Y. N. Chen,1D. S. Chuu,1and T. Brandes2
1Department of Electrophysics, National Chiao-Tung University, Hsinchu 300, Taiwan
2School of Physics and Astronomy, The University of Manchester, P.O. Box 88, Manchester, M60 1QD, United Kingdom 共Received 18 August 2005; published 26 October 2005兲
The shot-noise spectrum of a quantum dot p–i–n junction embedded inside a three-dimensional photonic crystal is investigated. Radiative decay properties of quantum dot excitons can be obtained from the observa-tion of the current noise. The characteristic of the photonic band gap is revealed in the current noise with discontinuous behavior. Applications of such a device in entanglement generation and emission of single photons are pointed out, and may be achieved with current technologies.
DOI:10.1103/PhysRevB.72.153312 PACS number共s兲: 73.63.⫺b, 73.50.Td, 71.35.⫺y, 42.70.Qs
Since Yablonovitch proposed the idea of photonic crystals 共PCs兲,1 optical properties in periodic dielectric structures have been investigated intensively.2Great attention has been focused on these materials not only because of their potential applications in optical devices, but also because of their abil-ity to drastically alter the nature of the propagation of light from a fundamental perspective.3Among these, modification of spontaneous emission is of particular interest. Historically, the idea of controlling the spontaneous emission rate was proposed by Purcell,4and enhanced and inhibited spontane-ous emission rates for atomic systems were intensively in-vestigated in the 1980s 共Ref. 5兲 by using atoms passed through a cavity. In semiconductor systems, the electron-hole pair is naturally a candidate to examine spontaneous emis-sion, where modifications of the spontaneous emission rates of quantum dot共QD兲 共Ref. 6兲 or quantum wire 共QW兲 共Ref.
7兲 excitons inside the microcavities have been observed ex-perimentally.
Recently, the interest in measurements of shot noise in quantum transport has risen owing to the possibility of ex-tracting valuable information not available in conventional dc transport experiments.8 With the advances of fabrication technologies, it is now possible to embed QDs inside a p–i–n structure,9 such that the electron and hole can be injected separately from opposite sides. This allows one to examine the exciton dynamics in a QD via electrical currents.10 On the other hand, it is also possible to embed semiconductor QDs in PCs,11where modified spontaneous emission of QD excitons is observed over large frequency bandwidths.
In this work, we present nonequilibrium calculations for the quantum noise properties of quantum dot excitons inside photonic crystals. We obtain the current noise of QD exci-tons via the MacDonald formula,12 and find that it reveals many of the characteristics of the photonic band gap共PBG兲.
Possible applications of such a device to the generation of entangled states and the emission of single photons are also pointed out.
Model. We assume that a QD p–i–n junction is embedded in a three-dimensional PC. A possible structure is shown in Fig. 1. Both the hole and electron reservoirs are assumed to be in thermal equilibrium. For the physical phenomena we are interested in, the Fermi level of the p共n兲-side hole 共elec-tron兲 is slightly lower 共higher兲 than the hole 共electron兲
sub-band in the dot. After a hole is injected into the hole subsub-band in the QD, the n-side electron can tunnel into the exciton level because of the Coulomb interaction between the elec-tron and hole. Thus, we may introduce the three dot states:
兩0典=兩0,h典, 兩↑典=兩e,h典, and 兩↓典=兩0,0典, where 兩0,h典 means there is one hole in the QD,兩e,h典 is the exciton state, and 兩0,0典 represents the ground state with no hole and electron in the QD. One might argue that one cannot neglect the state 兩e,0典 for real devices since the tunable variable is the applied voltage. This can be resolved by fabricating a thicker barrier on the electron side so that there is little chance for an elec-tron to tunnel in advance.13 Moreover, the charged exciton and biexcitons states are also neglected in our calculations, which means a low injection limit is required.14
Derivation of Master equation. We define the dot-operators nˆ ⬅兩↑典具↑兩, n↑ ˆ ⬅兩↓典具↓兩, pˆ⬅兩↑典具↓兩, s↓ ˆ ⬅兩0典具↑兩, s↑ ˆ↓
FIG. 1. Illustration of a QD inside a p–i–n junction surrounded by a three-dimensional PC.
