PRL 95, 056803 (2005)

PHYSICAL REVIEW LETTERS

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Electrical Pump-and-Probe Study of Spin Singlet-Triplet Relaxation in a Quantum Dot
S. Sasaki,1,* T. Fujisawa,1,2 T. Hayashi,1 and Y. Hirayama1,3
1

NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan
2
Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
3
SORST-JST, 4-1-8 Honmachi, Kawaguchi, Saitama 331-0012, Japan(Received 9 February 2005; published 28 July 2005)

Spin relaxation from a triplet excited state to a singlet ground state in a semiconductor quantum dot isstudied by employing an electrical pump-and-probe method. Spin relaxation occurs via cotunneling when
the tunneling rate is relatively large, confirmed by a characteristic square dependence of the relaxation rateon the tunneling rate. When cotunneling is suppressed by reducing the tunneling rate, the intrinsic spin
relaxation is dominated by spin-orbit interaction. We discuss a selection rule of the spin-orbit interaction
based on the observed double-exponential decay of the triplet state.
DOI: 10.1103/PhysRevLett.95.056803

PACS numbers: 73.61.Ey, 73.23.Hk, 73.63.Kv

Electron spin in semiconductors has been a focus ofresearch in the context of spintronics, in which spin is
manipulated with spin-orbit coupling [1,2], and of quantum computation, in which spin carries a quantum information [3]. In contrast to two-dimensional electron gas(2DEG) with continuum density of states, electron spinin a quantum dot (QD) is basically free from elastic scattering, and the resulting long-lived spin states are favorable
for spin-based applications. Indeed, relaxation times ofmore than 100 s have been reported in QDs between
Zeeman sublevels [4 –6], as well as between a spin triplet
and a singlet state [7,8]. These relaxation processes havebeen discussed in terms of either spin-orbit interaction or
the cotunneling effect. In this Letter, we study spin relaxation from a triplet state to a singlet state in a lateral QD, inwhich all the relevant parameters can be controlled withthe gate voltages. We observe smooth transition of therelaxation mechanism from the cotunneling regime to thespin-orbit regime by changing tunneling rates. The decayof the excited triplet state follows a single exponential
curve in most conditions, but double-exponential behavior
is observed at a particular magnetic field where the triplet
state crosses another state. This might be related to thelong-lived spin-entanglement state under strong spin-orbit
interaction.
Figure 1(a) shows a scanning electron micrograph(SEM) image of our QD device. The Alx Ga1x As 2DEG
is constricted by combined dry-etching and surface
Shottky gates. We use only the three gates on the righthand side to form a single QD as shown by the white circle.
All the measurements are performed in a dilution refrigerator at 90 mK with magnetic field B applied perpendicularly to the 2DEG.
The dot used in this study has charging energy of U 
2 meV and electron number N  8. When the magnetic
field is not very small (B > 0:4 T), electron orbitals in a
QD can be classified by the Landau level (LL) index, which
they approach in the high field limit [9]. Many-body correction of direct and exchange Coulomb interactions in0031-9007=05=95(5)=056803(4)$23.00

duces spin and orbital transitions associated with different
LLs in low magnetic field, but with the same LL in highmagnetic field [10]. Figure 1(b) shows an observed dc
current I through the dot as a function of the left gate
voltage VL and B. The two stripes show a pairwise motion

FIG. 1 (color). (a) SEM picture of the device together with aschematic of the measurement set-up. (b) dc current I as afunction of VL and B measured at Vsd  0:15 mV in the first
cooldown. I for the even N data is divided by 10. Evolution of
the current peak is sketched around the transition fields (triangles). (c) dI=dVL at Vsd  1:2 mV taken at the dashed squareregion shown in (b). ST scale shows the region where the spin
relaxation measurement is conducted. (d) dI=dVL at Vsd 
1:0 mV measured in the second cooldown. The vertical dashedlines denote spin transitions of the ground state. A blue circle
marks an anticrossing between two singlet states. Intensity forthe high B data is enhanced by multiplying some smooth
numerical function.

