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arXiv:0708.0468v2 [cond-mat.mtrl-sci] 15 Aug 2007

Direct observation of excitonic polaron in InAs/GaAs quantum dots

Ming Gong, Chuan-Feng Li ?, Geng Chen, Lixin He, F. W. Sun, and Guang-Can Guo Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, People’s Republic of China

Zhi-Chuan Niu, She-Song Huang, Yong-Hua Xiong, and Hai-Qiao Ni State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors,

CAS, P.O. Box 912, Beijing 100083, People’s Republic of China (Dated: February 1, 2008)

Excitonic polaron is directly demonstrated for the ?rst time in InAs/GaAs quantum dots with photoluminescence method. A new peak (s′) below the ground state of exciton (s) comes out as the temperature varies from 4.2 K to 285 K, and a huge anticrossing energy of 31 meV between s′ and s is observed at 225 K, which can only be explained by the formation of excitonic polaron. The results also provide a strong evidence for the invalidity of Huang-Rhys formulism in dealing with carrier-longitudinal optical phonon interaction in quantum dot. Instead, we propose a simple two-band model, and it ?ts the experimental data quite well. The reason for the ?nding of the anticrossing is also discussed.

PACS numbers: 60.20.Kr, 71.35.-y, 71.38.-k, 78.67.Hc

Electron-phonon interaction is a very important ingredient determining the physical properties of semiconductors, such as phonon-assisted hot carrier relaxation process[1, 2], light absorption and emission process[3, 4]. For electron-longitudinal optical (LO) interaction in the weak polar system (e.g., GaAs, αc = 0.067), the welldocumented Huang-Rhys model always gives good explanation in bulk material. However, in quantum dot (QD), this interaction is greatly enhanced owing to the discrete nature of energy levels with spacing comparable to the energy of LO phonon. Both theoretical and experimental ?ndings show that it may have entered the strong coupling regime[3, 5, 6, 7], which means that an accurate description of this interaction system should be the hybridation of electron state and the phonon state, thus the polaron as a new ground state will be formed[8]. Similar conclusion can also be drawn to hole and exciton interacting with LO phonons in QD.

Although Huang-Rhys parameter S and Fr¨ohlich coupling constant αc are both related to the average number of LO phonons, while the irreversible emission of LO phonons (0.1 ? 1 ps) provides an e?cient channel for energy relaxation[9], the formation of polaron will suppress the LO phonon contribution to the carrier decoherence process and hence leads to long polaron lifetime[5, 6, 7] and everlasting oscillation of survival probability[5, 10, 11].

In experiments, far-infrared (FIR) absorption results have evidenced the formation of electron-LO[10, 11], hole-LO [12] and exciton-LO[13] induced magnetopolaron in QDs under ultra-high magnetic ?eld (up to ? 28 T) at about 4 K, and anticrossing between the polaron levels di?ering by one LO phonon[6] is found. How-

?email: c?i@ustc.edu.cn

ever, hampered by the large full width at half maximum (FWHM), no direct observation of excitonic polaron has yet been reported.

In this letter, as we vary the temperature from 4 K to 285 K, the FWHM of the s peak decreases sharply, thus we can directly observe the excitonic polaron without applying magnetic ?eld. It is shown that at low temperature, the exciton-LO phonon interaction is weak and may be explained within the framework of Huang-Rhys model and enhanced S value is obtained. But at high temperature, due to the increasing of coupling strength, excitonic polaron is formed, and Huang-Rhys model is invalid.

The self-assembled InAs/GaAs QDs studied here were grown by molecular-beam epitaxy (MBE) on GaAs (001) substrate at 515? C. The sheet density, mean diameter, and height of the dots are ?85 ?m?2, 39 nm and 2.8 nm, respectively. The sample is mounted in a He cooled cryostat and the temperature is tuned from 4.2 K to 285 K, in the spacing of 7.5 K. Photoluminescence (PL) was performed with an He-Ne laser and a spectrometer with focal length of 0.75 m equipped with InGaAs line array. The excitation power varies from 5 mW to 0.5 ?W.

Figure 1(a) presents the PL results as a function of temperature from 4.2 K to 285 K at pump-power of 5 mW. s, p and d shells, origin of which have been well studied by Bayer et al[14], are also the zero-phonon line. Strikingly, there is a new peak (s′) on the lower-energy side of s shell with quite di?erent behavior. When T < 60 K, this peak is invisible even pumped with high power. As we increase the lattice temperature, this peak appears more and more clear, but the s peak is greatly suppressed. The intensity of s peak decreases monotonically, whereas, the intensity of s′ increases when T < 225 K, and then decreases a little from 225 K to 285 K as shown in Fig. 1 (b). The activation energy Ea of the thermal quenching process for s, p and d shells

are 110, 80, 56 meV, respectively, when ?tting with I(T ) = I0/(1 + A exp(?Ea/T ))[15]. Similar peaks have also been found in single quantum dot by Bayer et al, that arise from ”Hidden symmetry” e?ect[14], but as we reduce the pump-power from 5 mW to su?cient low conditions shown in Fig. 1 (c), the s′ still exist, implying that this new peak can not be accounted for this e?ect. It also can not from multicharge-exciton e?ect[16, 17], because it can not introduce so large redshift.

