Jun 15, 1994 - We calculate the Auger recombination rate of carriers in a quantum well taking intoaccount that due to the presence of the heteroboundaries.
PHYSICAL REVIE%'
8
15 JUNE 1994-II
VOLUME 49, NUMBER 24
NonthreshoM
Uni Uersi te de Montpellier
Auger recombination
in quantum wells
M. I. Dyakonov* II, Groupe d'Etudes des Semiconducteur, 34095 Montpellier
Cedex 5, France
V. Yu. Kachorovskii A
F. Io+. e Physico
Tech-nical Institute, 194021 St Pet. ersburg, Russia
(Received 23 November 1993) We calculate the Auger recombination rate of carriers in a quantum well taking into account that due to the presence of the heteroboundaries the momentum-conservation law is violated. As a result, nonthreshold Auger recombination becomes possible, i.e., low-energy carriers may recombine, the excess momentum being transferred to the well boundaries. This greatly enhances the Auger rate at low temperatures. We develop a method for calculating the overlap integrals in the limiting case which is of practical interest and obtain an explicit analytical formula for the nonthreshold Auger rate. The calculated recombination rate weakly depends on temperature but increases drastically with decreasing well width and may become more important for narrow wells.
I. INTRODUCTION
was assumed in Ref. 6, since the heavy-hole concentration is substantially greater than that of the light holes. Earlier, the importance of Auger transitions from the ground bound state in a quantum well to the continuum of unbound states, particularly at small well widths, was pointed out in Ref. 7, and supported by numerical calculations based on a simplified band-structure model (simple parabolic bands} and some assumptions concerning the values of the overlap integrals. The purpose of the present work is to calculate the rate of nonthreshold Auger recombination in a quantum well. We will show that the recombination rate weakly depends on the temperature, but increases drastically with decreasing well width and may become most important for narrow wells. We will obtain an explicit analytical formula for the recombination rate in the practically interesting limiting case.
Auger recombination is one of the most important recombination mechanisms for narrow-band-gap semiconductors especially at high excitation levels, which are characteristic for semiconductor lasers. The Auger process is generally considered to be the predominant nonradiative recombination mechanism for long-wavelength quantum-well lasers, and a number of papers was devoted to its theoretical investigation. ' In these works, the Auger process in a quantum well was considered quite analogously to the case of the bulk Auger recombination for which, as is well known, a threshold in the carrier energy exists. Accordingly, the same threshold was obtained for the quantum-well case. The calculated recombination rate depends exponentially on temperature and becomes extremely small at low temperatures. The threshold character of the Auger recombination is a consequence of the energy and momentumconservation laws. However, in a quantum well, the momentum-conservation law is violated, since one component of the momentum may be transferred to the well's heteroboundaries. As a result nonthreshold recombination becomes possible, i.e., low-energy carriers may recombine in a quantum well via the Auger process. Obviously this should greatly enhance the Auger rate at low temperatures. Thus, the heteroboundaries in a quantum well play the same role as phonons and defects in the bulk case, lifting the constraint imposed by the momenturn conservation. This idea was put forward exphcitly by Zegrya and Kharchenko who calculated the velocity of surface nonthreshold Auger recombination due to the presence of a single heteroboundary. However, their results are not exact because they were based on the incorrect assurnption that the main contribution to the Coulomb matrix element comes from the under-barrier region. It should be also mentioned that the dominant process is recombination with heavy holes, not with the light ones as it
FIG. 1. Auger recombination in a quantum well. Electron (1) recombines with a heavy hole (2) exciting another electron (3) to the final state (4).
0163-1829/94/49(24)/17130(9)/$06. 00
17 130
49
II. PROBLEM FORMULATION. MAIN PARAMETERS AND ASSUMPTIONS We consider the CCCH Auger process (see Fig. 1; we use the conventional notations for different Auger pro4ii
0
1
3 Eo t
'E I
Oh
1994
The American Physical Society
49
NONTHRESHOLD AUGER RECOMBINATION IN QUANTUM WELLS
cesses'): recombination
of an electron (1) with a heavy hole (2) is accompanied by excitation of another electron (3) to the final state (4). In the initial state, the carriers (1)—(3) are confined within the quantum well, their momenta being defined by the temperature (or Fermi energy) while the Auger electron (4) is ejected out of the well with a large momentum directed almost perpendicular to the plane of the well. Thus, the momentum is not conserved which is due to the presence of the heteroboundaries, characteristic of the nonthreshold process that we are interested in. It may be shown that, just as in the case of bulk Auger recombination, the CCCH process is the most important one. We consider for simplicity symmetrical quantum wells for holes and for electrons as shown in Fig. 1, and neglect the spin-orbit splitting of the valence band. The following assumptions will be made. (1) The relation between the electron quantization energy Eo, the mell depth U, and the band gap E is given by Eo «U«Eg (2) The band spectrum may be described by the Kane model and m, =m& «m
where m„mi, and mh are heavy-hole efFective masses, (3) The injection level is and hole concentrations are
the electron, light-hole, and respectively. high enough, so the electron approximately equal,
n=p .
