Hybrid exciton-polaritons in a bad microcavity containing the organic and inorganic quantum wells

3 Pages·0.08 MB·English

by  Yu-Xi Liu

Hybrid exciton-polaritons in a bad microcavity containing the organic and inorganic
quantum wells
Liu Yu-Xi and Sun Chang-Pu
Institute of Theoretical Physics, Academia Sinica, P.O.Box 2735, Beijing 100080, China
Westudy thehybrid exciton-polaritons in a bad microcavity containing theorganic and inorganic
quantum wells. The corresponding polariton states are given. The analytical solution and the
numerical result of thestationary spectrum for thecavity field are finished.
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PACS number(s): 42.50.Ct: Quantumdescription of interaction of light and matter; related exper-
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iment. 71.35. -y: exciton and related phenomena
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1. Introduction is the strong coupling regime which the exciton-photon
0
1 coupling is so strong that it no longer be treated as
Quantumwells(QWs)embeddedinsemiconductormi-
0 perturbation. In the strong coupling regime, the QWs
0 crocavity structures have been the subject of extensive excitons emit photons into the cavity. The photon are
0 theoretical and experimental investigation. We know
bouncedbackby the mirrorandreabsorbedby the QWs
/ that the excitons play a fundamental role in the opti-
h to create excitons again. So the Rabi oscillation are
p cal properties of the QWs. The photodevices of excitons formed. Theexciton-polaritonsmodesplittinginasemi-
- mayhavesmallsize,lowpowerdissipation,rapidnessand
t conductor microcavity is also observed by many experi-
n high efficiency. All of above these are required to the in-
mental group, such as [8,9]
a tegrated photoelectric circuits.
In this paper, we will deal with the hybrid exciton-
u
The excitons are classified as Wannier excitons (they
q polaritonsstatesfortheorganicandinorganicQWsina
: have large radius and weak oscillator strength) and badcavity. Inthesection2,weusethemotionequations
v
Frenkel excitons (they have smaller radius and strong
i includeddampingeffecttogivethemixedhybrid-exciton
X oscillator strength) by the size of the exciton model ra-
andcavityfieldmodes,thatis,hybridexciton-polaritons.
r dius. In the section 3, we will give the emission spectrum of
a Recently,anewexcitionicstate–hybridexcitonstatein
the system in the case of the stationary state and the
the composite organic and inorganic semiconductor het-
correspondingnumericalresultsalsoaregiven. Insection
erostructurehasbeendescribedbypioneeringwork[1–7].
4, a simple conclusion is given.
Sincethenfromquantumwell(QW)toquantumdot,the
hybrid exciton states due to resonant mixing of Frenkel 2. Model and exciton-polariton states
andWannier-Mottexcitonshavebeendemonstrate. The
We begin with the Hamiltonian of the organic and inor-
reference [1] shows that the hybrid excitons possess a
ganic quantum wells in a ideal microcavity [3]
strong oscillator strength and a small saturation density
(or large radius). The reference [3] proposes to couple H = [h¯ω A+A +h¯ω B+B +h¯Ωa+a ]
W k k F k k k k
Frenkel excitons and Wannier-Mott excitons through a X
k
ideal microcavity.
+ [h¯Γ (A+a +A a+)+h¯Γ (B+a +B a+)]. (1)
There are two different coupling regime into which in- 13 k k k k 23 k k k k
X
k
teraction between the cavity field and the optical transi-
tionoftheexcitonscanbeclassified. Oneistheweakcou- where A (A+), B (B+) and a (a+) are usual boson op-
k k k k k k
pling regime which the exciton-photon coupling is very erators for Wannier, Frenkel and cavity field. Γ =
13
small and can be treated as a perturbation to the eigen- 1P01E and Γ = 1P01E , P01 is the moment ma-
h¯ W 0 23 h¯ F 0 W,F
states of the uncoupled exciton-photon system. Another trix element for Wannier and Frenkel exciton from the
ground state. E is the amplitude of the vacuum elec- We have a(t) as following:
0
tric field at the center of the cavity. This Hamiltonian is
linear. For a bad microcavity, we could solve it by the a(t)= F(ω1)e−iω1t− F(ω2)e−iω2t+ F(ω3)e−iω3t (7)
motionequationincludedthedampingcoefficientandob- ∆1∆2 ∆2∆3 ∆1∆3
tainanymixedsolutionsofthehybridexcitonandcavity
with ∆ = ω −ω , ∆ = ω −ω and ∆ = ω −ω .
