Experimental investigation of the optical spin selection rules in bulk Si and Ge/Si quantum dots

0.53 MB·

by  N. Sircar

Experimental investigation of the optical spin selection rules in bulk Si and Ge/Si
quantum dots
N. Sircar1,2 and D. Bougeard1
1Institut für Experimentelle und Angewandte Physik,
Universität Regensburg, D-93040 Regensburg, Germany
2Walter Schottky Institut, Technische Universität München,
Am Coulombwall 4, D-85748 Garching, Germany
We study the relationship between the circular polarization of photoluminescence and the mag-
neticfield-inducedspin-polarizationoftherecombiningchargecarriersinbulkSiandGe/Siquantum
dots. First, we quantitatively compare experimental results on the degree of circular polarization
4 of photons resulting from phonon-assisted radiative transitions in intrinsic and doped bulk Si with
1
calculationswhichweadaptfromrecentlypredictedspin-dependentphonon-assistedtransitionprob-
0
abilities in Si. The excellent agreement of our experiments and calculations quantitatively verifies
2
these spin-dependent transition probabilities and extends their validity to weak magnetic fields.

n Such magnetic fields can induce a luminescence polarization of up to 3%/T. We then investigate
a phononlesstransitionsinGe/SiquantumdotsaswellasindegeneratelydopedSi. Ourexperiments
J systematically show that the sign of the degree of circular polarization of luminescence resulting

7 from phononless transitions is opposite to the one associated with phonon-assisted transitions in
Si and with phononless transitions in direct band gap semiconductors. This observation implies

] qualitatively different spin-dependent selection rules for phononless transitions, which seem to be
l
l related to the confined character of the electron wave function.
a
h
-
Silicon is an attractive materials platform for spin- a weak magnetic field [13] to Si, using recent theoret-
s
e based information processing devices [1] due to proper- ical predictions on the DCP in Si [9]. Our theoretical
m
ties favoring long spin lifetimes such as a weak hyperfine description is in excellent quantitative agreement with
. coupling [2], low spin-orbit coupling and the absence of our measurements of the DCP for phonon-assisted tran-
t
a piezo-electricity [3, 4]. For direct bandgap semiconduc- sitions both in intrinsic and degenerately doped Si. A
m
tors the optical orientation of charge carrier spins by an substantialDCPofupto3%/TatliquidHetemperatures
- interaction with circularly polarized light represents an highlights the impact of an applied magnetic field in op-
d
importanttoolforthestudyofcarrierspins[5,6]. Forin- tical orientation experiments in Si. Regarding the spin-
n
o directbandgapgroupIVmaterials,thisconcepthastrig- dependent selection rules of phononless transitions in in-
c gered recent work, highlighting for example the accessi- direct bandgap semiconductors, such as those occurring
[
bilityofopticalspinorientationviathedirectbandgapin in Ge/Si quantum dots (QDs), our comparative study of

1 Ge[7,8]. InbulkSi,however,opticaltransitionsrelevant experiments and calculations suggests that band-mixing
v for spin orientation experiments involve phonon-assisted effectsduetothequantumconfinementplayasignificant
5 transitions across the indirect bandgap. A quantitative role.
1
and contact-free optical spin detection and analysis of In the following, we use the definition DCP ≡ (I −
5 σ+
1 spindynamicsinthismaterialwilldependontheknowl- Iσ−)/(Iσ+ + Iσ−), where Iσ+ and Iσ− are the intensi-
. edge of spin-dependent optical selection rules for the in- ties of σ+- and σ−-polarized components of the PL, re-
1
volved phonon-assisted transitions. The foundations of spectively. Furthermore, the spin quantization axis and
0
4 such selection rules have been discussed only recently hereby the meaning of σ± are always defined by the
1 [9,10],providingatheoreticalframework. Moreover,the positive direction of the external magnetic field. Under
: spin-dependent mechanisms governing phononless tran- weak-field conditions, the magnetic field-induced DCP
v
i sitions in Si-based quantum confined structures, which for semiconductors of GaAs type can be separated into
X have been discussed in terms of enhanced optical prop- two additive contributions of polarized electrons, DCPe,
r erties compared to bulk Si [11, 12], have yet to be estab- on the one hand and holes, DCPh, on the other hand,
a
lished. which recombine with unpolarized holes and electrons,
In this contribution, we present a study on spin- respectively. In a first-order approximation, [13]
dependent transition probabilities for radiative recombi-
nations of photoexcited carriers in bulk Si and quantum DCP=DCPe+DCPh =−(cid:104)s (cid:105)+(cid:104)j (cid:105). (1)
z z
confinedGe/Sistructuresinphotoluminescence(PL)ex-
periments. The degree of spin polarization (DSP) of the (cid:104)s (cid:105) and (cid:104)j (cid:105) are the average projections of the electron
z z
photoexcitedcarriersisadjustedthroughstaticmagnetic andholeangularquantumnumbersalongthefielddirec-
fields. In order to quantitatively connect the DSP with tion.