(3)
(4) The electrons are degenerate and the holes are non-
degenerate,
E~„«T«E~,
(4)
,
where EF, =mR n lm, and Ert, =Er, m, lmz are the Fermi energies of electrons and holes and T is the temperature in energy units. (5) The electrons occupy only the first quantum level in the well, while the heavy holes occupy many quantum levels, E~~ &&Eo
~
T &&Eo
17 131
spin-orbital interaction into account will not change our results qualitatively provided that 6 is much less than Eg.
III. GENERAL
FORMULAS FOR AUGER RATE IN A QUANTUM WELL
The decay of the two-dimensional (2D) electron concentration in a quantum well due to Auger recombination is described by the equation dn
dt
= —R,
where the Auger rate R is given by the usual expression
R
1 = 2n. — g
—
S, 234 lMl 5[E, +E3 E2 E4]— X f;f Q';(1 f;), — 2
where M is the matrix element of the Coulomb interacand tion, are the Fermi functions for electrons and holes, and S is the normalization area in the plane of the well. The sum is taken over all the quantum numbers of the particles, spin included. Since we neglect spin-orbital interaction, the sum over the spins in Eq. (7) may be written as
f'
f"
g(u) IMI'= ',K (IMt+Mnl'+3IMt
—Mnl'),
where e is electron charge, ~ is dielectric constant, and
Mt=
f d3r d r'
M„=Jd
rd
q&i(r)%
r'0;(r)
3(r'),
+z(r)F14(r'),
1 qi3(r'),
qiz(r')@4(r)
1
.
(10)
Here M& and M&& are the direct and exchange Coulomb matrix elements, respectively. The two terms in Eq. (8) are due to contributions from the singlet and triplet states of the initial electrons. In the framework of Kane's model, the wave functions 4'; can be written as
qi=u(r)lS)+v„(r)lX)+v (r)lY)+v, (r)lZ),
(11)
where lS) and lX), Y), lZ) are the zone-center Bloch functions for the conduction band, and for the p-type valence band, u(r), v„(r),v (r), v, (r) are the corresponding envelope functions, which may be found by solving Kane equations for the case of a quantum well (see Appendix A). Since the in-plane momentum p is a good quantum number, me can choose the envelope functions in the form l
where Ep=m A l(2m, a ) and Ept, =Epm Iml, . Although our theoretical consideration will be quite general, we shall use for estimates the parameters of the with system InGaAsP/InP quaternary heterostructure band gap Eg =0. 8 eV, U =0.28 eV, m, =0.045, For n =p =2 X 10' we obtain m& =0.45. =0. =0. 1 01 eV. If take the mell width me eV, E~, Ezh 0 a =80 A, for the electron and heavy-hole quantization energies we have Eo =0. 13 eV, Eo& =0.013 eV. Thus, for T=0.026 eV, the above assumptions are reasonably well satisfied. We must note that ignoring the spin-orbital interaction in the valence band, while greatly simplifying the calculations, is generally not justified since the spin-orbital splitting 6 is normally not small compared to the relevant hole energies. However, we presume that taking the
cm,
u, (r)
1 =u;(z), exp(ip, p), 1/2 S &2
v, (r) =v, (z),1/2 exp(ip, p), &2 1
(12)
S
where i =1,2, 3. The normalization
condition for u;(z),
v;(z) ls
I
[lu;(z)l
+ Iv;(z)l ]dz=1 .
(13)
I. DYAKONOV
M.
17 132
AND V. YU. KACHOROVSKII
The electron in state 4 is not confined in the z direction and it is more convenient to use another definition for u4(z) and v4(z):
u4(r)
=u4(z), (SL„)'"exp(ip4p),
[(E /2)2+P2k2)1/2 (14)
(gL )1/2
[~u4(z)~
+ ~v4(z) ]dz=L,
.