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 2005 The American Physical Society

PHYSICAL REVIEW LETTERSPRL 95, 056803 (2005)

with B reflecting spin degeneracy. The lower stripe, corresponding to odd N, involves a level crossing (denoted by asolid triangle), associated with two orbitals in different
LLs. The upper stripe for even N involves two spin transitions (open triangles) under Coulomb interactions. The
ground state for even N is assigned to be spin triplet
between the two transition fields, otherwise spin singlet
state [10]. These spin states can be observed in the excitation spectrum of Fig. 1(c), in which the derivative of thecurrent, dI=dVL , with a large Vsd  1:2 mV, is plotted as afunction of VL and B. Some excited states as well as theground state that fall within the source-drain transport
window are observed. We study spin relaxation from thetriplet excited state to the singlet ground state (denoted by
the arrow) separated by energy, ST . This relaxation involves orbital change between different LLs (inter-LL
transition). Figure 1(d) is another excitation spectrum
taken with the same device but in the second cooldown.
In addition to the similar singlet-triplet transitions at B 
0:5 T, four spin-flip transitions are resolved at B  2–3 T
until the system enters a stable totally spin-polarized regime (  1), from which N  8 is estimated [9]. We also
study spin triplet-singlet relaxation (denoted by the arrow)that involves orbital change within the same LL (intra-LL
transition). The measurement in the 1st (2nd) cooldownwith relatively fast (slow) cooling speed resulted in moderate tunnel rates and spin transition fields suitable forstudying spin relaxation involving inter- (intra-) LL transition, but both sets of data show similar characteristics.
An electrical pump-and-probe measurement is performed by applying two-step square pulses to the plunger
gate [7]. First, the singlet and triplet states are emptied bylifting both states above the Fermi energy as shown in theinset to Fig. 2(a) (initialization). The duration of thisinitialization is tl . Next, both states are pulled down below
the Fermi energy as shown in the inset to Fig. 2(b). Then,
only one electron can enter the dot because of the Coulomb
blockade. This electron, if it populates the triplet state witha probability P, is allowed to relax to the singlet ground(a)t h =500ns(b)Γtot =1.2ns
-1

10

2 1
1
cot  ST @tot  

<nt>

nt

τ s=90 µs
T

5

µ

S

S

µD

µS

5

T

µD

S

0

0
0

5

10

tl (ns)15

state while the pulse height is kept at this condition during
the wait time, th . The three triplet sublevels with SZ 
1; 0 are presumably populated with an equal probability
since the Zeeman splitting is negligibly small and thetunneling probability does not depend on SZ . Finally, thepulse height is adjusted so that only the triplet state iswithin the transport window of 150 eV defined by the
Fermi energy of the left and right leads (readout). Then, theelectron can contribute to the current only if it remains
in the triplet state after th . This readout pulse width tm isfixed to 500 ns. Actually, several electrons [1=1  P]flow during this time for the unrelaxed case. Therefore,
the average number of tunneling electrons per one pulseP
cycle nt follows an exponential decay nt  1P
expth =s , from which the spin relaxation time s canbe determined [7].
Figure 2(b) shows observed nt as a function of th atST  300 eV. nt shows a single exponential decay withs  90 s. The relatively large nt th  0 ’ 8, corresponding to P ’ 0:89, comes from the fact that an injection into the triplet (singlet) state is more (less) effective
because an electron is added to the outer (inner) orbitalwith a larger (smaller) tunneling rate in this magnetic fieldregion [9].
We can also determine the total tunneling rate,tot  L R  by changing the initialization pulse
width, tl . Here, L (R ) is the tunneling rate for the left
(right) barrier, which is changed by VL (VR ). Figure 2(a)shows an example of such measurement. The rise timecorresponds to the escape time from the singlet groundstate through both tunneling barriers, and tot is estimated by fitting the data to the expression nt P
1P f1  exptl tot g. tot is successfully changed between 1  108 and 3  109 s1 by changing the gate
voltages.
When tot is large, higher-order tunneling, or cotunneling, is quite effective in causing an exchange of electrons
having opposite spins between the dot and the lead electrodes, resulting in a spin relaxation. According to thesecond-order perturbation theory, the cotunneling rate
1
cot is approximately given byt l =500ns

10

0

100

200

300

th( µs)FIG. 2. (a) nt as a function of the initialization time tl with
ST  300 eV (B  0:6 T in the first cooldown). The inset
shows the energy diagram of the initialization process. (b) nt as afunction of the wait time, th . The inset shows the energy diagramof the relaxation process.