thedegreeofcircularpolarizationofPL(DCP),weadapt To account for the different spin-dependent transition
the theory on the DCP of GaAs-type semiconductors in probabilities in Si as compared to GaAs-type semicon-
2
ductors, we modify Eq. (1) to
4
DCP =2η (cid:104)s (cid:105)+ γ (cid:104)j (cid:105). (2)
m m z 3 m z
Here m denotes either the transverse-acoustic (TA),
transverse-optical (TO), longitudinal-acoustic (LA) or
longitudinal-optical (LO) phonon-mode involved in the
considered radiative recombination. η and γ are
m m
the maximally attainable PL polarizations when elec-
trons and holes are fully oriented in the s = +1/2 and
z
j = +3/2 states, respectively. They contain the infor-
z
mation on the spin-dependent selection rules. We have
introduced the numerical prefactors 2 and 4/3 to ensure
that Eq. (2) equals Eq. (1) if η and γ were deduced
m m
from the spin-dependent transition probabilities of di-
rect transitions for GaAs-type semiconductors. As (cid:104)s (cid:105)
z
and (cid:104)j (cid:105) are well-known functions of the applied mag-
z
netic field, the spin-dependent selection rules of phonon- Figure 1. σ+- and σ−-polarized PL spectra of the intrinsic
Si sample in a magnetic field of -7 T. The inset is a close-
assisted radiative transitions in Si can thus be assessed
up for the TA-related luminescence. Both the “TO” and TA
by measuring DCP .
m free-exciton peaks exhibit a positive DCP. See the text for a
While η has been explicitly calculated in Ref. [9],
m definition of “TO”.
we derive the values of γ from the strict symmetry
m
arguments outlined in the same reference: The inten-
sity of the σ+-polarized component of luminescence aris-
ments on Ge QD structures fabricated with solid-source
ing from the recombinations of holes originating from
molecular beam epitaxy, which contained 80 QD layers
the j = +3/2 hole states is Ih,x =Ih,y = 9 Ix,y
z σ+,m σ+,m 16 0,m intotal,eachseparatedby25nm-thickSilayers. Details
for the transitions ending in the x- and y-valleys, which
on the growth, morphology, composition, and electronic
have their axis of revolution perpendicular to the spin
band structure of this specific QD sample may be found
quantization axis. The intensity of the of σ−-polarized
elsewhere.[14, 15]
light is Ih,x =Ih,y = 1 Ix,y for each of these four
transitionσs−.,mHere,σ−I,xm,y de1n6o0t,ems the total luminescence Figure 1 depicts the spectra of the σ+- and σ−-
0,m polarizedcomponentsofthePLoftheintrinsicSisample
intensity due to all transitions assisted by transverse
in a field of -7 T and at a temperature of 5 K. In these
phonons with final states in the x- and y-valleys, re-
spectra, one can distinguish three well-resolved main lu-
spectively. Transitions ending in the two z-valleys do
minescence peaks each accompanied by a broader side
not originate from the +3/2 state, and can thus be dis-
peak. The main peaks correspond to the TA, “TO”, and
regarded in the context of polarization. Nevertheless,
TO+OΓ phonon replica of free-exciton recombinations
these latter transitions will contribute to the overall lu-
minescence with an intensity Iz . We thus end up with as indicated in the figure [16]. In fact, for the present
0,m measurement the “TO” peak is a mixture of a TO and a
2Ih,x +2Ih,y −2Ih,x −2Ih,y
γ = σ+,TO σ+,TO σ−,TO σ−,TO = 0.293 and weaker LO component [17]. As Fig. 1 reveals, both the
TO 4Ix,y +2Iz
likewise with γ =0,0T.O325,0,wTOhere we employed the nu- “TO” free-excitonpeak,aswellastheTAfeature,clearly
TA
merical values of Ix,y and Iz as given in Ref. [9]. A display a positive DCP.