(15)
Here L, is the normalization length in the z direction. Expressing the Coulomb potential in Eqs. (9) and (10) as a Fourier integral after some simple transformations we obtain 4m
I
continuous spectrum, we introduce k4=m. s4/L, and replace the sum over s4 by the integral over k4. In the Lane model we have
1
1 v4(r) = v4(z), /2 exp(i p4p),
f
g(L
)
1/2
P2
P 1' P 3
00
77
1 z
g 5(E —E4)5p 4
= f " dk4'S[3E —[(E /2) +P k,']' '} „
3 23/2
I&2 q
I34 q
1
q'+(P2
(17)
—P1)'
I, 2 = f (u,'u2+v', v2)exp(iqz)dz, " ( u,'u4 + v,'v4)exp( z I34 iqz )d—
(19)
The expression for M&& has an analogous form with indices 1 and 3 interchanged. Further calculations may be greatly simplified if we make use of the inequalities (4) and (5), which ensure that the initial electron in-plane momenta p& and p3 are small compared to both ko = m. /a [as follows from Eq. (5)] and the initial-hole momentum p2 [as follows from Eq. (4)). Thus, we may put p, =p3 =0 in Eqs. (16)—(19). Then the wave functions %', and 43 become identical and consequently M1 =M11. It follows from Eqs. (8) and (16) that (20) (o)
Z
2
1
3
=p is the carrier 2D concentration in the well. The factor —,' appears because the summation does not include the spin. Since p and p3 are neglected, the integral does not depend on these variables, and finally we obtain the following formula for the Auger recombination rate: where n
I
&
2 1/2
(24) g
where
(25)
is the electron Bohr frequency and
Es
p ~' S'L,
»34 (21)
We have neglected the initial carrier energies in the argument of the 5 function. This may be done because these energies are small compared to the band gap E [see Eqs. (1)—(5)). We have denoted by e4 the electron energy in the final state 4 measured from the edge of the conduction band. We have also neglected the quantity 4 compared to unity, since this state is not occupied at the temperatures considered. The sum in Eq. (21) is taken over all quantum numbers except spin. These quantum numbers are the in-plane momentum p and the number of the energy level s. For heavy ho1es there is an additional quantum number related to the hole polarization (see Appendix C). We can now do the summation over the quantum numbers of state 4 in Eq. (21). Since these states belong to the
(23)
Xf1=Xfz=Xf3= 2
(2m, Eg )'
Equation (7) can be rewritten now as
P
The summation over indices 1, 2, 3 may now be performed using the relations
(18)
f
(22)
In Eq. (22) we have where P is the Kane parameter. neglected the quantity p4 since it is small corn(mh T)' /A while pared to k4, because ~p4~ = ~p2~ — )'/2/X. k, -(m, E, Using Eq. (22) we obtain
(16)
P4
where dg
49
(26)
The angular brackets in Eq. (24) denote averaging over the heavy-hole states [and also averaging over the different parity states of the electron 4, denoted by an overbar in Eq. (27), see discussion at the end of this section],
(27)
f
2
In deriving Eq. (24), the relation P=fi(E /2m, )' was used. We may also rewrite the expressions for I, I,2, and I34 as follows:
I„I34 2+ p2
(28)
~
I,2= f
(u,'u1, P+v,'vh )exp(iqz)dz,
(29)
NONTHRESHOLD AUGER RECOMBINATION IN QUANTUM WELLS
(u,'up+v, 'v2)exp(
I34= Jt
— iqz)dz .
(30)
Here we use the notations
(31)
Q=k4=v'2k
(32)
of k4 is determined by the 5 function in Eq. (23). In Eqs. (29} and (30), we took into account that where the value
the momenta denote Q1
Q3
p&, p3 may
Qc
be neglected.