field. 1 1 2 2 1 3 3 2 3
ω are determined by the pole equation. This equation
But here, we only deal with the case of a single mode i
indicatesthatthestrongcouplingofthetwokindsofthe
cavity field. That is, the above equation is simplified
QWs exciton states and cavity field results in three new
into:
eigenstates. Their eigenvalues are ω , ω and ω respec-
1 2 3
H =h¯ω A+A+h¯ω B+B+h¯Ωa+a tively. These states are just hybrid exciton-polaritons
W F
states. Their energysplitting are∆ ,∆ and∆ respec-
+h¯Γ (A+a+Aa+)+h¯Γ (B+a+Ba+). (2) 1 2 3
13 23 tively
So we have the motion equation
3. Stationary spectrum
∂a
=−iΩa−iΓ A−iΓ B−γ a (3a) For the case of the ergodic and stationary process, the
∂t 13 23 1
emission spectrum of the system is defined as follow-
∂A
=−iω A−iΓ a−γ A (3b) ing [11]:
∂t W 13 2
∂B ∞
∂t =−iωFB−iΓ23a−γ3B (3c) S(ω)=Z e−iωt <a+(t)a(0)>dt+c.c. (8)
0
Inordertodescribethe propertiesofthebadcavity,The
If the cavity field, Wannier exciton and Frenkel exciton
damping coefficient γ are added phenomenologically to
i areinitiallyinthenumberstates|n >,|n >and|n >
c W F
the above equation (3.a-3.c). In fact, when we write out
respectively, then
the interactionbetween the system andthe reservoir,we
could give a motion equation which includes the fluc- E(ω ) E(ω ) E(ω )
<a+(t)a(0)>=n¯ [−i 1 eiω1t+i 2 eiω2t−i 3 eiω3t]
tuation terms and dissipative terms by the Markov ap- c ∆∗∆∗ ∆∗∆∗ ∆∗∆∗
1 2 2 3 1 3
proximation. However, because we want to discuss the
(9)
exciton-polaritons in the strong coupling regime. We
aren’t interested in the noise properties of the system.
where n¯ is mean photon number of the cavity field
So the fluctuation terms may be neglected. c
and
By use of the Fourier transformation, we have:
E(ω)=(iγ +ω−ω )(iγ +ω−ω ) (10)
(iγ +ω−Ω)a(ω)=ia(0)+Γ A(ω)+Γ B(ω) (4a) 2 W 3 F
1 13 23
(iγ2+ω−ωW)A(ω)=iA(0)+Γ13a(ω) (4b) As the generalexciton-polaritons[12], If we assume that
(iγ +ω−ω )B(ω)=iB(0)+Γ a(ω) (4c) the damping is moderate, the process is almost ergodic
3 F 23
and stationary. It’s deserved to point out that all of the
Where, a(0), A(0) and B(0) are initial operators for the parameters excepting for ω are fixed by the properties
cavity field, Wannier excitons and Frenkel excitons re- of the organicand inorganic QWs as well as microcavity
spectively. In order to obtain a(t), we need solve the material. We always may choose some moderate param-
pole equation eters so that the stationary condition could be satisfied.