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1 2 =h;(1)where  and  are energies required to excite theN-electron triplet state to N  1- and N 1-electron virtual states, respectively [7]. tot is the effective tunnelingrate for the cotunneling process from the triplet to thesinglet state through either virtual state, while the experimentally obtained tot measures the tunneling rate from thesinglet state to N  1-electron state. Figure 3(a) showsobserved s as a function of tot for representative conditions. The data points at large tot are almost parallel2
with the dotted line having a slope 2, i.e., 1
cot  tot ,which is consistent with Eq. (1), assuming a linear relation tot  tot . Figure 3(b) compares the observed ST056803-2

2

6
4
2

10
2

4 6

2

4 6

0.1
1
9 -1
Γtot (10 s )

2

0.0
0

200
400
∆ST (µeV)

600

FIG. 3. (a) Log-log plot of s as a function of tot . s for interLL is measured at B  0:6 T (ST  300 eV) in the 1st
cooldown, while that for intra-LL is measured at B  1:2 T(ST  380 eV) in the 2nd cooldown. The solid lines are fitted
to the data. (b) The coefficient  for the cotunneling component
as a function of ST . The curve is calculated with an effective
tunneling rate, tot  tot .

dependence of  with calculated   fhST 1
1 2 2 g=42 . We take     U=2  ST  in thecalculation, which approximates the experimental conditions.  of 0.3 gives a reasonable fit to the experimental
results. Therefore, spin relaxation in the large tot regime
can be well explained by the standard cotunneling theory.
It is seen in Fig. 3(a) that, when tot is reduced, s
increases and eventually saturates. In this regime, thecotunneling process is suppressed, and inelastic spin relaxation is dominated by phonon emission under the spinorbit coupling effect rather than by coupling to nuclear
spins, etc. [6,8,11]. The solid lines in Fig. 3(a) are thecurves 1=so 1=cot 1 fitted to the data. Here, so is
the relaxation time due to the spin-orbit interaction, whichis independent of tot .
Figure 4(b) shows the ST dependence of s measured in
the small tot regime where spin-orbit interaction is dominant. s is almost constant in a wide ST regime (except at
a dip around ST  380 eV) and tends to increase whenST < 200 eV for both inter- and intra-LL data. Thisfeature might have arisen from the phonon emission spectra in a QD in the presence of spin-orbit coupling. The
phonon emission rate is maximized when the phonon
wavelength is comparable to the dot size (phonon energyof 300 eV for the dot size of 30 nm in the crystal growth
direction) [7,12]. The similarity of the ST dependence of
the inter- and intra-LL data supports this crude model.
Longer s is observed for the intra-LL case than for theinter-LL case in the whole ST region explored. Thismight reflect the different orbital quantum numbers involved in each case, which are relevant to the orbital effecton the phonon emission and spin-orbit interaction [13,14].
However, the precise mechanism is not clear yet.
Generally, in a one-electron system, the spin-up (down)state of one orbital is coupled with the spin-down (up) stateof the other orbital when spin-orbit coupling is consideredbetween the two orbitals [14]. This coupling gives rise to(a) 0.9

8
6
4

0.8

(c)

∆ ST =300 µ eV

nt

100

2

0.7
(b) 800
600

inter LL
intra LL

400
200

0.5
8
6
4
2

100

1

50

0.5
0
200

400
∆ST (µeV)