Thus, we may
u„v,
are the envelope functions for an electron where which is at the first quantum level and has zero in-plane momentum (the function v, is small compared to u, and describes the admixture of the valence band to the electron state). We also denote Q
=Q
~
V2=V
~
nonanalytical at some points. As a result of nonanalyticity, high Fourier components decrease according to a power law only. Thus, if the nth derivative of a function is discontinued having a finite jump at some point, then its qth Fourier component decreases for high q as C/q '+", where C is determined only by the jump magnitude. The wave functions in a rectangular quantum well are not analytical at the well boundaries (z = +a/2). We will see that their behavior near the boundaries determines the value of and that the leading term in the expansion of in powers of Q ' is on the order of Q The functions , VI", ~ change slowly on the scale 1/Q, defining the variation of u,~, vP. Thus as a function of q, the integral I34 has a maximum at q =Q, while the integral z is maximal at small q -p. Because of that, there are two regions of the variable q, which give the main contribution to I: q-p and q=Q. We may write
I
I
V) =V3=V~
Q4=Q~,
(33}
Here we ignore the dependence of the envelope functions in the final state 4 on p4, since p4= — Q. It is p2 and p2 more convenient for us to use as the quantum number for the heavy hole, the quantity
«
7r k= — s,
Q„V„QI,
I,
I —I(1)+I(2) where I'" is the contribution is the contribution
instead of s, the number of quantization level, having in mind to change the sum in Eq. (27) by integration over k and d p. We note that parity is a good quantum number for states in a rectangular quantum well. The function u, (z) is even, while v, (z) is odd. The parities of u& (z) and vzv(z) are determined by the quantum number k. As for the final state 4 for a given Q there are two degenerate states, one with even uP(z) and odd v'P(z), and the other one with odd uP(z) and even v~(z). As we will see in Sec. IV, the integral depends on the relative parities of states 2 and 4. This is why ~I is averaged over the parity of states 4 in Eq. (27).
I
~
IV. CALCULATION OF
I
In order to derive I, one must find the electron and heavy-hole wave functions in a quantum well using the Kane model, and then calculate the overlap integrals I&2, I34 This task is rather dif6cult and it seems to us impossible to obtain analytically the rigorous expression for in the general case. However, one can simplify the calculations making use of the large value of the momentum of the final electron Q-ks. We have developed a method for evaluating the overlap integrals by means of an expansion in inverse powers of Q. This allows us to obtain analytical expressions for and for the Auger recombination time ~, which are valid in the special case restricted by conditions specified above. Our method is based on the following well-known properties of high Fourier components of a slowly varying function. Such components are generally exponentially small, except for the case when the function is
I
of the region q = Q and I' ' of the region q -p. We will see that
of small q is the most important one.
the contribution
A. Calculation of
(34)
a
17 133
I"'
At first we will calculate the integral I, 2 This integral is the qth Fourier component varying expression (u,'u& v+v,'vt, v), which the first derivative at the boundaries of quantum well as it follows from the Kane Appendix A}. Rewriting exp(iqz) as exp( iqz) =
and integrating
I,2=
1
d dz
equations
(see
],
by parts, we have
exp(iqz) CC
[exp(iqz)
at large q =Q. of the slowly has a jump of a rectangular
(u,4'ul",k v+v,'vzv)dz .
dz
(36)
The jumps in the first derivative of the expression a /2) and (u,'uz v+ v,'vt", v) lead to terms with 5(z — 5(z+a/2) in the second derivative. Since q is large, these terms give the main contribution to I&2. In the expression for the second derivative in Eq. (36}, one should keep only the term u,'(u„"v)"+(v,', )"vt„v. Other terms may be shown to have higher powers of either k ' or m, /ml, . From the Kane equations (see Appendixes A and 8, it follows that the singular parts of (u„kv}"and in the lowest order in m, /m& are given by (
v„}"
iP(ul",
I
v)" = —U'(z) vs,
'
iP(v,', )"= —U'(z)u, .
(37}
Since for a rectangular quantum well
we
U'(z)= U[5(z — a/2} —5(z+a/2)], obtain the following result for I&2..
(38}
M.
17 134
I&z=
' —e' '
2iU
a/2)vz", i'( u,'( —
'
[e
2
[u,*(a/2)]
AND V. YU. KACHOROVSKII
—a/2)
u,'(a/2)vt"„(a/2)] .
Here, we have replaced q by Q. Replacing (q +p ) ' by Q ing over q we get
I"'=
I. DYAKONOV
(39)
in Eq. (28) and integrat-
(40)
for opposite parities of the functions v&,v(z} and u,~(z), for equal parities of these functions. and
I"'=0
Of
I'
g
f
S)2 =
)PQ
As a result, we have
. iqz)dz—
(v,', u,~+ u,"v,~ )exp(
(41)
J(al2)=
I
~= —
1
i(pv„"v)u,*
dz
t-')",
(u
fi(z
2p
uC
—e
'
'
~(
C
—a l2)
u,"(a/2)ug(a/2)]
.