The peak energy positions of s′, s, p and d are presented in Figure 2 with open cicle. Due to the scattering by phonons[18, 19, 20], s, p, d peaks redshift with increasing temperature, however, the s′ peak does not redshift from 60 K to 170 K. When T > 170 K, it begins to redshift, but the s shell seems to be ”frozen” at about 1.01 eV, which is quite similar to the theoretical results in the intermediate and strong coupling regimes (αc > 3)[8]. From 225 K to 285 K, s shell only redshifts less than 4 meV while the p, d shells redshift more than 25 meV (see below).

Generally, the peak position as a function of temperature can be well ?tted with Varshni formula in bulk material, and even in single QD when T < 100 K[21]. The empirical Varshni formula is

E(T ) = E(0) ? αT 2 ,

(1)

β+T

where α and β are ?tting parameters characteristic of a

given material. The solid line in Fig. 2 is the ?tting

results with this equation for s, p, and d peaks. For

p and d peaks, this formula ?ts the results quite well

with error less than 2 meV through the whole experi-

ment range. For s peak, it ?ts well when T <170 K,

while in the high temperature regime, huge deviation

is found. Interestingly, for these three peaks, we ?nd that α are almost the same (?0.590 meV/K), and for β,

βs(387.5 K) > βp(339.2 K) > βd(301.6 K), these values are larger than those in bulk material due to the con?nement e?ect (β = 93 K for InAs[22]). (α/β)s = 1.55×10?3 is also quite close to the results in single QD[21].

In Fig. 2, an anticrossing between s and s′, with

crossing energy about 31 meV, is found at 225 K, which indicates that the s′ peak can not be accounted for emis-

sion of one LO phonon from s[23, 24, 25]; and thus it is

believed to be a direct evidence for the formation of po-

laron. To understand this result, we propose a two-band

model following the spirit of Preisler et al[13]. We assume the two reference bands |s and |s′ satisfy (i) The energy of |s can be described by Varshni formula. (ii) The energy of |s′ is independent of temperature. Under

the strong coupling conditions, the polaron state should

be the entanglement of exciton levels and phonons, thus |s′ = i,j,nq ci,j,{nq}|eihj{nq} [6], where q is the di?erent modes of the LO phonon, ei and hj represent the electron and hole energy levels, respectively. We

adopt the same Fr¨ohlich Hamitonian as Refs. [6, 13], so the o?-diagonal matrix element can be read as s|(HFe +

2

HFh )|s′

= AF i,j,nq ci,j,{nq}q?1(√nq s|(e?iq·re ?

e?iq·rh )|eihj{nq ? 1} + eiq·rh )|eihj {nq + 1} ), where

nq AF

+

1

∝s|(eiq√·reαc

? [8],

nq = (e ωq/T ? 1)?1 is the number of LO phonons.

The nonvanishing coupling only occurs between the

factorized exciton-phonon states which di?er by one

LO phonon. The acoustic phonon is neglected for the

coupling strength to exciton is greatly suppressed in

QD[26, 27], while the e?ect of lattice vibrations on

the energy gap is included in Es[20]. Furthermore,

dispersionless LO phonon approximation is used, then

nq can be replaced by the average number of LO

phonon s|(HFe

n0, and the + HFh )|s′ =

m√antr0ivx?e+lem√enn0t

can be + 1v+,

simpli?ed where v±

to =

AF i,j,nq ci,j,{nq}q?1 s|(e±iq·re ?e±iq·rh )|eihj {nq ±1} .

For simplicity, we assume that v? = v+ = V0,

then the Vint(T ) =

(√o?n-0d+iag√onna0 l+

element can be reduced 1)V0. The total Hamitonian

to to

describe this anticrossing gives

H=

(√n0

E√s(T ) + n0 +

1)V0?

(√n0

+

√ n0

+

1)V0

Es′

,

(2)

where Es′ is chosen to be (Epolaron(225 K) + Eexciton(225 K))/2 = 0.996 eV, Es(T ) = Es(0) ? αT 2/(β + T ), α = 0.589 meV/K is the same as the

previous ?tting results, and β serves as the only ?tting

parameter here. crossing energy

toFrboem2(E√qnu0at+io√n n(20)+,

we have 1)|V0| =

the 31

antimeV.

