Energy bands and Wannier-Mott excitons in Zn(P_{1-x}As_{x})_{2} and Zn_{1-x}Cd_{x}P_{2} cry

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讲稿激子类别能量传递

EION – energy required to ionize the molecule n – exciton energy level
EEX = 13.6 eV μ/m∈ μ– reduced mass =memh / (me+mh)
Adapted from Electronic Processes in Organic Crystals and Polymers by M. Pope and C.E. Swenberg
semiconductors)
SEMICONDUCTOR PICTURE
Excitons
(bound electron-hole
pairs)
treat excitons as chargeless
particles capable of diffusion,
also view them as excited states
@ MIT
February 27, 2003 – Organic Optoelectronics - Lecture 7
Exciton
In some applications it is useful to consider electronic excitation as if a quasi-principle, capable of migrating, were involved. This is termed as exciton. In organic materials two models are used: the band or wave model (low temperature, high crystalline order) and the hopping model (higher temperature, low crystalline order or amorphous state). Energy transfer in the hopping limit is identical with energy migration.

第四讲-激子与发光

electric f第ie四ld讲a-n激d子te与nd发t光o ionize excitons.
8
Experimental data for free excitons in GaAs
•Exciton absorption of ultra pure GaAs at 1.2 K. •hydrogen-like energy spectrum of the exciton in the vicinity of the band gap.
•Mort density Nmott: the density at which the exciton-exciton distance is equal to the exciton diameter:
N Mott
1
4 3
rn3
•High density is achievable with a focussed laser beam.
•E1=1.5149 eV, E2=1.5180 eV, E3=1.5187 eV •Eg=l.5191 eV, agree with other measurements.
•The experimental Rx=4.2 meV is in good agreement with the
calculated value.
第四讲-激子与发光
3
The concept of excitons
•In semiconductors and insulators: photon absorption electrons in the conduction band and holes in the valence band. •Exciton: bound electron – hole pair by Coulomb interaction

矫顽电场强度的英文

矫顽电场强度的英文Coercive Electric Field Strength.The concept of coercive electric field strength, often referred to as the breakdown strength or dielectric strength, is crucial in the field of electrical engineering and materials science. It characterizes the ability of a material to withstand electric stress without experiencing electrical breakdown, which is typically manifested as arcing or current leakage. Understanding and optimizingthis property is vital for ensuring the reliability and safety of electrical components and systems.### Definition and Importance.Coercive electric field strength, often denoted as Ec or Eb, represents the maximum electric field that a material can sustain without experiencing dielectric breakdown. Dielectric breakdown occurs when the electric field becomes so strong that it ionizes the material'satoms, leading to a sudden increase in conductivity and potentially causing damage to the material. The higher the coercive electric field strength, the better the material's insulating properties and its resistance to electrical breakdown.The importance of coercive electric field strength lies in its application in various electrical devices and systems. For example, capacitors, transformers, and other electronic components rely on the dielectric properties of materials to store and transmit energy efficiently. Materials with high coercive electric field strengths are preferred for these applications because they can handle higher voltages without failing. Additionally, understanding the coercive electric field strength of materials is crucial for predicting their behavior in extreme electrical environments, such as in power generation, transmission, and distribution systems.### Measurement and Characterization.Measuring coercive electric field strength typicallyinvolves subjecting a material to gradually increasing electric fields until breakdown occurs. The maximumelectric field achieved before breakdown occurs is recorded as the coercive electric field strength. This process requires specialized equipment, such as high-voltage sources and sensitive measurement instrumentation, to accurately measure the electric field and detect any signs of breakdown.Characterizing the coercive electric field strength of a material involves examining its dielectric properties, such as permittivity, conductivity, and breakdown voltage. These properties are influenced by various factors, including the material's composition, structure, temperature, and exposure to external factors like moisture and contaminants. Therefore, it's crucial to control these variables when measuring and comparing the coerciveelectric field strengths of different materials.### Materials Considerations.The coercive electric field strength of a material isinfluenced by its intrinsic properties and microstructure. For example, ceramics and polymers are known for their excellent dielectric properties and high coercive electric field strengths. Ceramics, such as alumina (Al2O3) and titanium dioxide (TiO2), have high breakdown voltages due to their dense atomic structure and low porosity. Polymers, on the other hand, offer flexibility and processability advantages while maintaining good dielectric properties. However, their coercive electric field strengths are often lower than those of ceramics.Composite materials, which combine the properties of two or more materials, can also exhibit high coercive electric field strengths. For instance, polymer-ceramic composites combine the mechanical flexibility of polymers with the dielectric strength of ceramics, resulting in materials that are suitable for use in harsh electrical environments.### Applications and Challenges.The application of materials with high coerciveelectric field strengths is diverse and spans multiple industries. In the automotive sector, they are used in alternators, starter motors, and electric vehicle batteries to handle the high voltages and currents required for efficient energy conversion and storage. In the aerospace industry, materials with excellent dielectric properties are crucial for ensuring the reliability and safety of avionics and flight control systems. Additionally, theyfind applications in power electronics, telecommunications, and medical equipment, where they contribute to efficient energy transmission, signal processing, and patient safety.However, there are challenges associated with the use of materials with high coercive electric field strengths. For instance, they may be susceptible to electrical aging, which is a gradual degradation of dielectric properties over time due to exposure to electrical stress. This degradation can lead to reduced performance and shortened lifetimes of electrical components. Therefore, it's essential to monitor and manage the electrical stress experienced by these materials to ensure their long-term reliability.### Conclusion.In conclusion, coercive electric field strength is a crucial parameter in electrical engineering and materials science, governing the ability of materials to withstand electric stress without experiencing dielectric breakdown. Understanding and optimizing this property is vital for ensuring the reliability and safety of electrical components and systems across various industries. Future research and development efforts should focus on improving the coercive electric field strengths of materials, enhancing their resistance to electrical aging, and expanding their applications in emerging technologies, such as high-voltage direct current (HVDC) transmission systems and advanced energy storage technologies.。