1098-0121/2005/72共15兲/153312共4兲/$23.00 153312-1 ©2005 The American Physical Society
⬅兩0典具↓兩. The total Hamiltonian H of the system consists of three parts: H0关dot, photon bath Hp, and the electron共hole兲 reservoirs Hres兴, HT 共photon coupling兲, and the dot-reservoir coupling HV coupling strength with⑀andbeing the polarization vector of the photon and the dipole moment of the exciton, respec-tively. bkis the photon operator, X =兺kDkbk†, and cq and dq
denote the electron operators in the left and right reservoirs, respectively.
The couplings to the electron and hole reservoirs are given by the standard tunnel Hamiltonian HV, where Vqand Wqcouple the channels q of the electron and hole reservoirs.
If the couplings to the electron and the hole reservoirs are weak, it is reasonable to assume that the standard Born-Markov approximation with respect to these couplings is valid. In this case, one can derive a master equation from the exact time evolution of the system. The equations of motion can be expressed as共cf. Ref. 15兲
de-pends on the time interval only. We can now define the Laplace transformation for real z
C共z兲 ⬅
冕
0 and transform the whole equations of motion into z spacen↑共z兲 = − 关C共z兲 + C*共z兲兴n↑共z兲/z +⌫L These equations can then be solved algebraically, and the tunnel current from the hole- or electron-side barrier
IR
ˆ = − e⌫R具nˆ典↓ t, Iˆ = − e⌫L L关1 − 具nˆ典↑ t−具nˆ典↓ t兴 共5兲 can in principle be obtained by performing the inverse Laplace transformation on Eqs.共4兲. Depending on the com-plexity of the correlation function C共t−t⬘兲 in the time do-main, this can be a formidable task which can however be avoided if one directly seeks the quantum noise:
Shot noise spectrum. In a quantum conductor in nonequi-librium, electronic current noise originates from the dynami-cal fluctuations of the current around its average. To study correlations between carriers, we relate the exciton dynamics with the hole reservoir operators by introducing the degree of freedom n as the number of holes that have tunneled through the hole-side barrier16and write
n˙0共n兲共t兲 = − ⌫Ln0共n兲共t兲 + ⌫Rn↓共n−1兲共t兲,
n˙↑共n兲共t兲 + n˙↓共n兲共t兲 = 共⌫L−⌫R兲n0共n兲共t兲. 共6兲 Equations共6兲 allow us to calculate the particle current and the noise spectrum from Pn共t兲=n0共n兲共t兲+n↑共n兲共t兲+n↓共n兲共t兲 which gives the total probability of finding n electrons in the col-lector by time t. In particular, the noise spectrum SI
R can be calculated via the MacDonald formula12,17
SIR共兲 = 2e2
冕
0 In the zero-frequency limit, Eq.共6兲 reduces toSI
R共= 0兲 = 2eI
再
1 + 2⌫Rdzd关znˆ↓共z兲兴z=0冎
. 共9兲As can be seen, there is no need to evaluate the correlation function C共t−t⬘兲 in the time domain such that all one has to do is to solve Eq.共4兲 in z space.
Results and discussions. The above derivation shows that the noise spectrum of the QD excitons depends strongly on
C共z兲. Let us now turn our attention to the spontaneous emis-sion of a QD exciton in a three-dimenemis-sional PC, where the vacuum dispersion relation is strongly modified: An aniso-tropic band-gap structure is formed on the surface of the first Brillouin zone in the reciprocal lattice space. In general, the band edge is associated with a finite collection of symmetri-cally placed points k0i leading to a three-dimensional band structure.3In our study, the transition energy of the QD ex-citon is assumed to be near the band edgec. The dispersion relation for those wave vectors k whose directions are near one of the k0i can be expressed approximately by k=c
+ A兩k−k0i兩2, where A is a model dependent constant.18Thus, the correlation function C共z兲=兺k兩gDk兩2/关z+i共k−0兲兴 can be calculated around the directions of each k0i separately, and is given by
C共z兲 = − i0 23/2
冑
c+冑
− iz −共0−c兲, 共10兲 with 3/2= d2兺isin2i/ 8⑀0បA3/2.19 Here, ប0 is the transi-tion energy of the QD exciton, d is the magnitude of the dipole moment, andiis the angle between the dipole vector of the exciton and the ith k0i.The shot-noise spectrum of QD excitons inside a PC is displayed in Fig. 2, where the tunneling rates⌫Land⌫R are assumed to be equal to 0.1and, respectively. We see that the Fano factor关F⬅SIR共= 0兲/2e具I典兴 displays a discontinu-ity as the exciton transition frequency is tuned across the PBG frequency共c= 101兲. It also reflects the fact that be-low the band edge frequency c, spontaneous emission of the QD exciton is inhibited. To observe this experimentally, a dc electric field共or magnetic field兲 could be applied in order to vary the band-gap energy of the QD exciton. Another way to examine the PBG frequency is to measure the frequency-dependent noise as shown in the inset of Fig. 2, where the exciton band gap is set equal to 104. As can be seen, dis-continuities also appear as is equal to the detuned fre-quency between PBG and QD exciton.