0,m 0,m
similar analysis of the selection rules for the LO-assisted In the following, we focus on the “TO” replica peak in
recombinations yields γ =−0.375. more detail. DCP was measured for the “TO” peak
LO “TO”
In our experiments, the PL was excited with either energy for different external magnetic fields up to 7 T at
λ = 532 nm or 514 nm light with linear polarization. a temperature of 5 K. The resulting field dependences of
exc
The samples were mounted in a variable temperature DCP for the intrinsic and the heavily doped p-type
“TO”
magnet cryostat and the PL recorded in Faraday ge- and n-type samples are plotted in Fig. 2. Within the
ometry defined by the direction of the external mag- employedfieldrange,DCP isalinearfunctionofthe
“TO”
netic field. We first present measurements on bulk Si field for all samples. Also, as expected, it vanishes for
samples, cut from commercial, single-crystalline, high- zero-field. Interestingly, while the slope of the field de-
quality Si wafers. High-purity Si with a residual im- pendenceisnegativeforboththeintrinsicandthen-type
purity concentration below 3 × 1012cm−3, referred to sample, it is positive for the p-type sample. This behav-
as intrinsic Si, degenerately Sb-doped n-type Si (N = ior indicates that the dominant polarization mechanism
D
8×1018cm−3)anddegeneratelyB-dopedp-typeSi(N = oftheluminescenceisqualitativelydifferentinthesetwo
A
2.3×1019cm−3) were studied. We also discuss experi- cases.
3
average spin moment of the electron system in this case
is thus given by (cid:104)s (cid:105) = −3geµBB [13]. For the holes, on
z 8 Ee
F
the other side, Boltzmann statistics still apply, so that
the expression for their angular momentum does not dif-
fer from the one of intrinsic Si. According to Eq. (2),
the PL polarization in a magnetic field becomes here
DCP =−3geµBBη −5ghµBBγ . At the stud-
“TO” 4 EFe “TO” 3 kBT “TO”
ied doping concentration of about N = 8×1018 cm−3,
D
reasonablevaluesfortheelectronFermienergyliearound
Ee ≈ 20 meV [20]. Hence, the first term in the summa-
F
tion of DCP is negligible. For degenerate n-type Si
“TO”
the polarization of PL is therefore only a consequence of
thepolarizationoftheexcitedholes. Underthispremise,
and taking the same value for γ as for intrinsic Si,
“TO”
thefield-dependenceofthisexpressionofDCP isalso
“TO”
plotted in Fig. 2. As can be seen, the calculated curve
perfectly describes the measured field-dependence of n-
Figure2. MagneticfielddependenceoftheDCPofthe“TO” type doped Si.
PL peak of bulk Si of different doping type. The symbols
give experimental values while the solid lines represent the Forp-typeSithesituationisreversedcomparedtothe
theoretical calculations presented in the text. n-type case. The hole angular momentum (cid:104)j (cid:105) can now
z
beneglectedduetoconsiderationssimilartothoseforthe
n-type doping, but now for the Fermi energy of the hole
ForintrinsicSi,Boltzmannstatisticsdescribeboththe gas[21]. (cid:104)s (cid:105)remainsthesameasintheintrinsiccase,i.e.
z
aelreecgtrivoennainndfihrsotleosrydsetreamp.prHoexnimcea,t(cid:104)isozn(cid:105)aasn(cid:104)dsz(cid:104)(cid:105)jz≈(cid:105) o−f14Egqek.µB(BT2B) cDoCuPnt“TtOh”at=fo−r21agekhµBiBgTBhηp“-TtOy”p.eYdeotp,iwneg,hsapvient-oortbaiktecionutpoliancg-
and (cid:104)j (cid:105) ≈ −5ghµBB with the applied magnetic field B, comesintoeffectasaconsequenceoftheproximityofthe
thetemzperatu4rekTBT,theBoltzmannconstantk ,theBohr Fermi energy to the split-off band [9]. Given the doping
B
magneton µB, the electron g-factor ge, and the hole g- concentration of about NA =2.3×1019cm−3, the polar-
factor gh [13]. We use the values ge = 2 and gh = 0.56 izationfactorη“TO” thendecreasestoη“TO” =−0.035[9].