160 µ s1<nt>

spin-orbit

finite phonon emission probability between Zeeman sublevels. In the case of two-electron singlet/triplet states,
singlet state (jSi) is coupled with two of the triplet sublevels (jT i and jT i) having SZ  1, but not with theother sublevel (jT0 i) having SZ  0 [11,15]. Therefore,
relaxation from the triplet to the singlet state should havea selection rule in which the jT0 i state is still free from thespin-orbit relaxation mechanism as shown in Fig. 4(d).
This simple argument applies when the singlet-triplet energies are so close to each other that coupling with otherstates is negligible. Unfortunately, our pump-and-probe
technique is not available for the small ST regime because
the minimum energy resolution is about 100 eV. Whenanother singlet excited state is involved, however, clear
selectivity may appear in the vicinity of the level crossing
between the triplet and the singlet excited states.
Actually, as shown in Fig. 1(c), we do see a level crossing with an unknown state (X) at about B ’ 0:52 T (ST ’
400 eV) that could be a singlet state. When the dot
potential has no rotational symmetry, large anticrossing is
expected between the same spin states [16]. Typical anticrossing energy between the same spin states in our noncircular dot device is about 150 eV [for instance, marked
by the circle in Fig. 1(d)]. The unresolved anticrossing

nt

co

2

α (ps)

0.1
Γ tot g
-1 = α l i neτ cott u n n

τs(µs)

6
4

1st, inter LL
theory, β=0.3P(b)inter LL
intra LL

τs(µs)(a)1000

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PHYSICAL REVIEW LETTERSPRL 95, 056803 (2005)

TT0
T+

(d)

X

60 µ sS

810 µ s∆ ST =380 µ eV

100

200

th(µs)

300

FIG. 4 (color online). (a) ST dependence of the excitation
probability P for the inter-LL relaxation (1st cooldown). (b) ST
dependence of the s for the inter- and intra-LL relaxations. Datawith open symbols are obtained by fitting a single exponentialfunction, while the solid circle represents the fast component of
the double-exponential function. The dotted lines are guides forthe eye. (c) A logarithmic plot of nt vs th with tot  2:0 
108 s1 at ST  300 eV (B  0:6 T). The solid line is anexponential function fitted to the data. (d) A logarithmic plot of
nt vs th with tot  2:7  108 s1 at ST  380 eV (B 
0:55 T). The solid line is a double-exponential function fitted
to the data. The dotted and dashed lines are the fast and slow
components, respectively. The inset schematically shows allowed and forbidden transitions from the triplet sublevels tothe singlet state.

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PRL 95, 056803 (2005)

