VOL. 18, NO. 4
CHINESE JOURNAL OF PHYSICS
WINTER, 1980
Enhanced Direct-Free-Hole Absorption in Picosecond
Laser-Excited Germanium
C HUNG Y
EE
LEUNG (GE,& $$)
Departrrsnt of Physics, National Central University
Chung-Li, Taiwan, Republic of China
(Received January 19, 1981)
Direct-free-hole (DFH) transition between the subbranches of the valence band
normally has negligible contribution in the absorption of light by semiconductor
whose bandgap is smaller than the photon energy. However, in the interaction
of germanium with intense picosecond neodymium laser pulses, the DFH absorption rate is enormously enhanced by the high density and high distribution temperature of the free carriers created in the semiconductor. It is shown that this
process has an important role in the ultrafast optical response of photo-excited
Ge. Adding the DFH mechanism to a theoretical model proposed recently for the
generation and the temporal evolution of non-equilibrium electron-hole plasmas in
Ge produces better agreement between experiment and theory.
I. INTRODUCTION
PTICAL properties of the nonequilibrium electron-hole plasma in germanium, created by intense
ultrashort pulses from mode-locked laser systems, have been intensively studied recently(l-lOJ.
These investigations have led to advances in the upderstanding of ultrafast electronic processes in
Some measurements involve the interaction with single pulses, giving information
semiconductors.
on events that happen on a time scale comparable to the pulsewjdths. Fig. 1 shows the result of
one such experiment, where the transmission of picosecond pulses at 1.06 pm through a thin Ge
sample is measured as a function of incident pulse energy(l). The data show an increase in transmission with pulse energy. At high excitation levels, the enhanced transmission .appears to saturate
before sample damage occurs. In other experiments, the excite-and-probe technique is generally used.
The time delay ld of the weak probe pulse with respect to the strong excitation pulse (usually,
10 psec 3 td 7 1 nsec) allows the study of much slower processes.
Depicted in Fig. 2 is the result
of a typical experiment. Here, the transmission of a thin Ge -sample is measured with probe pulses
at various delay times after excitation’“).
The theoretical effort to explain the results of the above experiments has led to a comprehensive
theory by Elci, Scully, Smirl and Matter ~1 for both the generation and the subsequent transient
behavior of electron-hole plasmas created in germanium by intense picosecond optical pulses. As
shown by the theoretical curves in Fig. 1 and Fig. 2, this model (referred to as the BSSM model. in
___~ ~ ~~~_
( 1 ) C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu and M. 0. Scully, Phys.
Rev. Lett. 32. 419 (1974).
(2) D.H. Auston and C.V. Shank, Phys. Rev. Lett. 32, 1120 (1974).
(3) C. V. Shank and D. H. Auston, Phys. Rev. Lett. 34, 479 (1975).
(4) D.H. Auston, C.V. Shank and P. LeFur, Phys Rev. Lett. 35, 1022 (1975).
( 5) A. L, Smirl, J. C. Matter, A. Elci and M. 0. Scully, Opt. Commun. 16, 118 (1976).
(6) S. A. Jamison, A. V. Nurmikko and J. M. Gerritsen, Appl. Phys. Lett. 29, 640 (1976).
(7) H. M. van Driel, A. Elci, J. S. Bessey and M. 0. Scully, Solid State Commnn. 20, 837 (1976).
( 8) A. Elci, M. 0. Scully, A. L. Smirl and J. C. Matter, Phys. Rev. B16, 191 (1977).
158
a
4
CHUNG YEE LEUNG
159
lo-’ -,
I:ilTI;_L SAMPLE
TEMPERATURE
0 7: OK
INCIDENT
Q U A N T A ( A T 1.06pm)
Fig. 1. Single pulse transmission in Ge as a function of incident pulse energy
at two sample temperatures. The data are from Smirl ef 4l.15) T h e
dashed lanes are the predictions of Elci er al.(*) The solid lines are the
theoretical curves according to the present theory. Sample thickness is
5.2 pm, interaction volume is 10-a ems. and pulsewidth is 11 psec.
l.o[‘ - -
I
I
Ol
INITIAL LATTICE
TEMPERATURE
50
100
153
??O
D E L A Y T I M E (PSFC!
Fig. 2. Normalized probe pulse transmission. The data are from Smirl et 41.(6)
Excitation pulse contains 1014 quanta at 1.06,~~ interaction volume is
10-a crns and sample thickness is 5.2 pm. Dashed lines represent the predictions of Elci et al.(B) The solid lines are the theoretical curves according to the present theory.
L;.
