Semiclassical Time Evolution of the Holes from Luttinger Hamiltonian Page: 3 of 4
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3
1 6-
1 2-
08-04-
00-
10-
SX(t)
SZ(t)numerical
--- ki /k6.
4.
2.
--0SY(t)
4.
2 4 6 8 10
t (x1Q3s)FIG. 2: The mean values of spin three-component of a heavy-
hole (A - 3/2) vs. time. The red solid lines are numerical
results, and the blue dash lines are Ak/k.
Now let's study the applicability of Eq. (6). We have
used the approximation that both AE and 0 are slowly
varying functions of t, which is equivalent to AEdt < AE
and Odt < 0. They imply the same result, eEAt < hk,
which means that the approximation is valid when the
electric field has not brought large changes in momentum.
If we assume Ez 1 x 103V/m, and k 4 x 108m-1, we
get At < 2000T. So we have enough periods of oscilla-
tions in which Eq. (6) is applicable.
Fig. 2 indicates S, (t) Ak, (t)/k(t), which implies the
approximate conservation of HH's helicity. This can be
seen clearly in Fig. (5). The oscillations show that the
semiclassical spin vector always precesses around the mo-
mentum direction as the momentum changes in an elec-
tric field. The oscillation can be calculated with the sim-
ilar method above. The deep reason for the HH's helicity
conserving is the matrix element representing transition
between A 3/2 is zero. But the LH's helicity isn't
conserved as shown in the next section.
Time evolution of the light-hole: When we choose
the initial state as 1(0) U(k(0))t( 0, 1, 0, 0 )T, Eqs.
(2, 3) and the spins' equations describe the evolution of
a LH with helicity A 1/2. The trajectory is shown
in Fig. 3, and the evolution of spin is showed in Fig.
4. The anomalous shift in y-direction is not as large
as predicted from the abelian adiabatic theory of Ref.
[1], and the helicity is no longer as conserved as that of
HH. However, both the trajectory and the evolution of
spin can be explained in the non-abelian adiabatic theory
[1, 8], which properly takes into account the transition
between the two LH states.
If we confine the problem in the light hole's space, Eq.x(t)
z(t)numerical
.non-abelian
abeliany(t)
2 4 6 8 10
t (x10 3s)FIG. 3: A light-hole (A - 1/2) position three-component vs.
time. The red solid lines are numerical results, the blue dash
line is from the abelian adiabatic theory, and the black dot
lines are from the non-abelian adiabatic theory.
(4) is reduced to_ (f ) 0 - ) C
dt (C- /2 ) 0 \C2)
0.
(7)
It describes the evolution of two degenerate states. The
solution is[ cos(t 00) Cs(()t 0 j
-~t sin(Bt 00) cos(Bt 00) C()(8)
where Ot is the the polar angle at the time t. So we can
get the anomalous shift in y directions
yf (t) =CI(t)U(k) - ia,Uf(k)C(t)
3 cos(t 200) cos(3 - 200) 2 cos 00
4k0 sin 00
(9)
and the evolution of spin is (S(t))
Ct(t)U(k)SUt(k)C(t),3
Sst z(t) , sin(
SY, 2 (t) =0,1
200) + - sin(30t3 1
52, +(t) = [- cos(O1 200) - -cos(301(10)
200)].The results from Eq. (9) and (10) has been plotted in
Fig. 3 and Fig. 4, they describe the trends of numer-
ical curves very well except for the rapid oscillation on
the numerical curves, which have been explained in the
previous sections due to higher order corrections to the
adiabatic theory.
At last, we obtain the anomalous velocity in y-
direction,
Vy,az (t) = 4k2 [sin(Bt 200) -sin(30t 200)]. (11)WM"
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Jiang, Z. F.; Li, R. D.; Zhang, Shou-Cheng & Liu, W. M. Semiclassical Time Evolution of the Holes from Luttinger Hamiltonian, article, February 15, 2010; United States. (https://digital.library.unt.edu/ark:/67531/metadc927234/m1/3/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.