PHYSICAL REVIEW LETTERS

between T and X states (<100 eV) implies that X is asinglet state. As shown by the arrow in Fig. 4(b), a sharp
dip in s is observed at ST ’ 380 eV close to the X-Tcrossing point. The width of the dip is as narrow as about
20 eV in ST (10 mT in B) [17]. The dip could be
attributed to strong spin-orbit coupling around the crossing
point, resulting in the short relaxation time, as theoretically
predicted in Ref. [14]. It should be noted that the decay ofthe pulse-induced current shows a nonsingle exponential behavior around the dip as shown in Fig. 4(d), while
single exponential decay is always observed at other conditions, e.g., at ST  300 eV shown in Fig. 4(c). The
decay characteristic in Fig. 4(d) can be very well fittedwith a double-exponential function (the solid line);
C1 expth =so  C2 expth =cot . Here, C1 , C2 , andso are fitting parameters, and cot is an input parameter(810 s) determined by extrapolating the tot dependence of s in the cotunneling regime to the present valueof tot . The fast component of the double-exponential
decay can be assigned to the relaxation from jT i andjT i via spin-orbit coupling, while the slow component
to the relaxation from jT0 i via higher-order spin-orbit
coupling or the remaining cotunneling contribution. We
find that the obtained ratio of C1 =C2 is 2.0 and so is 60 s.
The ratio C1 =C2 obtained at slightly different tot ranges
between 1.6 and 2.0. These values of C1 =C2 are close to 2,
which is expected for an equal population of the three
triplet sublevels.
The above observations agree well with the selection
rule for spin-orbit coupling that is enhanced in the vicinityof the X-T crossing. However, we cannot safely rule out
other possibilities like populating an X state in addition to
the triplet state of interest. Indeed, a very small increase( ’ 1%) in the excitation probability P is noted at aroundB  0:55 T as shown by the arrow in Fig. 4(a), which
could be due to injection into the X state. However, thiseffect is too small to explain the observed doubleexponential behavior. Another possibility might be coupling to nuclear spins, which often appears at the level
coincidence of different spin states [18].
In summary, we have studied spin relaxation dynamics
from the triplet excited state to the singlet ground state in alateral QD. The dominant spin relaxation mechanism iscotunneling at a large tunneling rate, and it changes to spinorbit interaction when cotunneling is suppressed. The observed double-exponential decay characteristic could reflect the selection rule for the singlet-triplet transition
mediated by spin-orbit interaction. Further investigationis required to prove this is the case.
We can think of an ‘‘entanglement generator’’ using thisselection rule: The singlet ground state jSi  j"ia j#ia holdsa spin pair in an orbital a, while the triplet state contains
nonentangled states jT i  j"ia j"ib and jT i  j#ia j#ib ,week ending
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and entangled state jT0 i  p12 j"ia j#ib j#ia j"ib , with anelectron in each of the orbitals a and b. At a proper waitingtime after the electron injection [e.g., th  300 s in thecase of Fig. 4(d)], the system is left in the entangled triplet
state jT0 i with a probability of 18% [nt th  300 s 
1  P], or otherwise in the singlet ground state jSi. Our
pulse measurement is based on the extraction of an electron from the unrelaxed jT0 i state (the outer orbital withthe high probability P), and thus this scheme can be used
to generate or analyze an entangled spin pair by detecting the extracted electron with a sensitive electrometer[5,8].
The authors acknowledge valuable discussions with
Dr. A. V. Khaetskii and Y. Tokura. The work is financially
supported by SORST-JST and Grant-in-Aid for Scientific
Research from the Japan Society for the Promotion of
Science.

*Electronic address: satoshi@nttbrl.jp[1] Semiconductor Spintronics and Quantum Computing,edited by D. D. Awschalom, D. Loss, and N. Samarth(Springer, New York, 2002).
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[3] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120(1998).
[4] R. Hanson et al., Phys. Rev. Lett. 91, 196802 (2003).
[5] J. M. Elzerman et al., Nature (London) 430, 431 (2004).
[6] M. Kroutvar et al., Nature (London) 432, 81 (2004).
[7] T. Fujisawa et al., Nature (London) 419, 278 (2002);
T. Fujisawa et al., J. Phys. Condens. Matter 15, R1395(2003).
[8] R. Hanson et al., Phys. Rev. Lett. 94, 196802 (2005).
[9] M. Ciorga et al., Phys. Rev. B 61, R16315 (2000); A. S.
Sachrajda et al., Physica E (Amsterdam) 10, 493 (2001);
M. Ciorga et al., Physica E (Amsterdam) 11, 35 (2001).
[10] S. Tarucha et al., Phys. Rev. Lett. 84, 2485 (2000).
[11] A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12 639(2000).
[12] U. Bockelmann and G. Bastard, Phys. Rev. B 42, 8947
(1990).
[13] C. F. Destefani et al., Phys. Rev. B 70, 205315 (2004).
[14] D. V. Bulaev and D. Loss, Phys. Rev. B 71, 205324 (2005).
[15] S. Dickmann and P. Hawrylak, J. Supercond. 16, 387(2003).
[16] P. S. Drouvelis et al., J. Phys. Condens. Matter 16, 3633
(2004); Y. Tokura et al., Physica B (Amsterdam) 298, 260(2001); P. A. Maksym, Physica B (Amsterdam) 249, 233(1998).
[17] Because of the narrow feature, the dip was discovered onlyin the 1st cooldown, but reproduced well during the same
cooldown.
[18] K. Ono and S. Tarucha, Phys. Rev. Lett. 92, 256803(2004).

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