160
ENHANCED DIRECT-FREE-HOLE ABSORPTION
this paper) is successful in reproducing the basic features of the results of both the single (excitation)
pulse and the two (excite-and-probe) pulse transmission experiments. Yet, detailed analyses have
indicated that the model is incomplete. We list in the following some of its major limitations.
1. The theory fails to account for the details of the experimental saturation of the excitation
pulse transmission at high excitation energies (Fig. 1).
2. The experimental probe pulse transmission data show an early rise of transmission which
peaks at td- 100 psec, whereas the theory predicts a delayed, much steeper rise (Fig. 2).
3. This model predicts a strong pulsewidth dependence of the excitation pulse transmissionc8),
which is not found in recent experiments”‘ ).
4. The only free parameter used in the ESSM model is QJ, the electron-optical phonon coupling constant. The fitted values are found to be 6 x lo-’ erg cm” for lattice temperature 298 °K and
2 x lo+ erg cm” at 77°K. Although there is a wide spread in the Q, values reported in the literature(12) (ranging from 6.4 to 18.5 x 10” erg cm ”, with the most recently determined values(13’1E)
pointing to the interval of 7.7 to 10.7 x 10” erg cm”), these values used by Elci et al. are generally
considered as too low. Since the electron cooling (vis phonon emission) rate is proportional to Q%,
these small Q, values lead to relaxation times much longer than those reported by others(17).
4
4
In this paper we shall show that all of the disagreements between experiment and theory listed
above are essentially due to the neglect of an important process in the model: the direct-free-hole
(CFH) transition between the spin-orbit split branches of the valence band. We shall first give a
brief review of the ESSlM model by describing how it accounts for the interaction of picosecond
pulses with with semiconductor. The DFH transition process, enhanced under hot-carrier stituations,
is discussed in section III. In later sections (IV, V, VI), we shall revise the ESSM model and
demonstrate the importance of the DFH process in accounting for the observed nonlinear transmissions in Ge. Section VII describes the fast transient behavior of D F H absorption, and the final
section, Section VIII presents our conclusions.
II. GERMANIUM - PICOSECOND PULSE INTERACTION
The theory of Elci, Scully, Smirl and Matter c8) is the first comprehensive theory presented for
the generation and the temporal evolution of the dense, nonequilibrium electron-hole plasma produced
In spite of its limitations, the
in germanium by the absorption of intense, ultrashort optical pulses.
theory does represent a solid step toward an understanding of the complicated problem, thus providing a base for further developments. To provide the reader with the necessary background material,
the physical picture, according to the ESSM theory, for the interaction of picosecond pulses with
semiconductor is reviewed in this section. This will probably be best done by describing how it
accounts for the features observed in the two typical experiments mentioned in the last section.
As soon as an intense laser pulse enters the Ge sample, it is absorbed by direct interband
(valence-conduction) transitions, creating a large number of electrons (holes) in the central valley of
the conduction (valence) band. The electrons are rapidly (7 lO’L1 set) scattered to the conduction
band side valleys by long wave vector phonons. Since the electrons are emptied from the central
valley to side valleys at a rate comparable to the direct absorption rate, any decrease in the number
of states available in the central valley for direct absorption is ultimately determined by a buildup of
(9) H. M. van Driel, J. S. Bessey and R. C. Hansen, Opt. Commun. 22, 346 (1977).
(10) A. Elci. A. L. Smirl, C. Y. Leung and M. 0. Scully, Solid-State Electron. 21, 151 (1975).
(11) J. S. Bessey, B. Bosacchi, H.M. van Driel and A. L. Smirl, Phys. Rev. B17, 2782 (1978).
(12) W. P. Latham, Jr., A. L. Smirl and A. Elci, Solid State Electron. 21, 159 (1978).
(13) K. Seeger, Semiconductor Phytics, Springer-Verlag. Wien-New York (1973).
(14) D. C. Herbert, W. Fawcett, A. H. Lettington and D. Jones, in Proceedings of the 1Ith Infernational Conference on Semiconductor Physics, Warsaw (1972).
(I 5) M. Costato, S. Fontanesi and L. Reggianna, J. Phys. Chem. Solids 34, 547 (1973).
(16) W. Fawcctt and E. G.S. Pnige, J. Phys. C: Solid St. Phys. 4, 1801 (1971).
(17) D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen and 0. Tcschke, Solid-State Electron. 21; 147 (1978).