First-principle study of excitonic self-trapping in diamond

a rXiv:mtrl -th/9522v14Fe b1995First-principle study of excitonic self-trapping in diamond Francesco Mauri ∗and Roberto Car Institut Romand de Recherche Num´e rique en Physique des Mat´e riaux (IRRMA)IN-Ecublens 1015Lausanne,Switzerland Abstract We present a ﬁrst-principles study of excitonic self-trapping in diamond.Our calculation provides evidence for self-trapping of the 1s core exciton and gives a coherent interpretation of recent experimental X-ray absorption and emission data.Self-trapping does not occur in the case of a single valence exciton.We predict,however,that self-trapping should occur in the case of a valence biexciton.This process is accompanied by a large local relaxation of the lattice which could be observed experimentally.PACS numbers:61.80.−x,71.38.+i,71.35+z,71.55.−iTypeset using REVT E XDiamond presents an unusually favorable combination of characteristics that,in connection with the recent development of techniques for the deposition of thin diamondﬁlms,make this material a good candidate for many technological applications.Particularly appealing is the use of diamond in electronic or in opto-electronic devices,as e.g.UV-light emitting devices.Moreover,diamond is an ideal material for the construction of windows that operate under high power laser radiation or/and in adverse environments.It is therefore interesting to study radiation induced defects with deep electronic levels in the gap,since these can have important implications in many of these applications.Excitonic self-trapping is a possible mechanism for the formation of deep levels in the gap.The study of such processes in a purely covalent material,like diamond,is interesting also from a fundamental point of view.Indeed,excitonic self-trapping has been studied so far mostly in the context of ionic compounds,where it is always associated with,and often driven by,charge transfer eﬀects.In a covalent material the driving mechanism for self-trapping is instead related to the diﬀerence in the bonding character between the valence and the conduction band states.Both experimental data and theoretical arguments suggest the occurrence of self-trapping processes in diamond.In particular,a nitrogen(N)substitutional impurity induces a strong local deformation of the lattice[1–3]that can be interpreted as a self-trapping of the donor electron.The structure of a1s core exciton is more controversial[4–9].Indeed the similarity between an excited core of carbon and a ground-state core of nitrogen suggests that the core exciton should behave like a N impurity.However,the position of the core exciton peak in the diamond K-edge absorption spectra is only0.2eV lower than the conduction band minimum[4,7,8],while a N impurity originates a deep level1.7eV below the conduction band edge[10].On the other hand,emission spectra[8]suggest that a1s core exciton should self-trap like a N impurity.Finally,we consider valence excitations.In this case experimental evidence indicates that a single valence exciton is of the Wannier type,i.e.there is no self-trapping.To our knowledge,neither experimental nor theoretical investigations on the behavior of a valence biexciton in diamond have been performed,although simple scalingarguments suggest that the tendency to self-trap should be stronger for biexcitons than for single excitons.In this letter,we present a detailed theoretical study of excitonic self-trapping eﬀects in diamond.In particular,we have investigated the Born-Oppenheimer(BO)potential energy surfaces corresponding to a core exciton,a valence exciton and a valence biexciton in the context of density functional theory(DFT),within the local density approximation(LDA) for exchange and correlation.Our calculation indicates that the1s core exciton is on a diﬀerent BO surface in absorption and in emission experiments.Indeed X-ray absorption creates excitons in a p-like state as required by dipole selection rules.Subsequently the system makes a transition to an s-like state associated to a self-trapping distortion of the atomic lattice,similar to that found in the N impurity case.These results provide a coherent interpretation of the experimental data.In addition,our calculation suggests that self-trapping should also occur for a valence biexciton.This is a prediction that could be veriﬁed experimentally.Let us start by discussing a simple model[11,12].In diamond,the occupied valence and the lower conduction band states derive from superpositions of atomic sp3hybrids having bonding and antibonding character,respectively.Thus,when an electron,or a hole,or an electron-hole pair is added to the system,this can gain in deformation energy by relaxing the atomic lattice.Scaling arguments suggest that the deformation energy gain E def∝−1/N b, where N b is the number of bonds over which the perturbation is localized.This localization,due to quantum conﬁnement.The in turn,has a kinetic energy cost E kin∝+1/N2/3bbehavior of the system is then governed by the value of N b that minimizes the total energy E sum=E def+E kin.Since the only stationary point of E sum is a maximum,E sum attains its minimum value at either one of the two extrema N b=1or N b=∞.If the minimum occurs for N b=1,the perturbation is self-trapped on a single bond which is therefore stretched.If the minimum occurs for N b=∞,there is no self-trapping and the perturbation is delocalized.When N p particles(quasi-particles)are added to the system,one can showthat,for a given N b,E def scales as N2p,while E kin scales as N p.As a consequence,the probability of self-trapping is enhanced when N p is larger.This suggests that biexcitons should have a stronger tendency to self-trap than single excitons[12,13].In order to get a more quantitative understanding of self-trapping phenomena in dia-mond,we performed self-consistent electronic structure calculations,using norm-conserving pseudopotentials[14]to describe core-valence interactions.The wave-functions and the electronic density were expanded in plane-waves with a cutoﬀof35and of140Ry,respec-tively.We used a periodically repeated simple cubic supercell containing64atoms at the experimental equilibrium lattice constant.Only the wave-functions at theΓpoint were con-sidered.Since the self-trapped states are almost completely localized on one bond,they are only weakly aﬀected by the boundary conditions in a64atom supercell.The eﬀect of the k-point sampling was analysed in Ref.[3]where similar calculations for a N impurity were performed using the same supercell.It was found that a more accurate k-point sampling does not change the qualitative physics of the distortion but only increases the self-trapping energy by20%compared to calculations based on theΓ-point only[3].In order to describe a core exciton we adopted the method of Ref.[15],i.e.we generated a norm conserving pseudopotential for an excited carbon atom with one electron in the1s core level andﬁve electrons in the valence2s-2p levels.In our calculations for a valence exciton or biexciton we promoted one or two electrons,respectively,from the highest valence band state to the lowest conduction band state.Clearly,our single-particle approach cannot account for the(small)binding energy of delocalized Wannier excitons.However our approach should account for the most important contribution to the binding energy in the case of localized excitations.Structural relaxation studies were based on the Car-Parrinello(CP) approach[16].We used a standard CP scheme for both the core and the valence exciton, while a modiﬁed CP dynamics,in which the electrons are forced to stay in an arbitrary excited eigenstate[12,17],was necessary to study the BO surfaces corresponding to a valence biexciton.All the calculations were made more eﬃcient by the acceleration methods of Ref.[18].Weﬁrst computed the electronic structure of the core exciton with the atoms in the ideal lattice positions.In this case the excited-core atom induces two defect states in the gap:a non-degenerate level belonging to the A1representation of the T d point group,0.4eV below the conduction band edge,and a3-fold degenerate level with T2character,0.2eV below the conduction band edge.By letting the atomic coordinates free to relax,we found that the absolute minimum of the A1potential energy surface correponds to an asymmetric self-trapping distortion of the lattice similar to that found for the N impurity[3].In particular, the excited-core atom and its nearest-neighbor,labeled a and b,respectively,in Fig.1, move away from each other on the(111)direction.The corresponding displacements from the ideal sites are equal to10.4%and to11.5%of the bond length,respectively,so that the (a,b)-bond is stretched by21.9%.The other atoms move very little:for instance the nearest-neighbor atoms labeled c move by2.4%of the bond length only.This strong localization of the distortion is consistent with the simple scaling arguments discussed above.As a consequence of the atomic relaxation,the non-degenerate level ends up in the gap at1.5eV below the conduction band edge,while the corresponding wavefunction localizes on the stretched bond.The3-fold degenerate level remains close to the conduction band edge,but since the distortion lowers the symmetry from T d to C3v,the3-fold degenerate level splits into a2-fold degenerate E level and a non-degenerate A1level.In Fig.2we report the behavior of the potential energy surfaces corresponding to the ground-state,the A1and the T2core exciton states as a function of the self-trapping dis-tortion.Notice that the distortion gives a total energy gain of0.43eV on the A1potential energy surface.The same distortion causes an increase of the ground-state energy of1.29 eV.Our calculation indicates that the core-exciton behaves like the N impurity[3],support-ing,at least qualitatively,the validity of the equivalent core approximation.The similar behavior of the A1level in the core exciton and in the N impurity case was also pointed out recently in the context of semi-empirical CNDO calculations[9].The diﬀerences between the core exciton and the impurity[3]are only quantitative:in particular,the relaxationenergy and especially the distance of the A1level from the conduction band edge are smaller for the core exciton than for the N impurity.Our results suggest the following interpretation of the experimental data of Refs.[4,8]: (i)During X-ray absorption the atoms are in the ideal lattice positions.Dipole transitions from a1s core level to a A1valence level are forbidden,but transitions to the T2level are allowed.In our calculation the T2level is0.2eV lower than the conduction band edge,in good agreement with the core exciton peak observed in X-ray absorption spectra[4,8].(ii) On the T2BO potential energy surface the lattice undergoes a Jahn-Teller distortion which lowers its energy(see Fig.2).(iii)Since the LO phonon energy in diamond(0.16eV)is comparable to the energy spacing between the A1and the T2surfaces,which is less than 0.2eV after the Jahn-Teller distortion,the probability of a non-adiabatic transition from the T2to the A1surface is large.(iv)On the A1level the system undergoes a strong lattice relaxation resulting in a localization of the exciton on a single bond.(v)The self-trapping distortion induces a Stokes shift in the emitted photon energy.If the atomic relaxation were complete the Stokes shift would be equal to1.9eV,which correponds(see Fig.2) to the energy dissipated in the T2-A1transition(0.2eV),plus the energy gained by self trapping on the A1surface(0.43eV),plus the energy cost of the self-trapping distortion on the ground-state energy surface(1.29eV).The data reported in Ref.[8]show a shift of about1eV in the positions of the peaks associated to the1s core exciton in X-ray absorption and emission spectra.The emission peak is very broad,with a large sideband that corresponds to Stokes shifts of up to5eV.As pointed out in Ref.[8],this large sideband is likely to be the eﬀect of incomplete relaxation. This is to be expected since the core exciton lifetime should be comparable to the phonon period[8].As a consequence,the atomic lattice would be able to perform only a few damped oscillations around the distorted minimum structure during the lifetime of the core exciton.We now present our results for the valence excitations.While in the case of a single exciton the energy is minimum for the undistorted crystalline lattice,in the case of a biex-citon weﬁnd that the energy is minimized in correspondence of a localized distortion of theatomic lattice.This is characterized by a large outward symmetric displacement along the (111)direction of the atoms a and b in Fig.1.As a result the(a,b)-bond is broken since the distance between the atoms a and b is increased by51.2%compared to the crystalline bondlength.This distortion can be viewed as a kind of local graphitization in which the atoms a and b change from fourfold to threefold coordination and the corresponding hy-bridized orbitals change from sp3to sp2character.Again,in agreement with the model based on simple scaling arguments,the distortion is strongly localized on a single bond.As a matter of fact and with reference to the Fig.1,the atoms c and d move by1.2%of the bondlength,the atoms e and f move by2.3%,and the atoms not shown in theﬁgure by less than0.9%.The self-trapping distortion of the biexciton gives rise to two deep levels in the gap: a doubly occupied antibonding level,at1.7eV below the conduction band edge,and an empty bonding level,at1.6eV above the valence band edge.Both states are localized on the broken bond.In Fig.3we show how diﬀerent BO potential energy surfaces behave as a function of the self-trapping distortion of the valence biexciton.In particular,from thisﬁgure we see that,while for the biexciton there is an energy gain of1.74eV in correspondence with the self-trapping distortion,the same distortion has an energy cost of1.49eV for the single exciton,and of4.85eV for the unexcited crystal.We notice that,while DFT-LDA predicts self-trapping for the valence biexciton,it does not do so for the single exciton,in agreement with experiment.Similarly to the case of the core exciton the major experimental consequence of the self-trapping of the valence biexciton is a large Stokes shift in the stimulated-absorption spontaneous-emission cycle between the exciton and the biexciton BO surfaces.As it can be seen from Fig.3,this Stokes shift should be equal to3.23eV,i.e.to the sum of the energy gain of the biexciton(1.74eV)and of the energy cost of the exciton(1.49eV) for the self-trapping relaxation.The fundamental gap of diamond is indirect.Thus the spontaneous decay of a Wannier exciton in an ideal diamond crystal is phonon assistedand the radiative lifetime of the exciton is much longer than in direct gap semiconductors. However,after self-trapping of the biexciton,the translational symmetry is broken and direct spontaneous emission becomes allowed.As a consequence the radiative life time of the self-trapped biexciton is much smaller than that of the Wannier ing the DFT-LDA wavefunctions,we obtained a value of∼7ns for the radiative lifetime of the biexciton within the dipole approximation.This is several orders of magnitude larger than the typical phonon period.Therefore the self-trapping relaxation of the valence biexciton should be completed before the radiative decay.A self-trapped biexciton is a bound state of two excitons strongly localized on a single bond.Thus the formation of self-trapped biexcitons requires a high excitonic density.To realize this condition it is possible either to excite directly bound states of Wannier excitons, or to create a high density electron-hole plasma,e.g.by strong laser irradiation.In the second case many self-trapped biexcitons could be produced.This raises some interesting implications.If many self-trapped biexcitons are created,they could cluster producing a macroscopic graphitization.Moreover,since the process of self-trapping is associated with a relevant energy transfer from the electronic to the ionic degrees of freedom,in a high density electron hole plasma biexcitonic self-trapping could heat the crystal up to the melting point in fractions of a ps,i.e in the characteristic time of ionic relaxation.Interestingly,melting ofa GaAs crystal under high laser irradiation has been observed to occur in fractions of a ps[19].In Ref.[19]this phenomenon has been ascribed to the change in the binding properties due to the electronic excitations.Our study on diamond leads one to speculate that in a sub-picosecond melting experiment self-trapping phenomena could play an important role.In conclusion,we have studied excited-state BO potential energy surfaces of crystalline diamond within DFT-LDA.Our calculation predicts self-trapping of the core exciton and provides a coherent description of the X-ray absorption and emission processes,which com-pares well with the experimental data.Moreover,we also predict self-trapping of the valence biexciton,a process characterized by a large local lattice relaxation.This implies a strong Stokes shift in the stimulated absorption-spontaneous emission cycle of about3eV,whichcould be observed experimentally.It is a pleasure to thank F.Tassone for many useful discussions.We acknowledge support from the Swiss National Science Foundation under grant No.20-39528.93REFERENCES∗Present address:Departement of Physics,University of California,Berkeley CA94720, USA.[1]C.A.J.Ammerlaan,Inst.Phys.Conf.Ser.59,81(1981).[2]R.J.Cook and D.H.Whiﬀen,Proc.Roy.Soc.London A295,99(1966).[3]S.A.Kajihara et al,Phys.Rev.Lett.66,2010(1991).[4]J.F.Morar et al,Phys.Rev.Lett.54,1960(1985).[5]K.A.Jackson and M.R.Pederson,Phys.Rev.Lett.67,2521(1991).[6]J.Nithianandam,Phys.Rev.Lett.69,3108(1992).[7]P.E.Batson,Phys.Rev.Lett.70,1822(1993).[8]Y.Ma et al,Phys.Rev.Lett.71,3725(1993).[9]A.Mainwood and A.M.Stoneham,J.Phys.:Condens.Matter6,4917(1994).[10]R.G.Farrer,Solid State Commun.7,685(1969).[11]W.Hayes and A.M.Stoneham,Defects and defect processes in nonmetallic solids,(Wiley&Sons,New York,1985)pags.29-38.[12]F.Mauri,R.Car,(to be published).[13]The number of equal particles that can be accommodated on one bond of the crystal inthe same quantum state is limited by the Pauli principle.Thus no more than two holes or/and two electrons with opposite spins can be localized on one bond of a sp3bonded semiconductor.[14]G.Bachelet,D.Hamann,and M.Schl¨u ter,Phys.Rev.B26,4199(1982).[15]E.Pehlke and M.Scheﬄer,Phys.Rev.B47,3588(1993).[16]R.Car and M.Parrinello,Phys.Rev.Lett.55,2471(1985).[17]F.Mauri,R.Car and E.Tosatti,Europhys.Lett.24,431(1993).[18]F.Tassone,F.Mauri,and R.Car,Phys.Rev.B50,10561(1994).[19]orkov,I.L.Shumay,W.Rudolph,and T.Schroder,Opt.Lett.16,1013(1991);P.Saeta,J.-K.Wang,Y.Siegal,N.Bloembergen,and E.Mazur,Phys.Rev.Lett.67, 1023(1991);K.Sokolowski-Tinten,H.Schulz,J.Bialkowski,and D.von der Linde, Applied Phys.A53,227(1991).FIGURESFIG.1.Atoms and bonds in the ideal diamond crystal(left panel).Atoms and bonds after the self-trapping distortion associated with the valence biexciton(right panel).In this case the distance between the atoms a and b increases by51.2%.A similar but smaller distortion is associated with the core exciton:in this case the(a,b)distance is increased by21.9%.FIG.2.Total energy vs self-trapping distortion of the core-exciton.Theﬁgure displays the BO potential energy surfaces correponding to the ground-state,the A1,and the T2core exciton states.FIG.3.Total energy as a function of the self-trapping distortion of the biexciton.The BO energy surfaces correponding to the ground state,the valence exciton,and the valence biexciton are shown in theﬁgure.a b ce df(111)ground stateA 1−core excitonT 2−core excitonconduction ideal lattice distorted latticeground statebi−excitonexcitondistorted lattice ideal lattice。