The LO phonon energy we choose here is ωLO = 36

meV[5, 13, 23] (bulk GaAs-like), so we have |V0| = 10.1

meV, which is slightly larger than that obtained under

high magnetic ?eld[6, 10, 13]. The small discrepancy

may be accounted for the Coulomb interaction between

exciton and polaron, which is not included in Eq. (2).

β = 412 K is our ?tting result.

Es and Es′ are shown with dashed lines in Fig. 2, while the ?tting results with this two-band model are presented

by star. We can see that our theoretical calculation ?ts

the experimental results quite well. For the s shell, it

has recovered almost all the experimental results from

4 K to 285 K, especially at high temperature, perfectly reproducing the ”frozen” state. For s′ state, it ?ts the

results well from 185 K to 285 K; however, at low tem-

perature, the deviation is about 10 meV, because of the

simple model we used. Results show that this two-band

model only work well near the resonant condition.

It is worthy to underline that the position of this anti-

crossing di?ers greatly from that found with FIR magne-

tospectroscopy method[13]. In our results, the anticross-

ing is between the exciton state (s) and the polaron state (s′), but not between the polaron levels (intra-band) that

di?er with one LO phonon[6, 13]. It also implies that the

polaron and exciton may coexist in a single QD, and the

Coulomb interaction between them may play a role in

the formation of polaron (see below). Note that at 4.2 K, the level spacing of s′ and s is

3

67 meV, about twice the energy of ωLO[5, 23]. Actually, phonon replicas at low temperature in the PL spectra have been found and explained by enhanced S values[25, 28]. In this model, their integrated intensity is related to the transition probability wp = | dQχ?p(Q ? Q0)χ0(Q)|2 = Spe?S/p!, and the ratio between term p and term (p ? 1) gives S/p. At low temperature, it is reasonable to assume that the s′ is originated from exciton recombination by emitting 2LO phonons[23, 24], then we get S = 2Is′ /Is ≤ 0.2 (in bulk InAs, SInAs ? 0.0033[25]), in consistent with the experimental results in Table I of Ref. [24]. However, because of the large inhomogeneous broadening due to ?uctuation in the dot shape and/or size (FWHM = 32 ? 37 meV), it is hard to separate the -LO satelite line from the s shell at low temperature[24].

The FWHM of s shell with respect to temperature is presented in the inset of Figure 3. Di?erent from single QD, the FWHM of which increases monotonically with the increasing of temperature[27, 29], the FWHM of s shell decreases from 37 meV at 4.2 K to ?12 meV at 285 K. Many body scattering induced FWHM broadening is also observed[17], but quite small when T > 200 K. Similar anomalous decrease of FWHM is also found in quantum well due to the exciton thermalization e?ect[30]. This unexpected result is also one of the advantage in our experiment. The FWHM of s′ peak is less than 13 meV at 285 K, which is much smaller than the spacing between s and s′, so the emission line of s and s′ can be separated and identi?ed from PL spectra unambiguously.

Following the method used in Refs. [24, 28], we calculate the ratio of intensity of s′ to s. The experimental results are presented in Fig. 3 for di?erent pump-power (I = 0.5, 1.5 and 5 mW). It is shown that this value increases almost exponentially with respect to temperature, as shown in Fig. 3, in which the solid line is our ?tting result with A exp(αeT ) for I = 5 mW. For di?erent excitation power, αe varies from 0.041 to 0.045 K?1. On the other hand, as we increase the pump-power, Is′ /Is increases slightly, indicating that the Coulomb interaction between exciton and polaron plays a positive role in the formation of polaron. This ratio can exceed 8.0 at 285 K, which can not be explained by Huang-Rhys model and thus supports the argument that in QDs, excitonLO phonon interaction has entered the strong coupling regime although the system is electrically neutral[6].

To illustrate the essential physics of the polaron e?ect clearer, we calculate the oscillator strength (OS) of the given polaron state, which has mixed the ingredient of |s and |s′ . The OS from Fermi’s golden rule read as

O± ∝ | Φf (T )|H′|Φ± |2 = |α±ps + β±ps′ |2, (3)

where H′ is the Hamitonian of laser ?eld interacting with exciton, |Φ± = α±|s + β±|s′ is the eigenstate of Eq. (2), ps = Φf (T )|H′|s , and ps′ = Φf (T )|H′|s′ .