-
.,
.,
4
4
- -”
CHUNG YEE LEUNG
9
0
\ $I
161
the populations in the side valleys. Carrier-carrier scattering events, which also occur at a rate
comparable to the direct absorption rate, ensure that the electron and hole distributions will be
Fermi-like. They also ensure that the Fermi distribution for holes and the Fermi distribution for
electrons will reach a common temperature. Since the photon energy ho,, is greater than the band
gap EG, a direct absorption event followed by phonon-assisted scattering of the electron from the
central to side conduction band valleys results in an excess energy ho,-EE, being given to thermal
agitation. The result of the above processes is the generation of a large number of electrons (holes)
in the conduction (valence) band with a high distribution temperature.
There are other processes that need to be considered. Electrons (holes) located high (low) in a
conduction (valence) band valley can relax by emitting phonons.
The effect of this relaxation is to
lower the carrier distribution temperature, putting more carriers in the states near the bottom (top)
of the conduction (valence) band that are needed for direct absorption. Thus, phonon-assisted relaxation, which occurs on a time scale comparable to the optical pulsewidth, tends to increase the
transmission. However, as the carrier density increases due to direct absorption, the plasma frequency
of the carriers increases. When the plasma frequency is high enough, an electron in the central
conduction band valley can recombine with a hole near the top of the valence band while emitting
a plasmon. Normally, plasmon-assisted recombination can occur only if the plasma frequency, wp
is larger than the direct gap frequency Es/h, where E, is the direct band gap. However, the
plasma resonance is broadened considerably due to the strong perturbation of the solid by the optical
pulse. Thus plasmon-assisted recombination can occur at plasma frequencies below E,/h. These
collective plasma oscillations have lifetimes that are short compared to a picosecond. The energy
lost in their decay is transferred to electrons and holes, thus heating the carrier distribution. Therefore, as the carrier density increases, the plasmon-assisted recombination rate increases, which retards
the growth of the carrier number and raises the distribution temperature. Rapid increases in the
sample transmission is thus prevented as the electrons (holes) are heated and removed from the
optically coupled states. Free carrier absorption also serves to increase the distribution temperature.
Although not so effective as the other processes mentioned above, it is included in the model.
When the excitation pulse has passed through the sample, plasmon-assisted recombination is
essentially turned off. As time progresses, the carrier distribution will continue to cool by phononassisted relaxation. Thus a weak pulse that probes the electron-hole plasma at delayed times will
experience a lower absorption as time increases, since the electrons (holes) are now located lower
(higher) in the bands (Fig. 2).
III. DIRECT-FREE-HOLE TRANSITION
The valence band of germanium consists of three subbands which are separated by spin-orbit
interaction. The schematic band structure near the center of the Brillouin zone is shown in Fig. 3.
At the center of the Brillouin zone, the heavy hole band (E,,,) and the light hole band (Ek2) are
degenerate and the third band (Es) is separated from them with split-off energy A.
While quantum selection rules forbid direct transitions between the valence subbands at k - 0 ,
they are allowed at k#O. Direct transitions at 1.05 pm are indicated by arrows in Fig: 3. It is‘
seen that D F H transitions occur relatively far from the center of the Brillouin zone. Thus, unless
the free hole concentration is sufficiently dense, DFH transition rates are usually negligible, compared
to that of direct interband transitions, since both the initial and the final states Ii> and If> are
heavily populated. As shown in Fig. 3, even at the highest concentration (-2 x 10ZO cmeJ) of carriers
that can be reached in the experiments(18), the position of the Fermi quasi-level for holes is not
sufiiciently deep in the valence band to ensure by itself a significant depopulation of If>. However,
(18) Owing to processes (mainly plasmon-assisted recombination and direct-free-hole absorption) that compete with interband absorption at high carrier concentrations, our calculation indicates that the carrier
density is limited to a maximum of about 2x 10’0 cm-s. This value agrees well with the the estimation of Auston and Shank(*).
~-
_L-_
162
ENHANCED DIRECT-FREE-HOLE ABSORPTION
I
c
-3.fof
0.0
0.1
0.2
k (? )
Fig. 3, Schematic band structure of Ge near the center of the Brillouin zone for
298°K lattice temperature. The arrows indicate direct transitions at 1.06
pm @IT: direct interband transitions, DFH: direct free hole transitions).
The dashed line indicates the Fermi-level position of the holes at NC=2
x 10~0 cm-*, T, =298”K.
when the carrier temperature T, is high enough, the situation may be significantly altered. T, is
therefore a critical factor controlling the rate of DFH transitions.