基于CEEMDAN

第 39 卷第 1 期电力科学与技术学报Vol. 39 No. 1 2024 年 1 月JOURNAL OF ELECTRIC POWER SCIENCE AND TECHNOLOGY Jan. 2024引用格式：尚海昆，张冉喆，黄涛,等.基于CEEMDAN-TQWT方法的变压器局部放电信号降噪[J].电力科学与技术学报,2024,39(1):272‑284. Citation：SHANG Haikun,ZHANG Ranzhe,HUANG Tao,et al.Partial discharge signal denoising based on CEEMDAN‑TQWT method for power transformers[J]. Journal of Electric Power Science and Technology,2024,39(1):272‑284.基于CEEMDAN‑TQWT方法的变压器局部放电信号降噪尚海昆，张冉喆，黄涛，林伟，赵子璇（东北电力大学现代电力系统仿真控制与绿色电能新技术教育部重点实验室，吉林吉林 132012）摘要：针对传统方法处理局部放电信号时存在振荡明显、消噪不彻底等问题，采用基于自适应白噪声完备集成经验模态分解（complete ensemble empirical model decomposition with adaptive noise，CEEMDAN）与可调品质因子小波变换（tunable Q⁃factor wavelet transform，TQWT）相结合的方法对局部放电信号进行消噪处理。

采用CEEMDAN将含噪变压器局部放电信号分解成多个固有模态函数（intrinsic mode function，IMF）分量，并利用相关系数判断IMF分量与原始信号的相关度。

将弱相关者视为劣质IMF，对其进行TQWT分解，利用能量占比与峭度指标来筛选小波子带，提取IMF的有效细节信息，进行TQWT逆变换，从而得到新的IMF分量；将强相关者视为优质IMF，与变换后的新IMF分量共同进行信号重构，得到消噪结果。