2p
I
—oo dz e
— i2 (4S)
a/2)+
2
—P~z —a/2~
.d X i ~
we find
I' '=I'~'(a/2) —I'~'( —a/2),
(44)
where
—p lz —a/2l
+v,*vh&)
(45)
The expression u,*ul", +v,'vi, i' in Eq. (45) is proportional to 1/k~ since the functions u& and v, describe the small admixture of the conduction-band states to the heavyhole state and the valence-band states to the electron state, respectively. Thus, the integral in Eq. (45) is pro-
'
+p u,"vq,
For calculating the integral in Eq. (49) we may, taking into account the inequality U»Eo, use the wave functions for an infinitely deep well and accordingly restrict the integration limits by z = — a/2 and z =a/2. We will consider now the situation T »Eo& when the heavy holes are not quantized. In this case, the heavyhole in-plane momentum is inuch larger then ko =m. /a:
»ko
(50)
The main contribution to the integral in Eq. (49) comes from the narrow region with a width 1/p near the boundary z=a/2. The function u, varying slowly within this region it can be written approximately as
u, (z) = u,'(a /2)(z
—a /2),
(51)
since for an infinitely deep well u, (a /2) =0. The integral in Eq. (49) may be now easily calculated by using Eq. (51) and the expressions for the heavy-hole wave functions from Appendix C. This calculation gives
u,'(a /2)uP(a /2)
dz(u, *ul",
'
(pvt, ~)u,
(49)
p (43)
Eq. (43) in Eq. (28) and integrating over q,
I
I
J(a/2) =u,'(a/2)vz", i'(a/2)
(42)
'( —a/2)u
I = ik UP [e'~'
P
p
The result is
z
k
d e
w.
which may be derived from Eqs. (Bl) and (82) if we neglect terms of the order of U/Eg and m, /ml, . Again, integrating by parts, we pick up the singular term only, which comes from v,", =iU'(z)u, /P. The second term in the integrand may be transformed in a similar way We then though the calculations are more cumbersome. obtain
X
+
I'"
g
Substituting
(„»kpkp)
Note that the expression for I' ' in Eq. (47) has just the in Eq. (40). same order in k as Equation (48) may be rewritten by integrating by parts twice and using the simple formula — e plz a/2l = — — +e Plz —a~21 dz
'„u,
(47)
where
d
The functions u,~ and U,~ vary fast compared to the ', and exp( iqz) T—hus, e can again apfunctions v, ply the method described above. We transform the first term in the integrand of Eq. (41) as follows. We first use the relation u
~
'
We start with calculation of the integral I3& for small — the small in-plane q p. Since we have neglected momentum p compared to Q, we may write v,'v~= v,', v,~. Using the relations given by Eqs. (AS) and (A9), and integrating by parts, we obtain I34
portional to k '. To see this directly, one may use Eqs. (A10) and (Al 1) keeping in the expression for S&z the only oo: term which is finite as k
I' '(a/2) = —iU u,'(a/2)u ~(a/2)J(a l2),
vz,i'(a/2)uP(a/2),
B. Cglgglgtion
49
J(a/2) =u, (a/2)vt"„~(a/2) '(a/2)]* . (Sa)'~ [u, k +p 2
(52)
The first term in Eq. (52} gives a contribution to I' ', which is of the same order as [see Eqs. (40) and (47)]. Ho~ever, the main contribution is given by the second
I'"
NONTHRESHOLD AUGER RECOMBINATION IN QUANTUM WELLS
49
term. To see this, note that the first term tends to zero as U~00, since the wave functions are zero at the boundaries of an infinite well, while the second term remains finite. More precisely, it may be shown that the first term where eh is the may be neglected if eh U(m, /mz heavy-hole energy. We shall assume this condition to be
),
«
satisfied. Thus, only the second term in Eq. (52) should be kept. This means also that is small compared to I' ', and may be neglected. Finally we have =I'z',
I"'
I
U
I(2)—
(2a)'"k4P g X [u, (a/2)u, '(a/2)] uP(a/2)
2
2+p 2
(53)
for opposite parities of the functions v&,v(z) and u,~(z), and =0 for equal parities of these functions. The boundary values of the functions entering Eq. (53} are calculated in Appendix B. Using these values, we obtain the following expression for ~I ~:
I
I
g
1
o
1
3 ks (kga)
X
m,
o
Es sg(p, k)
m~
k k
+p
cos (Qa/2) sin
(54}
(Qa/2)
where cos (Qa/2) stands when u,~(z) is even and vz)'(z) is odd, and sin (Qa/2} stands when uP(z) is odd and v&v(z) is even; ~I~ =0 for equal parities of these functions. Note that I=O also for one of the two possible heavy-hole polarizations (see Appendix C).
