
mode, (ii) excite an electron-hole pair, or (iii) it can elastically scatter off one of the ionised dopant
impurities.
The total dielectric for the system is
-
where epl(&w ) is the dielectric function for the electrons and eph(w) the optical phonon dielectric
function. Even a t n = 1 0 ~ ~ c m
the- ~effective r: value for GaAs is less than unity and epl($ w) is
well approximated by the R.P.A. expression. We can write ePh(w) in the form
Figure 1shows the regions in q', w phase space for the various excitation modes of the system.
The imaginary part of el$, w) is non-zero in the shaded region. This is the single-particle excitation
region which is bounded by the parabolas w(q) = q2 f 29 where hw(q) is expressed in units of the
Fermi energy EF,and hq in units of the Fermi momentum hkF. Outside the shaded region the
zeroes in the function E($,w) determine the dispersion curves for the optical phonon/plasmon
modes. These are the curves labelled 1-3 in the figure. Because the plasmon frequency wpl and the
optical phonon frequency w~ are similar, the modes hybridise. For densities n > 8 x 10"crn-~,
the plasmon frequency wpl > WL, and Branch 1 consists of plasmon-like excitations interacting
(l)~elecomAustralia Research Laboratories. 770 Blackburn Road, Clayton Vic. 3168. Australia
(2'~ermanent address : Physics Department, Nanjing University, China
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987556
JOURNAL DE PHYSIQUE
C5-264
with phonons while those in Branch 3 are phonons screened by the electrons. For densities n <
8 x 10"c1n-~ the roles of Branches 1 and 3 reverse. In both density ranges Branch 2 is LO
phonon-like and approximately follows the uncoupled LO phonon mode. It is found only in the
small wave-length region of phase space.
2. Calculations
-
A hot electron of energy 300meV injected into the n-doped base region of a PDB transistor
encounters an electron gas typically with Fermi energy EF m 40meV. The incident momentum hpo
is much greater than the Fermi momentum hkF so that the Born approximation for the scattering
probability will be valid.[3]
Figure 1: Excitation modes of the electron gas which is confined in the n-type base region. The
plasmon and optical phonon modes hybridise into the Branches wl(q) and ws(q) as shown. The
shaded region is the permitted phase space for single-particle excitations. w2(q) is the nearly pure
LO phonon mode which occurs for large q.
The probability per unit time that the hot electron will couple t o one of the excitation modes
of the electron gas, transferring momentum hq and energy hw is then given by [3]
4ne2
P(q7 w) = -2 -1m
e-'(3 w ) .
q2
The corresponding differential energy loss rate with distance is
where the energy transfer is hw = (efi - efi-q).
We can also define the rate of Ioss of p l , the component of the momentum ~erpendicularto
the barrier.
1 d ~-l --,
PO dz
PO d z
Ifi-4>kr
ldP"]
where ~ ~ ' ( d p ~ / d zis) ,the
~ rate of loss arising from elastic scattering off ionised donor atoms
(Rutherford scattering).
2
.,
3. Results
In Eq. (1) there are contributions to Im
W ) from the three coupled optical phonon/
plasmon normal mode excitations [4] and also from single electron excitations. In Tables l(a)
and 2(b) we show the separate contributions to ~ i ' ( d ~ ~ / d
and
z )P~l(dpl/dz) for an electron of
incident energy Eo = 3OOmeV injected into the base region. The electron densities in the base
region range from 10" to 3 x 1 0 ' ~ c m - ~Table
.
l(a) shows clearly that the two dominant scattering
mechanisms for E ~ ' ( d ~ ~ / dare
z )the plasmon-phonon coupled mode i = 1 and the single-electron
all the modes and, as well, the elastic
excitations. In Table l(b) we see that for p;'(dpl/dz)
scattering off the dopant impurities can make a significant contributioq.
Table 1: Contributions to (a) energy loss rate ~ i ' ( d ~ ~ / dand
z ) (b) loss rate of perpendicular
momentum pol(dpl/dz) for a fixed incident energy ofEo = 3OOmeV. The dominant energy loss
mechanisms are the i=l mode and the single-particle mode over the entire range of densities shown.
In Figures 2(a) and 2(b) we plot the significant contributions to ~;l(dEo/dz) and p,'(dp~/dz)
as a function of the incident electron energy. The base electron density here is fixed a t l ~ ~ ~ c r n - ~ .
At this density mode i = 1 is predominantly plasmon-like, so the energy loss mechanisms are
completely electron dominated. At a much lower density such as n 10"cm-~ the single-electron
excitations are insignificant. The dominant mode for energy loss is still i=l, but this is now predominantly phonon-like. Phonon excitations thus dominate a t this density, and improved response
times could be expected by using a non-polar semiconductor.
to deduce the broadening of a pulse of current
We may use our results for p;'(dpl/dz)
traversing the barrier. For a thin barrier this estimation is straightforward. Table 2 gives the
width of a &function pulse of 3OOmeV electrons crossing a 1000Abarrier for different electron
densities in the base region.
-
Density ( ~ m -1 1
~ x lo'?
Width (ps) 1 0.01
1 3 x 1017 1 1 x 10" 1 3 X 10"
]
0.02
1
0.03
1
0.04
Table 2: Approximate broadening of a 6-function pulse after traversing a base region of thickness
ioook
Acknowledgements. One of us (J.S.) acknowledges the approval of the Director of Research,
Telecom Australia Research Laboratories, to publish this paper.
JOURNAL DE PHYSIQUE
Figure 2: (a) Total energy loss rate EG1(dEo/dz) , and contributions to it from the k = 1 mode
~ ~ ~ ( d E ~ / and
d z the
) ~single-particle
= ~
mode ~ ; ' ( d E ~ / d z ) ,as
~ a function of the incident energy
Eo. The density is n = 1 0 ' ~ c m - ~ .(b) Total perpendicular momentum loss rate p,'(dpl/dz) for
the same density. Also shown are the contributions from the individual modes and from elastic
scattering.
References
1. R. M a l i , T. AuCoin, R. Ross, K. Board, C. Wood and L.Eastman, Electron Lett. 16, (1980)
837.
2. S. Luryi and A. Kastalsky, Physica 134B,(1985) 435.
3. D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York) 1966.
4. S. E. Kumekov and V. I. Perel', Sov. Phys. Semicond. 16, (1983) 1291.