Excitons, Biexcitons and Trions

9Excitons,Biexcitons and TrionsIn Chap.8we deﬁned the bandstructure for electrons and holes as the solu-tions to the(N±1)-particle problem and later we saw that the number of electrons in a band can be increased or decreased by donors and acceptors,re-spectively(Sect.8.14).In contrast,the number of electrons remains constant in the case of optical excitations with photon energies in the eV or band gap region.What we can do,however,is to excite an electron from the valence to the conduction band by absorption of a photon.In this process,we bring the system of N electrons from the ground state to an excited state.What we need for the understanding of the optical properties of the electronic system of a semiconductor an insulator or even a metal is therefore a description of the excited states of the N particle problem.The quanta of these excitations are called“excitons”in semiconductors and insulators.We can look at this problem from various points of view.The ground state of the electronic system of a perfect semiconductor is a completelyﬁlled valence band and a completely empty conduction band. We can deﬁne this state as the“zero”energy or“vacuum”state.In addition it has total momentum K=0,angular momentum L=0and spin S=0. From this point E=0,K=0we will start later on to consider the dispersion relation of the excitons in connection with Fig.9.1b.Another point of view is the following.If we start from the above-deﬁned groundstate and excite one electron to the conduction band,we simultaneously create a hole in the valence band(Fig.9.1a).In this sense an optical excitation is a two-particle transition.The same is true for the recombination process.An electron in the conduction band can return radiatively or non-radiatively into the valence band only if there is a free place,i.e.,a hole.Two quasiparticles are annihilated in the recombination process.This concept of electron–hole pair excitation is also used successfully in other disciplines of physics.If one excites,e.g.,an electron in a(larger)atom or a neutron or proton in a nucleus from a deeper lying occupied state into a higher,empty one,a quantitative description concerning the transitions energies is obtained only if one takes into account both the particle in the2429Excitons,Biexcitons and Trionsexcited state and the hole left behind.This is even true if one excites an electron in a metal from a state in the Fermi sea with an energy below the Fermi energy E F to an empty state above.Excitons can be described at various levels of sophistication.We present in the next sections the most simple and intuitive picture using the eﬀective mass approximation.Other approaches are described in[57E1,62N1,63K1, 63P1,77B1,78U1,79E1,79R1,79S1,81F1,81K1,85H1]or[82M1,86U1,93P1, 96Y1,98E1,98R1,04O1]of Chap.1and references therein.In Chap.27we shall also see how excitons are described in semiconductor Bloch equations.The concepts of Wannier and Frenkel excitons were introduced in the second half of the1930s[31F1,37W1].There is some controversy concerning theﬁrst experimental observations.The author does not wish to act as the referee to settle this point.Instead we give some references to early work [50H1,52G1,56G1,58N1,62N1]and to Fig.13.9and leave the decision to the reader.9.1Wannier and Frenkel ExcitonsUsing the eﬀective mass approximations,Fig.9.1a suggests that the Coulomb interaction between electron and hole leads to a hydrogen-like problem with a Coulomb potential term–e2/(4π 0 |r e−r h|).Indeed excitons in semiconductors form,to a good approximation,a hy-drogen or positronium like series of states below the gap.For simple parabolicFig.9.1.A pair excitation in the scheme of valence and conduction band(a)in the exciton picture for a direct(b)and for an indirect gap semiconductor(c)9.1Wannier and Frenkel Excitons243 bands and a direct-gap semiconductor one can separate the relative motion of electron and hole and the motion of the center of mass.This leads to the dispersion relation of excitons in Fig.9.1b.E ex(n B,K)=E g−Ry∗1n2B+2K22M(9.1a)withn B=1,2,3...principal quantum number,Ry∗=13.6eVµm01ε2exciton Rydberg energy,(9.1b)M=m e+m h,K=k e+k h translational mass andwave vector of the exciton.(9.1c) For the moment,we use a capital K for the exciton wave vector to distinguish this two-particle state from the one-particle states.When we are more familiar with the exciton as a new quasi-particle we shall return to k.µ=m e m hm e+m hreduced exciton mass,(9.1d)a ex B=a H Bεm0µexcitonic Bohr radius.(9.1e)The radii of higher states can be considered on various levels of complexity.If one only takes into account the exponential term exp{Zr/na HB }in the radialpart of the wave function of the hydrogen problem appearing as the envelope function in(9.4a)and deﬁnes the(excitonic)Bohr radius by the decrease of this term to1/e,one obtains with Z=1for excitonsa B(n B)=a H Bεm0µn B,(9.2a)i.e.,a linear increase with n B.If,on the other hand,one takes the full radial function into account,i.e., also including the factorρl L2l+1n B+l(ρ)(9.2b)where l is the angular quantum number andρ=2Z r/n B·a H B and L2l+1n0B+l are the Laguerre polynomials and calculates the average distance between electron and proton or hole,respectively,one obtains[55S1,94L1]r(n B) =a B23n2B−l(l+1)(9.2c)i.e.,for the n B S states(i.e.l=0)a quadratic dependence starting with3a B/2 for n B=1.2449Excitons,Biexcitons and TrionsThe series of exciton states in(9.1a)has an eﬀective Rydberg energy Ry∗modiﬁed by the reduced mass of the electron and hole and the dielectric “constant”of the medium in which these particles move;n B is the principal quantum number.The kinetic energy term in(9.1a)involves the translational mass M and the total wave vector K of the exciton.The radius of the exciton equals the Bohr radius of the H atom again modiﬁed byεandµ.Using the material parameters for typical semiconductors oneﬁnds1meV≤Ry∗≤200meV E g(9.3a) and50nm a B≥1nm>a lattice.(9.3b) This means that the excitonic Rydberg energy Ry∗is usually much smaller than the width of the forbidden gap and the Bohr radius is larger than the lattice constant.This second point is crucial.It says that the“orbits”of electron and hole around their common center of mass average over many unit cells and this in turn justiﬁes the eﬀective mass approximation in a self-consistent way.These excitons are called Wannier excitons[37W1].In this limit,excitons can usually also be described in the frame of semicon-ductor Bloch equations,butﬁner details or corrections to this simple model, which we shall treat in Sect.9.2and which are necessary to understand optical spectra,are usually not incorporated in the semiconductor Bloch equations of Chap.27.It should be mentioned that in insulators like NaCl,or in organic crys-tals like anthracene,excitons also exist with electron–hole pair wavefunctions conﬁned to one unit cell.These so-called Frenkel excitons[31F1]cannot be described in the eﬀective mass approximation.As a rule of thumb,one can state that in all semi-conductors the inequalities(9.3a,b)hold,so that we always deal with Wannier excitons.A series of conferences devoted both to excitons in anorganic semicon-ductors and in organic ones as well as in anorganic and organic insulators is EXCON(excitonic process in condensed matter).The proceedings have been published in[94E1]To get an impression of the wavefunction,we form wave packets for elec-trons and holesφe,h(r e,h)in the sense of the Wannier function of(8.10)and obtain schematically for the exciton wavefunction(r e−r h),(9.4a)φ(K,n B,l,m)=Ω−1/2e i K·Rφe(r e)φh(r h)φenvn B,l,mwith the center of mass RR=(m e r e+m h r h)/(m e+m h),(9.4b) whereΩ−1/2is the normalization factor.The plane-wave factor describes the free propagation of a Wannier exciton through the periodic crystal similarly as9.1Wannier and Frenkel Excitons245 for the Bloch waves of Sect.8.1,and the hydrogen-atom-like envelope function φenv describe the relative motion of electron and hole.The quantum numbers l and m,with l<n B and−l≤m≤l have the same meaning as for the hydrogen atom in the limit that the angular momentum is a good quantum number in a solid(see Chap.26).As for the H atom,the exciton states converge for n B→∞to the ioniza-tion continuum,the onset of which coincides with E g.Excitons features are especially strong for regions of the electron and hole dispersions,where the group velocities of electrons and holesυe g andυh g are equal e.g.zero.See e.g.[96Y1]of Chap.1.This means,that in direct gap semiconductors excitons form preferentially around K=0if the direct gap occurs at theΓpoint.In indirect gap semiconductors exciton states form preferentially with the hole around k h=0and the electron in its respective minimum,as shown in Fig.9.1c schematically e.g.for Ge where the conduction band has minima at k e=0and k e=0.Electrons and holes with diﬀerent group velocities still“see”their mutual Coulomb interaction,but the dashed continuation of exciton dispersion from the indirect to the direct gap is an oversimpliﬁcation, among others because the states away from the band extrema are strongly damped and their binding energy varies.See e.g.again[96Y1]of Chap.1.The discrete and continuum states of the excitons will be the resonances or oscillators which we have to incorporate into the dielectric function of Chaps. 4and5.For direct semiconductors with dipole allowed band-to-band transitions, oneﬁnds an oscillator strength for excitons in discrete states with S(l=0) envelope function proportional to the band-to-band dipole transition matrix element squared and to the probability ofﬁnding the electron and hole in the same unit cell.For the derivation of this relation see[57E1].This lattercondition leads to the n−3B dependence of the oscillator strength for three-dimensional systems.f nB∝|H D cv|21n3B.(9.5)These f nBresult in corresponding longitudinal–transverse splitting as shown in connection with(4.26).Equation((9.5))holds for so-called singlet excitons with antiparallel electron and hole spin.Triplet excitons involve a spin ﬂip,in their creation which signiﬁcantly reduces their oscillator strength(spin ﬂip forbidden transitions).Since the singlet and triplet pair or exciton states will play some role in later chapters(see,e.g.,Chaps.13to16)we give some simpliﬁed information on this topic here.The crystal ground state,i.e.,completelyﬁlled valence bands and com-pletely empty conduction bands,has,as already mentioned above,angular momentum L,spin S and total angular momentum J equal to zero.If we excite optically an electron from the valence band to the conduction band,2469Excitons,Biexcitons and Trionse.g.,by an electric dipole transition,the spin of the excited electron does not change because the electricﬁeld of the light does not act on the spin. Consequently,the simultaneously created hole has a spin opposite to the one of the excited electron and the total spin S of the electron–hole pair state is still zero.Consequently,the electron–hole pair,and likewise the ex-citon,is said to be in a spin singlet state.The spin S=1of the photon is accommodated by the spatial part of the band-to-band matrix element in ((9.5))or by the envelope function of the exciton(see(9.4a)and[96Y1]of Chap.1).If the spinﬂips in the transition,e.g.,by interaction with the magnetic component of the lightﬁeld,one ends up with a total spin S=1corresponding to spin triplet(exciton)state.The triplet state is situated energetically below the singlet state and the splitting is due to a part of the electron–hole exchange interaction,essentially the so-called short-range or analytic(for K→0)part of the exchange interaction.There are names other than triplet and singlet used for certain materials like para and ortho exciton(e.g.,for Cu2O)or dark and bright excitons(in quantum dots).The last pair of names reﬂects the fact that triplet excitons have small oscillator strength because they are“spin-ﬂip forbidden.”Since spin and angular momentum are,strictly speaking,no good quantum numbers in a crystal(see Chap.26)it is not obligatory that the triplet exciton is threefold degenerate and the singlet is not degenerate.There are cases where the triplet or para exciton is nondegenerate,as in Cu2O, and the singlet is threefold degenerate,as in many zinc-blende type crystals like CuCl or ZnSe or GaAs.This is possibly the reason that in some older, especially French,literature the names singlet and triplet exciton states are interchanged.The oscillator strength of the continuum states is inﬂuenced by the so-called Sommerfeld enhancement factor.We come back to this point later in (9.3)and in Chaps.13and15,when we discuss the optical properties.In the picture of second quantization,we can deﬁne creation operators for electrons in the conduction band and for holes in the valence bandα+keandβ+kh ,respectively.The combination of both gives creation operators for elec-tron hole pairsα+ke β+kh.The exciton creation operator B+can be constructedvia a sum over electron–hole pair operators(see[81K1]or[93H1]of Chap.1).B+K=k e k h δ[K−(k e+k h)]αke,k hα+keβ+kh;(9.6)the expansion coeﬃcients a ke,k h correspond,in principle,to those also used inan(slightly old-fashioned)expansion into Slater determinantes of the manyparticle problem,which contains either all valence band statesϕkh (r i,h)forthe ground state or always has one line being replaced by a conduction band state[63K1].It can be shown that the B+K,B K,obey Bose commutation relations with a density-dependent correction term which increases with the number9.2Corrections to the Simple Exciton Model247of electrons and holes contained in the volume of one exciton4π(a exB )3/3[77H2,81K1].This has two consequences:in thermodynamic equilibrium for low densities and not too low temperatures,the excitons can be well described by Boltz-mann statistics with a chemical potential ruled by their density and temper-ature similar to(8.43).For higher densities they deviate more and more from ideal bosons until they end up in an electron–hole plasma made up entirely from fermions(see Sect.20.5).This makes the creation of a Bose-condensed state of excitons(or of biexcitons)a very complicated problem.Though the-ory predicts a region in the temperature density plane where an excitonic Bose–Einstein condensed state could occur[77H2],there are currently many hints but no generally accepted,clearcut observation of a spontaneous Bose condensation of excitons.There are some experiments that prove the Bose character of excitons[76L1,83H1,83M1,83P1,84W1,87F1].We come back to this problem in Sect.20.5where we shall discuss also more recent results. 