We assume that, over the portions of the energy bands that are important to our calculations, the
carrier energies may be described as quadratic functions of the wavevector k with the appropriate
effective masses. Thus the hole energies are given by
@
(1)
(2)
(3)
where Ar is the separation between E,,, a n d Eha (for k T 0 . 0 5 A-l); ml, m2, m, are the effective
masses of the heavy, light and split-off hole bands, respectively; and k is measured from the center
of the Brillouin zone. Following Elci et af.t8), we also assume that, as a result of fast carrier-carrier
scattering, the carrier populations are described by Fermi distribution functions with a common
distribution temperature T,. That is, for the holes,
(4)
(5)
(6)
where kg is the Boltzmann’s constant and E 11 is the Fermi quasi-level of the holes,
(I
CHUNG YEE LEUNG
163
The probability per unit time of a transition is given by the Fermi’s Golden Rule
(7)
8
where M/i is the matrix element of the interaction producing the transition between the initial (i)
and final (/) states of the system. ‘T he S-function ensures energy conservation. Th? electron-photon
interaction is described by the well-known Hnmiltonian
X CL, csr b,,i hermitian conjugate
(8)
w h e r e Isk) refers to Bloch states (s is the band index), &cl) is the unit phston polarization vector,
p is the electron momentum operator, csr and b,, are the electron and photon annihilation operators
respectively, e is the electron charge, m ths blr- electron mas;, and E, th? high-frequency dielectric
constant.
The diagrams in Fig. 4 describe direct optical transitions b-tween diff:r?nt branch-s of th:
valence band. For absorption, the initial state consists of N,,+l photons of frequency o a n d
wavevector q=r/T-,w/c, a hole in the upper valence band Zlj with wavevector k+q, and a ele:tron
in band vi with wavevector k. After the interaction, we have N,, photons of frequency w, an
electron in band vi with wavevector k+q, a;id a free hole in band vi with wavevector k. Thus
the absorption rate goes as
2n
tl- ~(probability of an electron in state vi, k)
~(probability of a hole in state zlj, k + q )
XNA,X I<v, k+qIH,,Ivi, k>l’
X
6[E*i(k)-E,j(k+q)-Aw]
(9)
For emission, we begin with N,, photons in the field, an electron in the upper band Zij with wavevector k+q, and a hole in vi with wavevector k. After the interaction, a photon of frequznzy o
is created and the hole makes a transition to state Vi, k+q. The emission rate thus goes as
2Tc
A ~(probability of an electron in state
~(probability of a hole in state
Vi,
Vj,
k+q)
k)
X(N~fl)x /<Vi, klH.,Ivj, kfq>la
(10)
X 6[E*i(k)-E,j(k+q)-ho]
We thus readily find the net absorption rate
“j .k+q
I\ ji
“i .k
cl
P
“i .k
AESOHPTIOPI
V
J.k+q
_ EMISSION
Fig. 4. Direct transitions between different branches (V,) of the valence
band. Solid lines reker to electrons, wavy lines to photons.
ENIIANCED
161
DIRECT-FREE-IIOLE ABSORPTION
x(~‘V,,+ 1 j [I -fhj(k+q)]f*,:k)d[E!,(k)-_E,,,:k+q)-_ho]i
(11)
Here, y; also includes spin summation. The photon mom~:nta hi are generally much smaller than
the elec’tron and hole momenta hk. Therefore,
(s’k’Ieir.l~;(q).pIs:i)_i,(q)*(s’k’jp/sk>
(12)
and the q’s in the matrix elements of p, in f ’s and in the &funztions can be omitted. Also, the
radiation is primarily at the circular frequency o, and the correspond:ng N,,))l. Therefore, the
spontaneous emission term can be neglected. M-e obtain
X
.
[hi(k) -fhj(k)l I
<vi, k I PI vi, k> I ’
where we have replaced the sum over k by an integral.
and yields
aN
ate
DFH
i-j
= -N(t)
X
(13)
The integral over the &-function is trivial,
( 3m&rz ) ($z$)
kij Ipij(kij) I ‘[fhi(kij)-fJj(kij)]
(14)
w h e r e /P,j(kij)l”=I(Uj, kijlPj”i. k;f>j*, and kii is obtained by the energy conservation relation
Ehz(kij) - Ehj(kij) a h@,
(15)
In our case, where AU,- 1.17 eV, the band structure allows only two DFH transitions; i=3, and
j=l, 2. Thus the DFH absorption coefficient is given by
aUFH = Lxs;!H + 0532’
ufi Ii
(16)
where
(17)
The matrix elements between different valence subbands in each sy;nmetry direction of the G2
has been calculated by KanetLB) with the k*p approximation including spin-orbit interaction.