V. RESULTS AND DISCUSSION We average the expression in Eq. (54) with the Boltzmann distribution for the heavy holes. It may be easily calculated that
If we
k
1
2
kz+pz
e (p k)
3T
p-(2m&T)'
«%kg. Thus, the most effective Fourier component of the Coulomb interaction is proportional to 1/p . On the other hand, the overlap integral I34 [see Eq. (30)] is equal to zero for q =0 since the states 3 and 4 are orthogonal. Because of that at small q we have I34-q-p [see Eq. (43)]. Hence the matrix element varies as 1/p and ( ~I ) decreases with temperature as
I
~
1/T. Equation (57) is not valid both for very low and for At low temperature (T very high temperatures. ) most of the heavy holes occupy the first quantum level. One can show that the structure of the overlap integral is such that in this case the Auger rate goes down to zero as T~O. Thus, the recombination rate should have a maximum at T-Eo&. On the other hand, at high temperature (T not only the holes but also the electrons occupy many levels of quantization. Qualitatively in this case one should replace the energy Eo in Eq. (57} by T, so that the Auger rate will increase proportional to T. In addition, at high enough temperature the conventional Auger mechanism should come into play. For the values of the parameters listed in Sec. II and T=300 K, Eq. (57} gives r=25 ns. Our results along with the bulk Auger rate (see Ref. 10) calculated for the same parameters are depicted in Fig. 2. The broken line represents qualitatively the low-temperature behavior. We will finally briefly discuss the role of other Auger processes. Processes in which there is a light hole in the initial state, such as CCCL or CLLL are not important, since the number of light holes is exponentially small un-
«E»
»Eo)
»
der the conditions considered (Eo& =Eo T). The probability of processes in which there is a heavy hole in the final state, such as CHHH, is small because the momentum of the Auger heavy hole would be very large [of the order of (m& /m, )' fikg ]. Thus, the only alternative process to the one that was considered in this work is the CHHL process, i.e., recombination of an electron with a
(55)
define the Auger recombination
time
v
0.20
as 0. 15—
(56) we finally obtain from Eq. (24) 1
32v2
n
+o
3m.
k
E
7/2
'co
U me
T
17 13S
0. 10—
(57)
m~
Equation (57) displays a very strong well-width dependence of the recombination time: r changes as a (since Eo is proportional to a ), so the role of the nonthreshold Auger process should greatly increase for narrow wells. Another interesting feature of Eq. (57} is the inverse temperature dependence for the Auger rate. This may be explained as follows. Unlike the case of recombination in the bulk, now the large momentum %kg of the Auger electron comes from the well boundaries, while the Coulomb interaction transfers the momentum of the order of the heavy-hole thermal momentum
0.05—
I
/
I I
I I I
I I
0.00
'
I
0
100
300
200
400
500
T (K) FIG. 2. Temperature dependence of Auger recombination time. Our results along with the bulk Auger rate calculated for the same parameters listed in Sec. II. The broken line represents qualitatively the low-temperature behavior.
M.
17 136
I. DYAKONOV
AND V. YU. KACHOROVSKII
heavy hole accompanied by excitation of another heavy hole to the light-hole band. However, the probability of this process is also relatively small, since in this case the matrix element would be proportional to the value of the heavy-hole wave function vt*„(a/2) at the well boundary instead of the value u,'{a/2) in Eq. (45). It may be shown that if m, «mz, the ratio vt'„(a/2)/u, '(a/2) is small.
S„=— iP[u„'v +u +
I
ACKNOWLEDGMENTS
%e are indebted to R. Kazarinov, F. Capasso, R. Suris, and G. Zegrya for stimulating discussions. This work was supported by AT&T Bell Laboratories.
In this work we calculate the electron and hole wave functions in the framework of the Kane model. For simplicity, we use a simplified version of the Kane model in which the spin-orbital interaction as well as the anisotropy of the energy spectrum are neglected and the electron mass is assumed to be equal to the light-hole mass. In this simplified version, the Kane equations may be written as follows:
"
[E+E /2+ U(z)]v+
$2
=—
iP divv=—[E Eg/2
—U— (z)]u
2mh
curlcurlv,
.