9.2Corrections to the Simple Exciton ModelThe simple model outlined in the preceeding section is,as already mentioned, adequate for non-degenerate,parabolic bands.We keep these assumptions for the moment and inspect aﬁrst group of corrections which are relevant for the parametersεandµentering in(9.1a–e).We already know from Chaps.4–7 thatεis a function ofω,resulting in the question of which value should be used.As long as the binding energy of the exciton E b ex is small compared to the optical phonon energies and,consequently,the excitonic Bohr radius((9.1e)) larger than the polaron radius(8.16)E b ex< ωLO;a B>a Pol⇒ε=εs,(9.7a) we can use for the static valueεs below the phonon resonances and the polaron masses and polaron gap.This situation is fulﬁlled for some semi-conductors for all values of n B, e.g.,for GaAs where Ry∗ 5meV andωLO 36meV.In many other semiconductors including especially the wide gap semicon-ductors(see Fig.9.3)the inequality(9.7a)holds only for the higher states n B≥2,while for the ground state exciton(n B=1)weﬁndE b ex ωLO;a B a Pol−→εs≥ε≥εb.(9.7b)Examples are CdS,ZnO,CuCl or Cu2O.In this situation a value forεbetweenεs andεb seems appropriate,because the polarization of the lattice can only partly follow the motion of electron and hole.A useful approach is the so-called Haken potential[55H1]which interpolates betweenεs andεb depending on the distance between electron2489Excitons,Biexcitons and Trionsand hole,where r eh,r p e and r h p are the distances between electron and hole, and the polaran radii of electron and hole,respectively:1ε(r e,h)=1εb−1εb−1εs1−exp(−r eh/r p e)+exp(−r eh/r ph)2.(9.8)The next correction concerns the eﬀective masses.The polarization cloudsof the polarons(Sect.8.6)have diﬀerent signs for electron and hole.If bothparticles are bound together in an exciton state fulﬁlling(9.7b)the polaronrenormalization is partly quenched,with the consequence that values for theeﬀective masses will lie somewhere between the polaron values and the onesfor a rigid lattice.The gap“seen”by the exciton in the1S state will likewisebe situated between the two above extrema.Fortunately,the above eﬀectstend to partly compensate each other.A transition from the polaron gap tothe larger rigid lattice gap shifts the exciton energy to larger photon energies.A transition fromεs toεb and a increase of the eﬀective masses increasesthe binding energy and shift the1S exciton to lower photon energies.Asa consequence oneﬁnds,even for many semiconductors for which inequality(9.7b)holds,that the1S excitonﬁts together with the higher exciton statesreasonably well into the hydrogen-like series of(9.1).We shall use this ap-proach in the future if not stated otherwise.In other cases like Cu2O or CuCl,the higher states with n B≥2follow a hydrogen-like n−2B series converging to the polaron gap,but the1S exciton shows,with respect to the polarongap,a binding energy E b ex,which diﬀers from and is generally larger thanthe excitonic Rydberg energy deduced from the higher states.We call theexperimentally observed energetic distance between the1S exction and thepolaron gap the exciton binding energy E b ex,in contrast to Ry∗in(9.1).Thisdiscrepancy introduces some ambiguity when comparing theoretical resultswith experimental data,since in theory one often normalizes energies withthe excitonic Rydberg energy Ry∗,but the1S exciton has a diﬀerent valuefor E b ex.There is a general trend of the material parameters m eﬀandεwith E gwhich results in an increase of the exciton binding energy with increasing E gas shown in Fig.9.3.This is an analogous consequence of increasing the band width or in otherwords decreasing eﬀective mass with decreasing width of the gap as described,e.g.,in the band structure model of Sect.8.5.The next complication comes from the band structure.If the bands aredegenerate,as is theΓ8valence band in T d symmetry,it is no longer possibleto separate the relative and the center of mass motion–they are coupledtogether.Similar eﬀects stem from k-linear terms and other sources.We getlight-and heavy-hole exciton branches and splittings,e.g.,between the2Sand2P exciton states partly induced by the envelope function.An examplefor Cu2O will be given in Sect.13.2.Furthermore it should be mentioned,without going into details,that thesplitting between singlet and triplet excitons∆st and the splitting of the9.2Corrections to the Simple Exciton Model249 singlet state into a transverse and a longitudinal one∆LT,are both due to exchange interaction between electron and hole caused by their Coulomb in-teraction[73D1,88F1,03G1]if we consider the N-electron problem in the form of Slater’s determinant where the ground state consists only of valence-band states and the excited state of a sum of determinants in each of which one valence-band state is replaced by a conduction-band state.This aspect is treated in detail in[73D1]or in[93H1,93P1]of Chap.1.Usually the following relation holds for Wannier excitons∆st ∆LT with0.1meV ∆LT 15meV.(9.9)For the AΓ5-excitons in CdS or ZnO oneﬁnds,e.g.,∆CdSLT =1.8meV,∆CdSst =0.2meV and∆ZnOLT=1.2meV,∆ZnOst=0.17meV.Since theshort range interaction increases with decreasing exciton radius as a−3B thesituation begins to change for1S excitons with a value of a exB exceeding onlyslightly the lattice constant,and leading thus also to the limit of the conceptWannier excitons.Oneﬁnds,e.g.,for CuCl or Cu2O,∆CuCLLT =5.5meV,∆CuClst =2.5meV,∆Cu2OLT≈50µeV,∆Cu2Ost=12meV(see[85H1]).Theextremely low value of∆LT in Cu2O comes from the fact,that the band-to-band transition is parity forbidden and the1S singlet or ortho exciton is only allowed in quadrupole approximation(see Sect.13.2).The singlet-triplet splitting is even enhanced by the compression of the excitonic wave function in quantum wells and especially in quantum dots(see, e.g.,[98F1,03G1]).We come back to this aspect in Sect.9.3and Chap.15.Fig.9.2.The splitting of the1S exciton with the notation of the various contri-butions to exchange splitting for bulk samples(a)and in a quantum dot(b).The abreviations have the following meanings:S=singlet,T=triplet,∆lr=long range or nanoanalytic(for k→0)part of the exchange interaction,∆sr=short range or analytic part of the exchange interaction[03G1]2509Excitons,Biexcitons and TrionsFig.9.3.The exciton bind-ing energy E b ex as a functionof the band-gap for various di-rect gap semiconductors([82L1,93H1,93P1]of Chap.1) It should be mentioned that the excitation of an optically(dipol-)allowedexciton is accompanied by a polarization as detailed in Chaps.5and27onpolaritons and on(semiconductor)Bloch equations,respectively.The concept of exciton-phonon boundstates has been introduced in[68T1,72K1,02B1].Finally we mention that excitons can also be formed with holes in deepervalence bands.These so-called“core-excitons”are usually situated in the VUVor X-ray region of the spectrum and have a rather short lifetime.An exampleand further references are given in Sect.13.1.7and[87C1,88K1].9.3The Inﬂuence of DimensionalityIf we consider the exciton again as an eﬀective mass particle with parabolicdispersion relations,as given by(9.1),we expect aﬁrst inﬂuence of the di-mensionality on the density of states analogous to the situation shown in Fig.8.20for every exciton branch n B=1,2,3...Another eﬀect of the dimensionality manifests itself in the binding energy,the Rydberg series and the oscillator strength.We consider an exciton,forwhich the motion of electron and hole is restricted to a two-dimensional plane,but the interaction is still a3d one,i.e.,proportional to e2/|r e−r h|and ﬁnd(9.10a),(9.10b)in comparison to the3d case of(9.1)(see e.g.[93H1]ofChap.1):3d:E(K,n B)=E g−Ry∗1n2B+2(K2x+K2y+K2z)2M(9.10a)and for the oscillator strength f for the principal quantum number n B in the limit of(9.2a):f(n B)∝n−3B;a B∝a H B n B;n B=1,2,3....(9.10b)9.3The Inﬂuence of Dimensionality2512d:E(K,n B)=E g+E Q−Ry∗1(n B−12)2+2(K2x+K2y)2M(9.10c)with E Q quantization energyf(n B)∝n B−12−3;a B∝a H Bn B−12,n B=1,2,3....(9.10d)Essentially n B has to be replaced by n B−1/2when going from3d to 2d systems and the quantization energies E Q of electrons and holes must be considered.Actually E Q diverges for conﬁnement in a mathematically2d plane,see Sect.8.10.The excitonic Rydberg Ry∗is the same in both cases with the consequence that the binding energy of the1S exciton is Ry∗in three,and 4Ry∗in two dimensions.The oscillator strength increases and the excitonic Bohr radius decreases when going from three-to two-dimensional systems.The usual realization of quasi-2d excitons is via(M)QW of type I.In this case the motion in the z-direction is quantized,but the width of the quantum well l z is non-zero.In Fig.9.4we show the exciton binding energy for GaAs as a function of l z for inﬁnitely high barriers,where the curve reaches4Ry∗for l z=0and for ﬁnite barrier height,where the binding energy converges to the value of the barrier material for l z=0passing through a maximum of about2to3times Ry∗depending on the material parameters.It is possible to describe the binding energy of the exciton in the quasi two-dimensional case of a QW in terms of an“eﬀective”dimensionality d eﬀthat ranges between three and two and interpolates thus between the limiting cases of(9.10a)and(9.10c)by[91H1,92M1].E b ex(1S)=Ry∗1+d eff−322(9.10e)with d eﬀ=3−exp−L W2aβ(9.10f)where aβis the three-dimensional excitonic Bohr radius and L W the width of the quantum well increased by the penetration depth of electron and hole into the barrier.The increase of the oscillator strength of the1S exciton comes from the fact that the quantization in the z-direction increases both the overlap between electron and hole and their attraction,which results in turn in a reduction of the two-dimensional Bohr radius.The Sommerfeld factor F,which describes the enhancement of the os-cillator strength of the continuum states and which arises from the residual electron–hole correlation,depends also on the dimensionality[66S1,91O1]. It reads3d:F3d=πW1/2eπW1/2sinh(πW1/2)with W=(E−E g)Ry∗−1;(9.11a)2529Excitons,Biexcitons and TrionsFig.9.4.The calculated bind-ing energy of n z=1hh excitonsin AlGaAs/GaAs quantum wellsas a function of the well thick-ness l z([81M1,84G1,88K2])2d:F2d=eπW1/2cosh(πW1/2)with W=(E−E g+E Q)Ry∗−1.(9.11b)In the three-dimensional case it has a square-root singularity at E g and decreases gradually to unity for E>E g.In two dimensions it decays only from two to one with increasing energy and in a one-dimensional system it is even below unity above the gap quenching thus the singularity in the(combined) density of states.We shall see the consequences in Chap.13.The corrections which we mentioned in Sect.9.2hold partly also in the quasi-2d case.The2d valence band structure may be even more complex, than the3d one,comparing e.g.Figs.8.17and Fig.8.24.However,it should be noted that the most widely investigated(M)QW are based on AlGaAs and InGaAs,which fulﬁll the inequality(9.6).The most striking feature of excitons in quasi-two-dimensional systems and in systems of even lower dimensionality,however,are the facts,that ex-citon series(n B=1...∞)exist for every combination of electron-and hole subband,though partly with small or vanishing oscillator strength and that the exciton splits into light-and heavy-hole excitons which results from the corresponding splitting of the valence-band states(Figs.8.23and8.24).To describe the exciton states we thus need more quantum numbers.Apart from the principal quantum number n B in(9.1)or(9.10a),we must state which of the quantized conduction-and valence-band states are involved.The sim-9.3The Inﬂuence of Dimensionality253 plest optical interband selection rule is∆n z=0,so that we shall see in optical spectra mainly excitons which obey this rule.Finally we must specify whether we are speaking of the light-or the heavy-hole plete information might thus be the n h z=2hh,n e z=2,n B=1exciton ually one uses the abbreviation n z=2hh exciton involving the above selection rule and the fact that excitons in MQW with n B>1are usually diﬃcult to resolve due to broadening eﬀects.For examples see Sect.15.1.As already discussed with one particle states in Sect.8.10,minibands also form for excitons in superlattices.In type II structures,the Coulomb interaction is decreased due to the spa-tial separation of electrons and holes in the two materials with wave function overlap reduced to the interface region.Excitons in such structures are also said to be indirect in real space.In strictly one-and zero-dimensional cases the binding energy for the ex-citon diverges.So it is not possible to give general formulas like(9.10c)for these situations.One is always limited to numerical calculations which have to explicitly include theﬁnite dimensions of the quantum wire or quantum dot.The various possibilities to produce quantum wells,superlattices,quan-tum wires and dots have been discussed already in Sects.8.11to8.13.In the following,we give some information relevant for approximately spherical quan-tum dots,as they occur frequently for semiconductor nanocrystals in glass or organic matrices.For details see e.g.[93B1,97W1,98G1,98J1]of Chap.1.Three regimes of quantization are usually distinguished in quantum dots in which the crystallite radius R is compared with the Bohr radius of the excitons or related quantities:weak conﬁnement:R a B,E Q<Ry∗;(9.12a) medium or intermediate conﬁnement:a e B≥R≥a h B;E Q≈Ry∗,(9.12b)with a e,hB =a H Bεm0m e,hstrong conﬁnement:R≤a h B;E Q>Ry∗.(9.12c) In theﬁrst case(9.12a)the quantum dot(QD)is larger than the exciton.Asa consequence the center-of-mass motion of the exciton,which is described in(9.1a)and(9.4a)by the term e i K·R,is quantized while the relative motionof electron and hole given by the envelope functionφnB,l,m (r e−r h)is hardlyaﬀected.This situation is found,e.g.,for the Cu halides where a B is small,or for CdSe QD with R≥10nm.In the second case(9.12b)R has a value be-tween the radii of the electron orbit and the hole orbit around their common center of mass.As a consequence,the electron state is quantized and the hole moves in the potential formed by the dot and the space charge of the quan-tized electron.This case is the most demanding from the theoretical point of view since Coulomb eﬀects and quantization energies are of the same order of magnitude.However,it is often realized for QD of II–VI semiconductors.The regime given by(9.12c)becomes easier again.The Coulomb energy increases。