The reader is referred to Kane ’s paper for these lengthy expressions for the matrix elements. To
evaluate the absorption coezcient, we compute the matrix elements for each of the four symmetry
directions: (loo), (11 l), (110) and (r), a n d c o m b i n e them to give iPij(kij)j’ by weighting the
directions appropriately. (I ’h e (r) direction makes equal angles with (I;j3), (ill), an,l (113) directions.) Following a procedure by Kanetla’, the weighting is m,tdc by considering the surface of a
sphere in k-space and assigning a small element of area to whichcvcr of the four directions it is
closest. The weightings are then made proportional to the areas and are normalized. It results in
‘the following weights: (IOD): 0.09, (III): 0.16, (110): 0.21, (j.): 0.53. ‘I’he energy b:tnd parameters
____.___~~ ~~ -~~~---~- - ~.
(19) E.O. Kane, J. Phys. Cbem. Solids 1, 82 (1956).
crystal
CHUNG;YEE
LEUNG
165
used in the calculation of the absorption coefficient a.re taken from Fawcett(zo’. At k near k,, and
k 3% we estimate m,-0.34 m, m,-0.34 m, m,-0.0665 m for the effective masses and A~~O.14 eV for
the separation between E,, and E.,2. .A, the split-off energy at lc=O, is taken to be 0.295 eV.
Fig. 5 depicts the direct-free-hole aborption coefficient (at 1.06pm) as a function of carrier
density NC for various carrier distribution temperatures T,. It is seen th3t aDFH rises steeply with
b o t h NC a n d T,. A t l o w tcmpcrat:;res, ~~~~~ is negligible unless NC is well above 10zO cm”.
However, with distribution temperatures above IOOPK, and carrier densities higher than 5 :: 1Ol9 cm+
(a carrier density easily reached in oilr esperiments), a.,F1l is of the order of lOI c m ” . T h i s i s
significant compared to the direct interband (\aIence-conduction) absorption coe”rcient a.,, which is
reduced due to band filling at such high carrier densities.
The absorption coefficient of heavily doped p-type germanium at thermal equilibrium (T,=T,)
has been studied by Newman and Tyler(“).
Although their measurements 6s not extend to 1.0; ,~m,
by extrapolating their results we can estimate, for example, that at T,=77”K, NC= 1O:‘cm” t h e
intervalence band absorption coefficient is about 10’ cm”, w i t h w h i c h OX computation resu!t, p r e sented in Fig. 5, agrees well.
_
lo4
‘T
-
T;
. . .._._.....
298°K
.-.-.-.
503°K
p 1ClOO'K
_ --____ 2300°K
3
I
::
d
.
IO3
lo*
(
19
10
CARRIER
I
20
13
DENSITY
I
III
10
21
lCt4-3
Fig. 5. Direct-free-hole absorption coefficient versus carrier densi:y for
various carrier temperatures, at lattice tempcrsture 198°K.
IV. SATURATION OF ENHANCED SINGLE PULSE TRANSMISSION
The direct-free-hole mechanism described above is added to the throry of Elci Ed al.(B). In addition, we also make the following revisions, which are found to be necessary from our calculations.
( a) Electron-optical phonon coupling constants within the range of previously determined values
found in literature are used. As discussed in Sect. I, the values of Q, used by Elci et u/.'~J are too
low compared to the experimentally and theoretically determined values found in literature’“ ). This
is especially true for the lattice temperature T,=77”K case, nhere Q,==2 x lo-* erg cm” w a s u s e d .
The necessity of using higher Q, values is seen immediately when we simply add the 37% process
in the ESSM model. Numerical computatiol:s show that the theoretical transmissions tl;us produced
are too low. This is because we now ilave t;vo processes: plasmon-assisted recombination and
direct-free-hole transition, that enhance absorption as carrier density increases. Both of these processes
raise the carrier distribution temperature. W C thus cxpcct that the phonon-assisted relaxation mechanism, which enhances transmission by filling the optically-coupled states via carrier distribution
cooling, should be stronger than was predicted by Elci et al.(8).
( b) A more realistic conduction band structure, which consists of a direct valley r, four s i d e
(20)
W. Fnwcett, Proc. Phys. Sot. 85, 931 (1965).
(21) R. Newman and W.W. Tyler, Phys. Rev. 105, 885 (1957).
166
ENHANCED DIRECT-FREE-HOLE ABSORPTION
I-alkys (referrzd to a~ &valleys) with an indirect energy gap &-,, lower than the direct gap 6,.