[18, 19]. To account for the admixture of the LO- Weincludedthestraightlinerepresentingthefielddepen-
component to the “TO” peak we calculate the effective dence of the corresponding DCP“TO” in Fig. 2. Again,
polarizationfactorη byaweightedaveragingoverthe the matching of the experiment and theory is excellent.
“TO”
contributing polarization factors η and η . The rel- In this case, the slope is now positive, revealing a domi-
TO LO
ative intensities I and I of the TO and LO peak nant contribution of the recombination of spin-polarized
0,TO 0,LO
[9], respectively, are taken as weighting factors. This re- electrons to the luminescence polarization.
sults in η“TO” = ηTOII00,,TTOO++Iη0L,OLOI0,LO = −0.133. Using the For each studied case — intrinsic, degenerate n-type
γ and γ values that we have calculated earlier, an and degenerate p-type Si — our respective calculations,
TO LO
effective polarization factor γ =0.285, attributed to which are based on selection rules presented in [9], show
“TO”
polarizedholes,isobtainedbyasimilarweightingproce- an excellent quantitative agreement with the experimen-
dure. Thus according to Eq. (2), the general expression tally observed magnetic field dependence of the DCP.
for the field and temperature dependence of DCP of Our measurements represent the first direct and quanti-
“TO”
intrinsic Si is then DCP =−0.067· µBB. A straight tative experimental confirmation of the spin-dependent
“TO” kBT
line representing this latter expression for a temperature transition probabilities from Ref. [9]. Additionally, we
of T = 5 K is shown in Fig. 2. The experimental data show here that the validity of these transition probabili-
points for intrinisc Si are in excellent agreement with tiescanbeextendedtoweakmagneticfields. Theconsis-
our theoretical prediction. The experimental observa- tency between experiment and theory in the presence of
tionandtheoreticalfindingofanegativesignoftheslope a weak external field suggests that merely the band pop-
of DCP for intrinsic Si congruently hint toward the ulation depending on the angular momentum is changed
“TO”
importance of the polarization of the hole angular mo- by the field, whereas the matrix elements of the dipole
mentum in a magnetic field, which dominates the PL andthephonon-assistedtransitionsbetweentheindivid-
polarization. ual bands, which mainly rely only on the band symme-
Compared to intrinsic Si, the slope of heavily n-type- tries,arenotaffected. TheobservedDCPofupto3%/T
doped Si is also negative, but steeper. This can be con- represents a significant and entirely field-induced contri-
nected to the degenerate character of the electron sys- bution to the luminescence polarization of optical spin
tem, which now obeys the Fermi-Dirac statistics. The injectionordetectionexperimentsinSiwhencarriedout
4
Figure 3. Circular-polarization-resolved spectra of the phononless PL of the Ge QD sample. The spectra were acquired at a
temperature of about 10 K and in a magnetic field of (a) +7 T and (b) -7 T.
in the presence of an external magnetic field, such as for served polarization of luminescence. Since the electrons
example in spin LEDs [22–25]. are weakly confined in Si at the interface to the Ge QD,
Ourexperimentsshowthatdopingplaysacriticalrole the magnetic field dependence of the average spin mo-
in the magnitude of the field-induced polarization effect. mentum (cid:104)sz(cid:105) of these electrons is similar to electrons in
Inthecasesofstrongp-andn-typedegeneracywecould bulkSi. However,theorientationoftheangularmomen-
completelydisregardrespectivelyeitherholesorelectrons tum (cid:104)jz(cid:105) of the holes confined within the Ge is reversed
in the analysis of the data. In this context, it is interest- from the bulk Si case, because the holes now possess a
ingtohighlightthebroadside-peaks,eachapproximately negative hole g-factor [29] .