2.有机半导体的基本知识

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Electronic processes in molecules
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Nonradiative energy transfer
Energy transfer processes: 1. Radiative transfer 2. Förster transfer 3. Dexter transfer
An exciton “hoping” along several chains is also called “migration”.
21
Fluorescence vs Phosphorescence
• • • Fluorescence emitter: No change for e- spin direction Phosphorescence emitter: 100% spin state change, e.g. Eu or Ir-complex TADF: Low energy splitting materials (~0.5 – 1.0 eV) Phosphorescence TADF: thermally activated delayed fluorescence
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Förster transfer
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Förster transfer - example
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Dexter transfer
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Nonradiative energy transfer
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Diffusion of excitons
Exciton can move as a pair – highly undesirable for any process which relies on charge extraction, e.g., organic PV Exciton can only move in a order of ~10 nm long, the area along an interface is extremely small. Förster transfer: shorter chains have a higher gap energy than long chains. It is energetically favorable for an exciton to move to longer chains.

雅思作文wave energy

雅思作文wave energyThe wave energy is so huge and exists so widely that it has attracted skilled craftsmen in the coastal areas since ancient times. They have tried every means to control the waves for human use.The energy contained in waves mainly refers to the kinetic energy and potential energy of the ocean surface waves. Wave energy is generated by the wind transmitting energy to the ocean, which is essentially formed by absorbing wind energy. The energy transfer rate is related to the wind speed and the distance between wind and water (i.e. wind zone). When the water mass displaces relative to the sea level, the waves have potential energy, while the movement of water quality points makes the waves have kinetic energy. The stored energy is dissipated by friction and turbulence, and the dissipation speed depends on the wave characteristics and water depth.The energy dissipation speed of big waves in deep sea area is very slow, which leads to the complexity of wave system, and it is often accompanied by local wind and the impact of stormsgenerated in the distance a few days ago. Waves can be described by such characteristics as wave height, wavelength (distance between two adjacent peaks) and wave period (time between two adjacent peaks).The energy of the wave is proportional to the square of the wave height, the movement period of the wave and the width of the wave surface. Wave energy is one of the most unstable energy sources in the ocean. The power density of huge waves caused by typhoons can reach thousands of kW per meter of the wave surface, while the annual average wave power of the North Sea area in Europe, which is rich in wave energy, is only 20-40kw / m. The annual average wave power density of most Chinese coasts is 2-7kw / m.The theoretical estimated value of wave energy in the world is also in the order of 109kw. According to the data of China coastal ocean observation station, the annual average theoretical wave power in China's coastal area is about 1.3x107kw. However, the actual coastal wave power is greater than this value because the observation sites of many ocean stations are located in the inner bay or where the wind and waves are small. Among them, the coastalareas of Zhejiang, Fujian, Guangdong and Taiwan are rich in wave energy.。