3:ld sis si32 vnl!cy; ;.f-\..:llcy:;) with an indirect gap &.* higher than E‘, is used. In ths ES311
mod-l, the con.luction band of Ge was assumed to have ten equivalent side valleys with an averaged
Such an approximation simplifies the calculations. However, it may also
indirect band gap Eo.
Icad to erroneous results (see (c) below), particularly when the carrier distribution temperature is
Detailed calculation reveals that, for distrirclativcly l o w . Eu.f is about 0.2 eV higher than lZoL.
bution temperatures below 2000°K and carrier densities that are generally reached in our experiments,
most of the free electrons are in the L-valleys. For example, at TL=298”K, T,- 1003”K, iV,- 1010 cm-‘,
\ve estimate -99.5% of the free electrons are located in the L-valleys. We therefore, in our calcuAs we shall see, the use of more
lation, may consider only the_ L-valleys whenever T,< 10X1’-<(.
acceptable (higher) Q, values leads to lower distribution tenperdtures (generally less than 2tXXYK).
Much higher temperatures were predicted by ESSIM.
In the calculation of
( c) The rates of the plasmon-assisted recombination (PAR) are reduced.
t h e P A R rate by Elci et a1.(8), the plasmon resonance broadening was cjtimated to be proportional
to ten times (because there were ten equivalent side valleys) the rate for the scattering of an electron
from the r-valley to a side valley accompanied by the emission of a phonsn. However, the transition of an electron to a X-valley is impossible unless its energy is higher than Eom. Thus, at relativcly low distribution temperatures (<2Oav;c), when most of the electrons have energies less than
&s, the plasmon resonance broadening should be considered as proportional to four times the I’-*
L-valley scattering rate. l ’h e expression for the PAR rate is@)
(18)
where tn, is the effective mass of an electron in the central valley, mk=ml=m2 is the effective mass
of holes in the heavy or light hoie bands. The plasmon broadening in the revised form is given by
(19)
where Q, is the electron-acoustical phonon coupling constant, p is the Gz mass density, m, is the
effective mass of a free electron in the L-valley, and 0, is th= angular frequency of the optical
phonon. The resulting rR thus evaluated is smaller (approximately l/5 to 2/5) than that given by
Elci er ~l.‘~).
One effect of including the split between the heavy and light hole bands is on the expression
for the direct interband absorption coefficient a3. It now contains two terms,
&+A’--&+ mm:- (E-E,-A/)] -gpH-Af+ “m: ( E - & h ’ ) ] }
w h e r e m;‘= (mgl+m;‘) ,
(20)
E-ho,,
(21)
(22)
(23)
The effect of the split-off valence band on direct interband absorption is completely negligible. The
effect of the changed band structure on other processes such as free-carrier absorption, phonon-assisted
relaxation are insignificant.
The theoretical single pulse transmission predicted by the revised model is shown in Fig. 1. It
is clear that the inclusion of the DFH mechanism successfully reproduces the saturation of the en-
CHUNG YEE LEUNG
167
hanced transmission, thus removing a major limitation of the ES.%M model. This result is physically
intuitive, since we now have a mechanism which competes more efficiently than both free-carrier
absorption and plasmon-assisted recombination in opposin g the trend towards transparency caused by
the filling of the optically coupled states. Fig. 6 shows the instantaneous absorption coefficients as
functions of incident energy, immediately after the excitation pulse transverses the Ga sample. The
104-
dCA
-
-
$103 s
-
!L
5
E
g102
*
5
Iz
6
111110
2
.
loo -
v
.’
,013
I’N CICENT
./’
‘._-A--1
11
i-_.-
10’4
Q U A N T A (PT 1.06;im
)
Fig. 6. Instantaneous absorption coefficients versus energy in excitation pulse
immediately after its passage through the Ge sample. Lattice temperature
is 77X, interaction volume is 10-scm3 and pulsewidth is 11 psec. ao:
direct interband absorption, (IDFB: direct free hole absorption and aFCA:
intraband free carrier absorption.
Fig. 7. Single pulse transmission versus pulse energy for diflerent electron-optical
phonon coupling constants, 80. Pulse duration= 11 psec, sample thickness
=5.2 pm and interaction volume= 10-S cm3.
168
ENHANCED
DIRECT-FREE-HOLE
ABSORPTION
total absorption coefficient cz, which is the sum of three terms: a~, &DFR and &.?‘-A (intraband freecarrier absorption), remains relatively constant at high incident energies, where the DFH m e c h a n i s m
becomes important. Although ~~~~~ is always lower than the direct interband (valence-conduction)
absorption coefficient au, when its rate of increase with incident energy balances that of the decrease
of LY~ with energy, saturation of the enhanced transmission occurs.