(Al) (A2)
Here U(z) is the confining potential for the quantum well assumed to be the same for electron and holes, and
P=A
E
1/2
(A3)
2me
is the Kane parameter (interband optical matrix element). The term (fi /2m&) curlcurlv in Eq. (Al) accounts for the finite heavy-hole mass. The conventional energy spectrum is obtained from Eqs. (Al) and (A2) in the bulk case [ U(z) =0]:
E ( K ) —+ [(E /2)2+ P2K 2]1/2 E (K)= —[(E /2) +P K ]'
(A4)
Eg(K)= —E /2 —A' K /2m„. For the electron and light-hole states we have
=0,
v~~K and
while
f [u„u'+v„v']d r=0 .
(A5)
This means that the integrand in Eq. (AS) may be written as a divergency of some vector
S„
u„*u
+ v„*v =
The expression for
1
E —E„divS„
S„ follows
[v„'X curlv
2mh
]
—[v
X curlv„*]
(A7)
Equations (A6) and (A7) are used for calculating the integral I' ' in Sec. IV B. For our purposes these equations may be further simplified. For two electron states 3 and 4 the second term in Eq. (A7) is equal to zero, since curlU, =curlu, ~=0. The in-plane momenta p3, p4 are small compared to Q and we may write dS34 dlvS34
dz
+i (p4 —p3 }S=
+V
Q
Q
dS34 dz
dS34 V
{A6}
from Eqs. (Al) and (A2):
(A8)
z
g
where
S34= iP[u,'— vg +univ, ", ] .
(A9)
Here we used the approximation E4 — E3 As for states 1 and 2, we have
kp+
c
h
1
+Vkp
c
dS(2
h
Eg.
(A 10)
+iPS&2
g
In using Eq. (A10) for the transformation of Eq. (45) of the main text, it is suScient to keep only the lowest-order terms in 1/k in the expression Eq. (A7) for S|2. Since v, -u, /k and ul", v vt", v/-k, the only term in Eq. (A7) ao is u, Uh, and we may write which is finite as kg
~
S)2
(Al 1)
LPQc Vh
APPENDIX B: ELECTRON STATES IN A QUANTUM WELL
For the electron states considered in this paper, the inplane momentum may be neglected. In this case Eqs. (Al) and (A2) have a simple form:
iP du /dz
= [E+— E~ /2+
iP dv, /dz=—[E Eg/2
for the heavy-hole states vlK and divv=O. Note that there are two different polarizations for the heavy-hole state. Solutions of Kane equations with different values of the energy E are orthogonal, curlv
v„*]
Thus, Eq. (A6) may be rewritten as
APPENDIX A: KANE EQUATIONS
iPVu
49
(B 1)
U (z ) ]v, ,
. U(z)]u —
—
(B2)
The solution of these equations for a rectangular quantum well may be easily obtained. However, what we need are only the boundary values u (a/2) and u'(a/2), which enter Eq. (53). For the low-energy ground state (states 1 and 3), Eqs. (Bl) and (B2) may be reduced to the onedimensional Schrodinger equation for u (z). The solution of this equation for the case of a deep well (U&)Eo) gives 1/2
u, (a/2)
=
2EO
aU
u,'(a /2)
= ——2 a
a
1/2
(B3)
For the high-energy state 4 of the Auger electron with momentum Q =k v'2, we may neglect U(z) in Eqs. (Bl) and (B2). Taking into account our normalization [see Eq.
NONTHRESHOLD AUGER RECOMBINATION IN QUANTUM WELLS
49
(15)], we easily obtain
cos(ga/2)
u,~(a/2)= 3i~2 X sin(Qa/2}
for even uP(z)
'
(B4)
for odd u,~(z)
These results may create an impression that the lowenergy electron states may be described by the Schrodinger equation, and that in calculating the wave function in state 4, the well potential U(z) may be neglected. In fact, this is true only for determining the boundary values of u (z) and u'(z) in Eq. {53). However, for the calculation of the singular parts of u" and v,", which are used in the derivation of Eqs. (40) and (53), these approximations are not valid. APPENDIX C: HEAVY-HOLE WAVE FUNCTIONS IN A QUANTUM WELL
The low-energy tion 2mh
hv h
1
V divvy+ U(z)vi,
=si, vt,
,
L
(Cl) which may be derived from Eqs. (Al) and (A2) by excluding the function ul, and neglecting terms of the order of ei, /Es and U/Es. Equation (Cl) describes the light and heavy holes in the case when spin-orbit interaction is absent and is analogous to the t.uttinger equation. '2 The light- and heavy-hole quantization in an infinitely deep well was studied on the basis of the Luttinger equation in
„and
1. Solutions For an infinitely deep
well we have
Vhx
—k coskz
Vhz
ip sinkz
coshyz
k
(C2)
(C3)
where k and y are related to the energy eh by the equation
Permanent
address:
A.F. Ioffe Physico-Technical
tanh(ya/2),
Institute,
194021, St. Petersburg, Russia. G. P. Agrawal and N. K. Dutta, Long-8'avelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
(C5)
p k cos(ka/2) p cosh(ya /2)
{C6)
(C4) and (C6) define the energy spectrum.