小波包频带能量分英文

小波包频带能量分英文Here's a piece of writing in English, following the requirements you mentioned:Wavelet packet analysis is a really cool tool for frequency band energy analysis. It allows us to dig deep into the signal and extract energy information fromspecific frequency ranges. You know, sometimes you're just interested in a certain part of the spectrum, and wavelet packets are perfect for that.The way wavelet packets break down signals is kind of like zooming in on a photo. You can start with a broad overview and then gradually focus on smaller and smaller details. This makes it super flexible for analyzing signals with complex frequency content.One of the great things about wavelet packets is that they can adapt to changes in the signal. Unlike some other methods, wavelet packets don't assume the signal is static.They can handle signals that vary over time, which is often the case in real-world applications.When it comes to frequency band energy, wavelet packets give you a really fine-grained view. You can see exactly how much energy is in each frequency band, and how that energy changes over time. This can be really useful for diagnosing problems or optimizing systems.So if you're dealing with signals that have complicated frequency content, and you want to get a detailed picture of the energy distribution, wavelet packet analysis is definitely worth considering. It's a powerful tool that can give you the insights you need to make informed decisions.。

自然科学对话英语作文

自然科学对话英语作文标题，A Dialogue on Natural Science。

自然科学对话。

---。

Introduction。

Natural science plays a crucial role in understanding the world around us. Through observation, experimentation, and analysis, scientists uncover the mysteries of nature and contribute to the advancement of human knowledge. In this dialogue, two students, Alex and Sarah, engage in a conversation about various aspects of natural science, including biology, physics, and environmental science.---。

Alex: Hey, Sarah! Have you ever wondered how plantsconvert sunlight into energy?Sarah: Yeah, it's fascinating! They use a processcalled photosynthesis, where they absorb sunlight through chlorophyll and convert it into glucose and oxygen.Alex: Exactly! And did you know that photosynthesis not only produces food for plants but also generates the oxygen we breathe?Sarah: Wow, nature is truly amazing! Speaking of which, have you heard about the discovery of the Higgs boson particle in physics?Alex: Yeah, it's a groundbreaking achievement! The Higgs boson is often referred to as the "God particle" because it gives mass to other particles through the Higgs field.Sarah: I find quantum physics so intriguing. The idea that particles can exist in multiple states simultaneously challenges our understanding of reality.Alex: Definitely! Quantum mechanics opens up a whole new realm of possibilities and has practical applications in various fields, such as computing and cryptography.Sarah: Switching gears a bit, let's talk about environmental science. Climate change is a pressing issue facing our planet today.Alex: Absolutely. Human activities, such as burning fossil fuels and deforestation, release greenhouse gases into the atmosphere, leading to global warming and unpredictable weather patterns.Sarah: It's alarming to see the impact of climate change on ecosystems and biodiversity. We need to take action to reduce our carbon footprint and protect the environment for future generations.Alex: Agreed. Sustainable practices, renewable energy sources, and conservation efforts are essential steps toward mitigating the effects of climate change andpreserving Earth's natural resources.Sarah: I'm glad to see more people becoming aware of these issues and advocating for positive change. Education and awareness are key to addressing environmental challenges.Alex: Absolutely. By working together and implementing effective solutions, we can create a healthier and more sustainable planet for all living beings.---。

The power of the wave Wave energy harvesting

The power of the wave Wave energyharvestingWave energy harvesting, also known as wave power, is a form of renewable energy that has been gaining attention as a potential solution to the world's growing energy needs. The power of the wave is a promising source of clean and sustainable energy that has the potential to significantly reduce our reliance on fossil fuels and mitigate the impacts of climate change. This form of energy harvesting involves capturing the energy from ocean waves and converting it into electricity, offering a unique opportunity to harness the power of the ocean to meet our energy demands. One of the key advantages of wave energy harvesting is its abundance. The world's oceans are a vast and virtually untapped source of energy, with the potential to generate large amounts of electricity. Unlike other renewable energy sources such as solar or wind power, waves are constant and predictable, making wave energy a reliable and consistent source of energy. This reliability makes it an attractive option for meeting the base load energy demands of communities and industries. Furthermore, wave energy harvesting has the potential to significantly reduce greenhouse gas emissions and mitigate the impacts of climate change. By harnessing the power of the ocean, we can generate electricity without burning fossil fuels, thereby reducing our carbon footprint and contributing to a cleaner and more sustainable future. This is particularly important in the face of the growing threat of climate change, as we urgently need to transition to low-carbon energy sources to limit global temperature rise andits associated impacts. In addition to its environmental benefits, wave energy harvesting also presents economic opportunities. The development and deployment of wave energy technologies can create jobs and stimulate economic growth in coastal communities. As the demand for clean energy continues to rise, investing in wave energy infrastructure can not only provide employment opportunities but also contribute to the development of a new industry that has the potential to drive innovation and technological advancement. Despite its potential, wave energy harvesting also faces challenges and limitations. The technology for capturing and converting wave energy is still in the early stages of development, and there aresignificant technical and engineering hurdles that need to be overcome to make it commercially viable. The harsh marine environment, including the corrosive effects of saltwater and the destructive power of storms, poses significant engineering challenges for the design and maintenance of wave energy devices. Furthermore, the high upfront costs of deploying wave energy technologies and the lack of established regulatory frameworks and incentives for wave energy projects present barriers to widespread adoption. Without the necessary policy support andfinancial incentives, the wave energy industry may struggle to attract the investment needed to scale up and compete with other forms of renewable energy. In conclusion, the power of the wave holds immense potential as a clean and sustainable source of energy. Its abundance, reliability, and environmental benefits make it an attractive option for meeting our energy needs whilemitigating the impacts of climate change. However, the development and deployment of wave energy technologies require concerted efforts to overcome technical, economic, and regulatory challenges. With the right support and investment, wave energy harvesting could play a significant role in shaping a more sustainable energy future.。

The power of the wave Wave energy for irrigation

The power of the wave Wave energy forirrigationWave energy has been gaining attention as a potential renewable energy source for various applications, including irrigation. With the increasing demand for water in agriculture, especially in arid and semi-arid regions, findingsustainable and cost-effective ways to provide water for irrigation is crucial. Wave energy has the potential to address this need by harnessing the power of ocean waves to pump water for irrigation purposes. This innovative approach offers several benefits, but also presents challenges and considerations that need to be carefully evaluated. One of the key advantages of using wave energy forirrigation is its sustainability. Unlike finite resources such as fossil fuels, wave energy is abundant and renewable, making it a reliable long-term solution for powering irrigation systems. By tapping into the natural motion of ocean waves, farmers can access a consistent and clean energy source to pump water for their crops. This not only reduces reliance on non-renewable energy sources but also minimizes the environmental impact associated with traditional irrigation methods. In addition to sustainability, wave energy for irrigation can also contribute to energy independence for agricultural communities. Many farmers rely on grid-connected electricity or diesel generators to power their irrigation systems,which can be costly and prone to supply disruptions. By harnessing wave energy, farmers can become more self-sufficient in meeting their energy needs, reducing their reliance on external energy providers and mitigating the risks associated with power outages or fuel shortages. This can lead to greater resilience and stability for agricultural operations, particularly in remote or off-grid areas. Moreover, the use of wave energy for irrigation has the potential to enhance water efficiency in agriculture. Traditional irrigation techniques, such as flood irrigation or sprinkler systems, can be inefficient in water distribution, leading to wastage and overconsumption. By integrating wave energy-powered pumping systems, farmers can implement more precise and targeted irrigation methods, such as drip irrigation or micro-sprinklers, which deliver water directly to the root zones of plants. This not only conserves water but also promotes optimal growing conditions,potentially improving crop yields and resource utilization. Despite these promising benefits, there are several challenges and considerations that need to be addressed when implementing wave energy for irrigation. One of the primary concerns is the technological and infrastructural requirements for wave energy conversion and distribution. Wave energy converters, which capture the energy from ocean waves and convert it into usable power, need to be robust and reliable in marine environments. Additionally, the integration of wave energy systems with irrigation infrastructure demands careful planning and engineering to ensure compatibility and efficiency. Another consideration is the variability of wave energy, which can pose operational challenges for irrigation systems. Unlike traditional energy sources that can be dispatched on demand, wave energy is dependent on the natural rhythms of the ocean, which can fluctuate in intensity and frequency. This variability requires the development of energy storage and management solutions to ensure a consistent and reliable power supply for irrigation, especially during periods of calm or turbulent seas. Balancing the intermittency of wave energy with the continuous demand for water in agriculture is a complex yet critical aspect of its implementation. Furthermore, the economic viability of wave energy for irrigation is a significant factor that needs to be evaluated. While the potential long-term benefits of sustainable energy and water efficiency are compelling, the initial investment and operational costs of wave energy systems can be substantial. Farmers and agricultural stakeholders may require financial incentives, support mechanisms, or access to funding to adopt wave energy technology for irrigation. Additionally, the regulatory and policy frameworks governing renewable energy and water resource management play a crucial role in facilitating the integration of wave energy into agricultural practices. In conclusion, the use of wave energy for irrigation presents a promising opportunity to address the water and energy challenges facing agriculture. Its sustainability, potential for energy independence, and water efficiency benefits make it a compelling option for powering irrigation systems. However, careful consideration of technological, operational, and economic factors is essential to realize the full potential of wave energy in agriculture. By addressing these challenges and leveraging the opportunities presented by wave energy, farmers canenhance the resilience, sustainability, and productivity of their irrigation practices, contributing to a more secure and sustainable food supply.。