In our numerical computations, the electron-optical phonon coupling constant is treated as a free
parameter. The fitting with experiment is found with Q,-7.2x 10”ergcn” for T,-77°K a n d
Q,-9.8 x IO’*erg cm-l for T,==298°K.
These values are not far from those reported recently(13*16)
thus removing another limitation of the ESS.M model. Fig. 7, which shows single pulse transmission
for different Q0 values, illustrates that the transmission sensitively depends on the electron-optical
phonon coupling constant. In fact, if all other physical parameters associated with the model were
precisely known, the experiment would represent a sensitive method for determining Q,.
V. PULSEWIDTH DEPENDENCE OF SINGLE PULSE TRANSMISSION
One feature of the theory of Elci et al. (81 is its prediction of a strong dependence of the nonlinear
transmission of single pulse through GB on the width of the pulses. This dependence on pulsewidth
is depicted in Fig. Sa, which shows single pulse transi;&ion in a 5.2 pm-thick Ge sample as a function of incident energy at initial lattice temgcrature 298”K, and for, pulsewidths of i and 1.5 t.
However, as was pointed out in the Introduction, these predictions are in substantial disagreement
with the experimental measurements(llJ
We have pointed out in Section II that, according to the ES%4 model, intravalley phononassisted relaxation and plasmon-assisted recombination will cool and heat, respectively, the carrier
distribution. Thus the pulsewidth dependence is determined by which of these two processes is
stronger in the following way. If phonon-assisted cooling dominates, more electrons (holes) will be
located lower (higher) in the conduction (valence) band. Thus the states needed for direct transition
across the valence-conduction band gap will be more com?lztely filled and the transmission will be
enhanced. However, if plasmon-assisted recombination is the stronger process, more electron (holes)
will be found high (low) in the conduction (valence) band. Therefore, this process will tend to free
the states coupled by direct interband transition, thus decreasing the transmission. When a longer
pulse interacts with the crystal, there is more time for bath processes. If plasmon-assisted recombi-
+& 45
INCIDENT
Q U A N T A [ AT 1 . 0 6
pm 1.
Fig. 8. Single pulse transmission in Ge at 298°K for pulsewidths ; and 1.5;.
Sample thickness=5.2 ,um, inertaction volume=lO-6crn8, ;=I1 psec.
CHUNG YEE LEUNG
B
0
169
nation dominates, the longer pulse will create an electron-hole plasma with a higher distribution
temperature and lower carrier density and, consequently, have a lower transmission. If phononassisted relaxation is stronger, similar arguments show that the transmission will have the opposite
pulsewidth dependence. The theoretical curves of Fig. 8a show that, in the ESSM model, plasmonassisted recombination is the dominant factor that determines the pulsewidth dependence of single
pulse transmission. Fig. 9 shows the theoretical curves, according to the ESSM mode!, of the tem
poral behavior of the distribution temperature during the passage of an energetic optical pulse of
2 x 10’ quanta. The strong dependence on pulsewidth is obvious.
After adding the direct-free-hole transition process, as was shown in the last section, we discovered
that in the ESSM mode!, the plasmon-assisted recombination (PAR) rate had been overestimated while
the phonon-assisted relaxation’ rate had been understimated. Thus the PAR process is actually not as
dominant as it had been estimated to be, and we will expect that the pulsewidth dependence to be
much weaker than that was predicted. This is in fact the case, as can be seen from the theoretical
curves for the temporal behavior of carrier temperature according to the revised mode! in Fig. 9,
and from Fig. gb, which depicts the pulsewidth dependence of transmission in the present theory.
The difference between the transmissivities for the two pulsewidths is smaller than the error associated
with typical experimental data, thus making the pulsewidth dependence of transmission hard to be
observed in an experiment.
r
I
I
I
I
PULSEWIDTH
t
_-___,.5 i
U
0.z
/
/
/
/’
1’
0.L
0.6
0.6
TIME / PULSEWIDTH
1.0
Fig. 9. Temporal behavior of carrier temperature in Ge at 298°K lattice
temperature during interaction with an excitation pulse of energy
2x1014 quanta. Interaction volume is 10-S cm3 and i is 11 psec.
VI. PROBE PULSE TRANSMISSION
Using the theory of the present paper, w e o b t a i n t h e theorctica! curves for the probe pulse
transmission depicted in Fig. 2. Although the rise in transmission is still slower than observed in
experiment, the agreement with experiment result is obviously better than that achieved by the ESSM
_ mode!. Following are the two causes for this result.
