Itinerant Spin Dynamics in Structures of ... - Jacobs University

58 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems
all wave vectors, K = Q/Q SO , is given by
with
E B,2D,1 /D e Q 2 SO
= 1+K 2 x, (3.100)
E B,2D,2 /D e Q 2 SO
= E B,2D,1 + f 1
E B,2D,3/4 /D e Q 2 SO = E B,2D,1 − 1 2
+ 1 6
f 1/3
2
− 1 3 f1/3 2 , (3.101)
(
1±i
√
3
)
f1
f 1/3
2
(
1∓i √ 3
)
f 1/3
2
, (3.102)
f 1 = ˜B 2 −4Kx 2 −1, (3.103)
( √3 √ )
f 2 = 3 108Kx 4 +f3 1 −18K2 x , (3.104)
˜B = gµ B B/D e Q 2 SO, (3.105)
Thus, there are spin states with the same real part of the Cooperon energy, so that they
decay equally in time, but the imaginary part is different, so that they precess with different
frequencies around the magnetic field axis. A significant change of the Cooperon spectrum
appears when gµ B B/D e exceeds Q 2 SO
, as can be seen in Fig.3.16(b). All states with a low
decay rate do precess now, due to a finite imaginary value of their eigenvalue. Associated
with this change is also a change of the dispersion of the real part of E B,2D,3 which changes
for K y = 0 from a nearly quadratic dispersion in K x , a 0 + a 1 K 2 x for ˜B < 1 to one which
changes more slowly as a 0 +a 1 K 2/3
x +a 2 K 4/3
x +a 3 K 2 x for ˜B ≥ 1 [see Fig.3.16 (a)].
Weak Field
In the case of a weak Zeeman field, ˜B ≪ 1, the singlet and triplet sectors are
still approximately separated. A finite ˜B ≪ 1 lifts however the energy of the singlet mode
to E B,2D,2 (K = 0)/D e Q 2 SO
= ˜B 2 /2 + O(˜B 4 ), thus the singlet mode attains a finite gap,
corresponding to a finite relaxation rate. The absolute minimum of two of the triplet
modes is also lifted by E B,2D,2 (K = ± √ 15/4)/D e Q 2 = 7/16 + (3/4) ˜B 2 SO
+ O(˜B 4 ), while
their value is independent of ˜B at K = 0. In contrast, the minimum of the triplet mode
E B,2D,3 , whichapproachesE T+ inthelimitofnomagneticfield(seeFig.3.4)isdiminishedto
E B,2D,3 (K = 0)/D e Q 2 = 2− ˜B 2 SO
/2+O(˜B 4 ). So, in summary, a weak Zeeman field renders
all four Cooperon modes gapfull and that gap can be interpreted as a finite relaxation rate