For our purposes we must consider the situation when k -p I/a and mt mi, . It follows from Eq. (C4) that in this case y is very close to p. Equations (C4) and (C5) may be now simplified to give
»
«
tan(ka/2)
=k/p,
(C7)
8 =22 sin(ka/2)exp( —pa/2) .
(C8)
From Eq. (C7) we find ka
=2ns+5,
, tan(5,
/2) =2trs/(pa),
(C9)
»1.
where s is a positive integer, 0 & 5, & n, and s Thus, for the high heavy-hole quantum levels that we are interested in, we may ignore the phase 5, . By normalizing the wave function (C2), we find 1/2
2
(C 10)
a(p +k } 2. Solutions with odd vq„ The derivation The results are vh
vh
tan(ka /2)
is quite similar to the previous
=C
k sinkz ip coskz
where s
» 1, 0
+D
,
case.
—p sinhyz i y coshyz
= —p /k,
={2s+1)m. +5,
tan5,
vt, (a /2) =0,
=
+tan(ka/2)
ka
p +8 — iy sinhyz
(C4)
-k.
D =— 2C cos(ka/2)exp(
with even vq„
(p' —y') .
—
Ref. 13.
If we direct the in-plane hole momentum p along the x axis, we can see that vs is decoupled from vi„and vi, and satisfies the Schrodinger equation for a particle with mass mh. Heavy holes with the y polarization do not participate in the Auger process since such states do not overlap with the electron states. The functions vi„and vi„satisfy a system of two coupled equations, which follow from Eq. (Cl). For a symmetric with respect to the z=0 plane potential U(z), soluthere are solutions with even vt and odd vt, tions with odd vh„and even vs.
2ml
The second term in Eq. (C2) describes the admixture of the light-hole state due to the rejections at the well boundaries. The quantity y is real if p &k(milmi, }' a condition we shall assume to be fulfilled since we are interested in the case when p From Eqs. (C2) and (C3) we derive the following relations:
Equations
hole states are described by the equa-
+
(p'+k') =
2mh
17 137
(C12)
—pa/2),
(C13) (C14)
& 5, & m,
= m{2s+1)
(C15)
Qp
2
a(p +k }
1/2
(C16)
2L. C. Chiu and A. Yariv, IEEE J. Quantum Electron. QE-18, 1406 (1982). N. K. Dutta, J. Appl. Phys. 54, 1236 (1983). ~A. Sigimura, IEEE J. Quantum Electron. QE-19, 932 {1983}.
M.
17 138
I. DYAKONOV
AND V. YU. KACHOROVSKII
R. A. Abram, and M. G. Burt, J. Phys. C 16, L171 (1983). 6G. G. Zegrya and V. A. Kharchenko, Zh. Eksp. Teor. Fiz. 101, 327 (1992). 7C. Smith, R. A. Abram, and M. G. Burt, Superlatt. Microstruc. 1, 119 (1985) 8In fact not only the first derivative but the expression itself is discontinued at the boundaries, since the functions u, and uz have small jumps proportional to (m, /mz )kg ', which are ab5C. Smith,
~
sent in the limit mz = It may be shown that taking these jumps into account results in an additional term in Eq. (40), which is small due to the inequality m, ((mz. During these transformations one must take into account terms
~.
49
of the order of U(z)/E~
(which give singularities after differentiation) and also the jump in the second derivative of uc. ~~V. N. Abakumov, V. I. Perel, and I. N. Yassievich, NonradiaAmtive Recombination in Semiconductors (North-Holland, sterdam, 1991), p. 202. E. O. Kane, in Handbook on Semiconductors, edited by W. Paul (North-Holland, Amsterdam, 1982), Vol. 1, pp.
193-217.
J. M. Luttinger, M. I. Dyakonov 1584 (1982).
Phys. Rev. 102, 1030 (1956). and V. A. Khaetskii, Zh. Eksp. Teor. Fiz. 82,