First, the rise in probe pulse transmission with delay time is due to filling of optically-coupled
(by direct transitions at 1.06pm) states by free carriers cooled via phonon emission. As we have
seen in previous sections and in Fig. 9: the carrier temperatures reached in the laser pulse-germanium
.,‘.
_.
170
ENHANCED DIRECT-FREE-HOLE ABSORPTION
interaction is considerably lower than were predicted bofare. Thus a slower increase in probe transmission can be expected.
Second, since the states coupled by direct interband transitions are located low (high) in the
conduction (valence) band near the band edge, they will not be effectively clogged until the distribution has been cooled to rather low temperatures. Thus, the ESSM model, in which the only direct
transition considered is that across the valence-conduction band gap, predicts that the drop in absorption as seen by the probe pulse is delayed to relatively long times. On the other hand, enegry states
that are coupled by direct-free-hole transitions are located much lower in the valence band. A little
drop in distribution temperature will quickly reduce the occupancy of the coupled states by holes,
thus drastically decreasing the absorption rate. Thus, when free hole transition is included, the rise
in probe transmission can be expected to happen sooner.
VII. THE ULTRA-TRANSIENT BEHAVIOR OF
DIRECT-FREE-HOLE ABSORPTION
An interesting feature of the direct-free-hole absorption mechanism is its speed. As pointed out
in Sect. III, since the hole density is generally not su.?rcient to deplet the final states of the llFH
transitions, the process is actually controlled by the temperature of the carrier distribution. It is
activated at excitation levels for which sufficiently high temperatures are reached, and is automatically
switched off as the distribution cools down. The time during which the channel of the DFH process
stays open is thus predicted by the cooling of the heated carriers. It is therefore an extremely fast
process (of the order of 10 psec), as can be seen from Fig. 10, which shows the three absorption
coefficients LY~, aDFH and aFcd during, and after the interaction with a 11 psec-long pulse containing
LATT ICE TQdFZWTLR~
77OK
EXCITATION QUANTA
10’5
PULSE BURATI ON
11 P S E C
I NtEf?ACTl ON VOLUME
IO-’ CM3
TIME
(PSEC)
Fig. 10. Time evolution of absorption coefficients during and after the interaction
with an excitation pulse of 10” quanta. no: direct interband absorption, CI,WH: direct-free-hole absorption and UFCA: intraband free-carrier
absorption.
CHUNG YEE LEUNG
10” quanta at 1.06pm. It shows that LX~,..~~
171
vanishes in less than IO psec after the excitation
VIII. CONCLUSIONS
One process neglected in Elci et al’s. (q) description of the photo-generation of electron-hole
plasmas in germanium was the direct transition between the spin-orbit split subbranches of the
valence band. We have shown (Section III) the rate of this transition rises rapidly at sufficiently
high carrier distribution temperatures and carrier densities. Incorporation this process into the original
model by Elci ef al. ($1 eliminates its most significant limitations. We are now able to account for
the experimental saturation of single-pulse transmission in germanium at high excitation levels (Section
IV) and the rapid rise in probe pulse transmission observed (Section VI). The addition of the D F N
mechanism also points to the under-estimation of the phonon-assisted cooling rate in the ESSM
theory, due to its choice of electron-optical phonon coupling constants (Q,) that were too low. By
using higher Q, values chosen by the revised model, the strong pulsewidth dependence of single-pulse
transmission predicted by the original model is largely reduced (Section V), thus explaining the experimental results of Bessey et al. (l‘), where this dependence was not found.
In the last section, we pointed out the ultrafast behavior of the D F H m e c h a n i s m . W e h a v e
recently shown that the D F H transition may also be responsible for the reflectivity enhancement
The availability of a process that entails a
observed in Ge under picosecond laser excitation(23-e4).
substantial variation of the optical constants of a semiconductor over such a short time may have
interesting potential applications.
ACKNOWLEDGEMENTS
The author gratefully acknowledge helpful discussions with Proof. M. 0. Scully and Dr. B.
Bosacchi. This work was partially supported by the National Science Council, Republic of China.
(22) B. Bosacchi, C.Y. Leung and M. 0. Scully, in Picosecond -Phenomena, ed. by C. V. Shank and E. P.
Ippen, (Springer-Verlag, New York, 1978), p. 244.
(23) B. Bosacchi. C. Y. Leung and M. 0. Scully, Opt. Commun. 27, 475 (1978).
(24) C.Y. Leung and M.O. Scully, to be published.