17 meV lower in energy than the corresponding phonon- So far, calculations of spin-dependent probabilities of
replica peak, as shown in Fig. 1. These are due to the phononless transitions in Ge QDs do not exist. As
recombination of electron-hole pairs within electron-hole highlighted by Fukatsu et al. [35], phononless PL can
droplets(EHD)atlowtemperatures[26]. Ascanbeseen, be reduced to an effective matrix element of the form
thesefeaturesdonotexhibitanysignsofacircularpolar- (cid:10)Γ+|pˆ|∆ (cid:11),wherepˆ denotesthedipoleoperatorandΓ+
8 1 8
izationuptofieldsof7T.Sincethequasi-Fermienergies and∆ therepresentationsoftheholeandelectronwave
1
of both the electron and hole system are relatively large functions, respectively. The Γ+ valence and ∆ conduc-
8 1
in the EHD [27], this behavior is fully in line with our tion band states in the Ge/Si QD system share, at least
previous description of degenerate systems: i.e. now in partly,thesamep-ands-symmetriesas,respectively,the
the EHD, both carrier species do not exhibit a relevant valence and conduction band states in GaAs-like semi-
angularmomentumpolarizationandnoDCPisobserved conductors. Thisfactmotivatesustoqualitativelyapply
in the experiment. theselectionrulesforGaAs-likeradiativerecombinations
Having discussed the spin-dependent phonon-assisted toourGeQDsinafirstapproximation. Doingsoimplies
selection rules in Si, we now address spin-dependent that phononless recombinations would correspond to a
phononless radiative recombinations. These occur in negativevalueforη andapositivevalueforγ . Under
m m
indirect band-gap materials whenever momentum mis- thispremise,apositiveslopeofthedegreeofpolarization
match in optical transitions may be removed, e.g. by of the Ge QD PL as a function of the external magnetic
quantum confinement. We first present the PL spectra field is then always expected from Eq. (2), regardless of
for the Ge QD sample which are dominated by phonon- the exact values of η , γ , (cid:104)s (cid:105), and (cid:104)j (cid:105). On the con-
m m z z
less radiative recombinations [28]. trary, however, the measurements shown in Fig. 3 yield
Figure 3a and 3b depict the polarization-resolved a negative slope.
phononlessPLoftheGeQDsampleinamagneticfieldof Further evidence for the unexpected behavior of
+7and-7T,respectively. Theyclearlyexhibitacircular phononless luminescence in indirect band gap materials
polarization, the field dependence of the DCP having a comes from our PL measurements of doped bulk Si. As
negative gradient. a representative example, Figure 4a and 4b depict the
For these undoped Ge QDs neither the spin polariza- polarization-resolved PL spectra of the heavily doped n-
tion of electrons, which are localized in Si, nor that of type Si sample in a magnetic field of +7 T and -7 T, re-
holes, which are localized in Ge [15], can be neglected. spectively. Duetothehighdopantconcentration,aweak
Recombinationofbothspeciesthuscontributetotheob- phononless PL is observed in addition to the phonon-
5
Figure4. Circular-polarization-resolvedPLspectraofann-typedopedSisample. Thespectrawereacquiredatatemperature
ofabout5Kandinamagneticfieldof(a)+7Tand(b)-7T.Theinsetsareclose-upsofthehigh-energyregionsofthespectra,
containingtheTApeakaround1110meVandthephononlesspeak,labeledNP,at1130meV.ThephononlessNPpeakhasan
opposite polarization compared to the TA and “TO” peak.
replica as highlighted in the insets of Fig. 4. Interest- The authors would like to thank Jaroslav Fabian for
ingly, the phononless luminescence feature at 1130 meV helpful discussions, Gerhard Abstreiter for support and
exhibits a sign of the DCP that is opposite to all the the Deutsche Forschungsgemeinschaft DFG for funding
transverse-phonon-related luminescence features. How- via SPP 1285.
ever,ifGaAs-likeselectionruleswereapplicableforthese
phononless recombinations, their DCP would have the
same sign as the DCP of transverse-phonon-assisted re-
combinations, according to Eq. (2), because of the same
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