So the spectrum of the system is :
(iγ +ω−Ω)(iγ +ω−ω )(iγ +ω−ω )−
1 2 W 3 F
A B C
(iγ2+ω−ωW)Γ223−(iγ3+ω−ωF)Γ213 =0 (5) S(ω)= (ω−ω′)]2+Γ2 + (ω−ω′)]2+Γ2 + (ω−ω′)]2+Γ3
1 1 2 2 3 2
Thesolutionsofthiscubicequationcouldbeobtainedan-
(11)
alytically by using any mathematics handbook, such as
[10]. But usually, a cubic equation can be solved more with
quicklywithnumericalmethodsthanwithanalyticalpro-
Re[E(ω )∆ ∆ (ω−ω )]
cedures. So, we set the form of the analytical solutions A(ω)=2n¯ 1 1 2 1 (12a)
ofthe equation(5)areω1 =ω1′ −iΓ1, ω2 =ω2′ −iΓ2 and c |∆1|2|∆2|2
ω =ω′ −iΓ respectively. If we make Re[E(ω )∆ ∆ (ω−ω )]
2 3 3 B(ω)=2n¯ 2 3 2 2 (12b)
c |∆ |2|∆ |2
3 2
F(ω)=i(iγ +ω−ω )(iγ +ω−ω )a(0)+
2 W 3 F Re[E(ω )∆ ∆ (ω−ω )]
+iΓ23(iγ2+ω−ωW)B(0)+iΓ13(iγ3+ω−ωF)A(0) (6) C(ω)=2n¯c |3∆ |23|∆1|2 3 (12c)
3 1
2
We find that when the system reaches stability, the hy-
brid exciton-polaritons spectrum is superposition three
Lorentzian lines which are expected. The exciton-
polariton splitting may be measured at the peak points
of the emission spectrum which are determined by the
[1] V. Agranovich, R. Atanasov and F.Bassani, Solid State
condition dSd(ωω) =0. Communications 92, 295(1994)
Weapplythegeneraleq.(11)togiveanumericalsketch [2] V.I.Yudson,P.ReinekerandV.Agranovich,Phys.Rev.
map. We firstly adopt to the assumption of the refer- B52, R5543(1995)
ence [3] namely ω =Ω, ω =ω (1+δ), δ =10−2. [3] V.Agranovich,H.Benisty andC. Weisbuch,Solid State
F W F
Now we give a set of the possible values for the above Communications 102, 631(1997)
parameters. n¯ only determines the amplitude of the [4] V. Agranovich, D. M. Basko, G. C. La. Rocca and F.
c
spectrum, so we set n¯ = 1. We assume that ω = Ω = Bassani, J. Phys.: Condens. Matter 10, 9369(1998).
c F
1562meV, Γ2 = 16meV, Γ2 = 8meV, γ = 0.1meV, [5] A.Engelmann,V.I.YudsonandP.Reineker,Phys.Rev.
23 13 1 B57, 1784(1998)
γ = 0.18meV, γ = 0.12meV. By using these numbers
2 3 [6] D. Basko, G. C. La. Rocca, F. Bassani and V. Agra-
and eq.(11), we give the sketch map of the spectrum for
novich, Eur. Phys.J. B8, 353(1999)
the system. (Fig.1).
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mat/9812307
[8] C. Weibuch, M. Nishioka, A. Ishikawa and Y. Arakawa,
Phys. Rev.Lett. 69, 3314(1992)
[9] H. Cao, PhD Thesis (Stanford University1998)
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Optics (Springer-Verlag, Berlin 1991)
1
[12] Stanley Pau, Gunnar Bjork, J. Jacobson, Hui Cao, and
Y. Yamamoto, Phys.Rev.B51, 14437(1995)
0.5
1550 1560 1570 1580 1590 1600
-0.5
-1
FIG.1. Schematic drawing for theemission spectrum
This sketch map shows there are sudden change near
the three peaks. If we choose moderate parameters,
A(ω), B(ω) and C(ω) are slowing varing functions of ω
nearthepeaksandcanbeconsideredasconstant. Sothe
sudden varing points will disappear, the precise station-
ary spectrum is given.
4. Conclusion
In conclusion, the hybrid exciton-polariton states in
a bad microcavity containing the organic and inorganic
quantum wells is given. Although we only discuss a sin-
gle mode model, this approach also could be apply to
general case. This paper shows that the hybrid exciton-
polaritons decay at three difference rate. The analytical
and numerical results of the emission spectrum for the
exciton-polaritons are also given.
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