When the atomic resonant transition frequency is very close to the edge of the band and the band gap is relatively
large, the above one-band model is a good approximation. If the band gap is narrow, one must consider both upper and lower bands. For a three-dimensional anisotropic PC with point-group symmetry, the dispersion relation near two band edges can be approximated as
k=
再
cc12+ C− C12兩k − k兩k − k10i20j 兩 共兩 共kk⬎⬎cc11兲,兲.冎
共11兲Here, k10i and k20j are two finite collections of symmetry related points, which are associated with the upper and lower band edges,20and C1and C2are model-dependent constants.
Following the derivation for the one-band PC, the correlation function can now be written as
C共z兲 =
兺
collections of anglesi共n兲, n = 1, 2.
Figure 3 illustrates the frequency-dependent noise of QD excitons embedded inside a two-band PC. The two-band edge frequenciesc1andc2 are set equal to 101and 99, respectively. There are three regimes for the choices of the exciton band gap: 0⬎c1, 0⬍c2, and c1⬎0⬎c2. When0is tuned above the upper band-edgec1 共or below the lower band-edgec2兲, the QD exciton is allowed to de-cay, such that the shot noise spectrum 共gray curve兲 is sup-pressed in the range of 兩兩⬍兩0−c1兩. On the other hand, however, if0 is between the two band edges, spontaneous emission is inhibited. As shown by the dashed curve, the current noise in the central region is increased with its value equal to unity. Similar to the one-band PC, the curves of the shot noise spectrum reveal two discontinuities at 兩兩=兩0
−c1兩 or 兩0−c2兩, demonstrating the possibility to extract information from a PC by the current noise.
A few remarks about the application of the QDs inside a PC should be mentioned here. As is known, controlling the FIG. 2. Current noise共Fano factor兲 of QD excitons in a
one-band PC as a function of the exciton one-band gap0. The PBG fre-quency c is set equal to 101. The inset shows frequency-dependent noise, in which0is fixed to 104.
FIG. 3. Shot-noise spectrum of QD excitons in a two-band PC withc1andc2set equal to 101 and 99, respectively. To dem-onstrate the ability of extracting information from the PC, the exci-ton band gap0in gray and dashed curves is chosen as abovec2
共0= 101.5兲 and between the two band edge frequencies 共0
= 100.5兲, respectively.
propagation of light共waveguide兲 is one of the optoelectronic applications of PCs.21 By controlling the exciton band-gap
0 across the PBG frequency with appropriate tunneling rates of the electron and hole, one may achieve the emission of a single photon at predetermined times and directions 共waveguides兲,22which are important for the field of quantum information technology. Furthermore, it has been demon-strated recently that a precise spatial and spectral overlap between a single self-assembled quantum dot and a photonic crystal membrane nanocavity can be implemented by a de-terministic approach.23One of the immediate applications is the coupling of two QDs to a single common cavity mode.24 Therefore, if two QD p–i–n junctions can also be incorpo-rated inside a PC共and on the way of light propagation兲, the cavitylike effect may be used to create an entangled state
between two QD excitons with remote separation.13The ob-vious advantages then would be a suppression of decoher-ence of the entangled state by the PBG, and the observation of the enhanced shot noise could serve in order to identify the entangled state.10
In summary, we have derived the nonequilibrium current noise of QD excitons incorporated in a p–i–n junction sur-rounded by a one-band or two-band PC. We found that char-acteristic features of the PBG can be obtained from the shot noise spectrum. Generalizations to other types of PCs are expected to be relatively straightforward, which makes QD p–i–n junctions good detectors of quantum noise.25
This work is supported partially by the National Science Council of Taiwan under Grant Nos. NSC 94-2112-M-009-019 and NSC 94-2120-M-009-002.
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