|Φf (T ) is the ?nal state depending on the temperature. When T → 0, the allowed transitons are the components containing zero phonon, so |ps| ? |ps′ |, and this can explain why s′ is hard to be observed at low temperature (Fig. 1 (a)). In the dark state limit, we can choose ps′ = 0, then Is′ /Is = O?/O+ = |α?/α+|2. The ?tting results under this limit is presented in Fig. 3 with dashed line. At low temperature, it works quite well, while at 285 K, Is′ /Is ? 3 is found. Although this value is smaller than the experimental results, the tendency agrees with the experimental results quite well and the transformation from exciton to polaron can still be indicated clearly. In fact, ps′ do not close to zero, for the lack of knowledge about this quantity, we further present our ?tting result with ?xed ps/ps′ = ?9.96 through the whole experimental range with dash dotted line in Fig. 3, and it gives good ?tting especially at high temperature. At 285 K, the OS of |Φ? is much stronger than |Φ+ (Is′ ? Is), which is also responsible for the suppression of s peak shown in Fig. 1 (a). More importantly, this result is also a signi?cant evidence that the anticrossing found in Fig. 2 is not arti?cial, but indeed exists.

To sum up, we have demonstrated the formation of excitonic polaron directly from PL results. A new peak (s′) is found below the s shell, and the optical properties of this state with respect to temperature are studied in detail. The origin of s′ is not from emission of one LO phonon and/or many body e?ect. We propose a simple two-band model, and it ?ts both the peak positions and the ratio of integrated intensity of s′ to s quite well. The reason for the ?nding of the anticrossing is also discussed.

In QDs, the enhancement of S up to ? 0.02 can be explained by 8-band k · p model taking the piezoelectric e?ect into account[25]. But this estimation is still one or two orders of magnitude smaller than the experimental results[24]. Other ways to achieve large S values may include exact diagonalization method[3] and strong Fr¨ohlich coupling e?ect[23]. In this letter, the formation of polaron provides an unambiguous evidence for the invalidity of Huang-Rhys formulism (adiabatic theory) in dealing with exciton-LO phonon interaction in QD. Recently, T. J. Devreese et al[31, 32] emphasize that the e?ect of non-adiabaticity should be taken into account to interpret the surprisingly enhancement of the phonon replicas in the PL spectra, and it is also expected that this method can pave the way for a deeper understanding of our results. In the strong coupling regime (e.g., high temperature), we also believe that our results are important to the understanding of the carrier decoherence and relaxation process in QD[1, 2, 3, 4].

This work was supported by National Fundamental Research Program, the Innovation funds from Chinese Academy of Sciences, National Natural Science Foundation of China (Grant No.60121503 and 60625405) and Chinese Academy of Sciences International Partnership Project.

4

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PL Intensity (a.u)

5

(a)

p shell

s shell

d shell

30K s’

Intensity

10 8 s Ea=110meV 6 (b)

4

2 s’ 0 100 150 200 250

Temperature (K)

(c) s 15 s’ 10

5?W 5mW

T = 255K

PL Intensity

5

285K

x30

0

1.0 1.1 1.2 1.3 1.00 1.20 1.40

Energy (eV)

Energy (eV)

FIG. 1: (Color online). (a) The PL spectra of InAs/GaAs QDs ensemble with respect to temperature from 4.2 K to 285 K at pump-power 5mW, the dashed lines are just for guide. The curves have been o?set for clarity. (b) The integrated intensity of s′ and s as a function of temperature, Ea is the activation energy of s shell. (c) The PL spectra at 255 K for di?erent pump-power, the 5 ?W PL results have been multiplied by a factor of 30.

1.20 α=0.592, β=301.61

1.15 1.10

d

α=0.591, β=339.15

p

Ref. bands Varshini Experiment

Theory

Energy (eV)

1.05 1.00

α=0.589, β=380.72

s

67meV

2Vint~31meV

s’

0

50 100 150 200 250

Temperature (K)

FIG. 2: (Color online). The emission lines with respect to temperature at pump-power 5mW. The open circle is the experimental results for s′, s, p and d, while the solid line is the results ?tted with Equation (1), α and β are the corresponding ?tting results. The star is the ?tting results with our two-band model, while the dashed line is the corresponding reference bands (see the text). The huge anticrossing energy is about 31meV at 225K.

Is’ / Is FWHM (meV)

6

9

8

40 I=5.0mW

7 35

6

30 I=1.5mW

25

5

20

4

15

3 2

100 100 200

Temperature (K)

1

I=0.5mW I=1.5mW I=5.0mW

ps’=0

ps/ps’=-9.96

exponential

0 100

150

200

250

Temperature (K)

FIG. 3: (Color online). Is′ /Is with respect to temperature for di?erent pump-power (I=0.5, 1.5 and 5 mW), the dashed line is the ?tted results with our two-band model in the dark state limit (ps′ = 0), the dash dotted line is the ?tted result assuming ps/ps′ = ?9.96, the solid line is the ?tted results with AeαeT for I=5 mW. The inset is the FWHM of s shell as a function of temperature for I=1.5 and 5 mW.

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