The power of the wave Wave energy harnessing

The power of the wave Wave energyharnessingWave energy harnessing is a promising and innovative technology that has the potential to provide a sustainable source of power for the future. The power of the wave is immense, and harnessing this energy could significantly reduce our dependence on fossil fuels and mitigate the impacts of climate change. However, there are also challenges and considerations that need to be addressed in order to fully realize the potential of wave energy harnessing. One of the key benefits of wave energy is its abundance. Waves are a constant and renewable resource, making them an attractive option for generating electricity. Unlike solar or wind power, which can be intermittent, waves are consistently present in the ocean, providing a reliable source of energy. This reliability makes wave energy a valuable addition to the renewable energy mix, offering a stable and predictable source of power. In addition to its abundance, wave energy is also highly concentrated. The power of the waves can be harnessed in a relatively small area, making it a space-efficient option for energy production. This is particularly important as land availability becomes increasingly limited. By utilizing the vast expanse of the ocean, wave energy has the potential to meet a significant portion of our energy needs without competing for valuable land resources. Furthermore, wave energy has the advantage of being less visually intrusive compared to other forms of renewable energy. While wind turbines and solar panels can be seen for miles, wave energy devices are mostly submerged and out of sight. This can help alleviate concerns about the visual impact of renewable energy infrastructure, making wave energy a more socially acceptable option for coastal communities and beyond. However, despite its potential, there are several challenges that need to be addressed in order to fully harness the power of the wave. One of the main obstacles is the high cost of developing and deploying wave energy technology. The harsh marine environment and the need for durable, corrosion-resistant materials make wave energy devices expensive to manufacture and maintain. Additionally, the installation and maintenance of these devices in the ocean can be complex and costly, further adding to the overall expenses. Another challenge is thepotential environmental impact of wave energy devices. While wave energy is a clean and renewable source of power, the installation and operation of wave energy devices can have ecological consequences. These devices may disrupt marine ecosystems, interfere with migratory patterns of marine animals, and pose a risk of entanglement for marine life. It is crucial to carefully assess and mitigate these potential impacts in order to ensure that wave energy remains a sustainable and environmentally friendly option. Furthermore, the variability of wave energy presents a significant technical challenge. Waves are influenced by a multitude of factors, including wind, tides, and ocean currents, making them inherently unpredictable. This variability can pose challenges for integrating wave energy into the existing power grid, as it requires sophisticated energy storage and grid management systems to ensure a consistent and reliable power supply. Despite these challenges, the potential of wave energy harnessing cannot be overlooked. With ongoing advancements in technology and increasing global commitment to renewable energy, there is a growing opportunity to overcome these obstacles and fully realize the potential of wave energy. Research and development efforts are focused on improving the efficiency and reliability of wave energy devices, as well as minimizing their environmental impact. Additionally, policy support and financial incentives can help drive the growth of the wave energy industry, making it a more economically viable option for energy production. In conclusion, the power of the wave holds great promise for the future of renewable energy. Wave energy harnessing offers a reliable, space-efficient, and visually unobtrusive source of power that can significantly contribute to our transition towards a more sustainable energy future. While there are challenges to overcome, the potential benefits of wave energy make it a valuable investment in the pursuit of clean and renewable energy sources. With continued innovation and support, wave energy has the potential to play a significant role in meeting our energy needs while mitigating the impacts of climate change.。

节能发电调度模式下火力发电单元的月度电能计划编制方法

第39卷第4期电力系统保护与控制Vol.39 No.4 2011年2月16日Power System Protection and Control Feb.16, 2011 节能发电调度模式下火力发电单元的月度电能计划编制方法张慧琦1，常永吉2，唐大勇1，王松岩2，陈骞1，李志刚1，王玉奇1，于继来2（1.黑龙江省电力有限公司，黑龙江哈尔滨 150090；2.哈尔滨工业大学，黑龙江哈尔滨 150001）摘要：编制月度电能计划的传统方法主要基于平均分配发电量及利用小时数的方式，相对简单和粗略，与新形势下实施节能减排发电调度不相适应。

提出了一种节能发电调度模式下制定火力发电单元月度电能计划的方法。

该方法在给出节能发电调度序位表和组合优先级方案的同时，即可直接获得发电单元月度电能子空间。

计划制定过程除考虑机组装机容量、检修、负荷系数等基本信息外，还侧重计及了火电机组的能耗、排放指标信息。

算例表明，由新方法制定的月度电能计划，具有节约燃煤、减少污染物和CO2排放的效益，但也可能会引起电网公司购电成本的上升。

关键词：电力网络；火力发电单元；节能；减排；月度电能计划；发电调度模式A monthly electric energy plan making method of thermal power generation unit in energy-savinggeneration dispatching modeZHANG Hui-qi1，CHANG Yong-ji2，TANG Da-yong1，WANG Song-yan2，CHEN Qian1，LI Zhi-gang1，WANG Yu-qi1，YU Ji-lai2（1. Heilongjiang Electric Power Company Limited，Harbin 150090，China；2. Harbin Institute of Technology，Harbin 150001，China）Abstract：The traditional method of making monthly electric energy plan is mainly based on the average distribution of generating capacity and using hours, which is relatively simple and crude，and is incompatible with the energy saving and emission reduction dispatching mode of generation units．A method for energy-saving generation dispatching mode is proposed for making monthly electric energy plan of thermal power generation unit．The method gives generation dispatching sequence table and programs about combination priority，and at the same time can obtain the monthly electric energy subspace of each generating unit．The planning process considers the unit capacity，maintenance plan，load factor and other basic information，and also focuses on considering the thermal power unit energy consumption and emission index information．Example shows that the monthly electric energy plan made by the new method has the benefits of saving coal and reducing emissions of pollutants and CO2，but may also cause rise of purchasing electricity costs of grid company．This work is supported by National Natural Science Foundation of China(No. 50877014).Key words：electric power network；thermal power generation unit； energy saving； emission reduction；monthly electric energy plan；generation dispatching mode中图分类号： TM73 文献标识码：A 文章编号： 1674-3415(2011)04-0084-060 引言建设“资源节约型、环境友好型”[1]社会的总体战略，迫切要求对现行发电调度方式进行改革。

Unit2EnergyinTransition（补充汉译英）

Unit2EnergyinTransition（补充汉译英）Unit 2 Energy in Transition ( 补充汉译英 )1.汉普顿-悉尼学院以其诚信制度与其军事化管理体系一样儿享有盛名。

而且此诚信制度扩展到学生在校内和校外的所有活动中。

并且认为对违规行为的包容本身就是一种违规行为。

( on a par with )Hampden-Sydney College is reputed for an honor system on a par with military systems, and this honor system extends to all student activities both on and off campus, and considers tolerance of a violation itself a violation.2.虽然全球变暖对地球构成威胁，但是人类或许可以通过提高大气层中二氧化碳含量（值）来缓和其所导致的气候威胁。

( pose a threat on sth/sb. )Although global warming poses a threat to the earth, humans can probably ease the climate threat brought on by rising levels of carbon dioxide in the atmosphere.3.对于厄尔尼诺潜在的破坏性人们已了解许多，但其现象本身却仍是令人沮丧的费解之谜。

( enough is known about sth )Enough is known about Elnino’s destructive potential, but the phenomenon itself remains a frustrating mystery.4.中国就生态和环保已形成全社会共识并正在率先行动起来。

The power of the wave Wave energy for desalination

The power of the wave Wave energy fordesalinationWave energy has been gaining attention as a potential renewable energy source for desalination. Desalination, the process of removing salt and other impurities from seawater to make it suitable for human consumption or agricultural use, is a critical need in many parts of the world facing water scarcity. Wave energy, harnessed from the motion of ocean waves, has the potential to power desalination plants in a sustainable and environmentally friendly manner. This essay will explore the power of wave energy for desalination from various perspectives, including its benefits, challenges, and potential impact on communities and the environment. From a technological standpoint, wave energy presents an exciting opportunity for desalination. Unlike traditional desalination methods that rely on fossil fuels or electricity from the grid, wave energy offers a clean and abundant source of power. Wave energy converters, which capture the kinetic and potential energy of ocean waves, can be integrated with desalination systems to provide a continuous and reliable energy supply. This integration not only reduces thecarbon footprint of desalination plants but also increases their energy resilience, making them less vulnerable to power outages and fluctuations. As a result, wave-powered desalination has the potential to provide a sustainable and secure source of fresh water for coastal communities around the world. In addition to its technological promise, wave energy for desalination also holds socioeconomic benefits. Many regions that could benefit from desalination, such as arid coastal areas or small island nations, often face economic challenges and lack access to reliable electricity. By harnessing wave energy for desalination, these communities can address their water scarcity while also creating local employment opportunities and stimulating economic development. Furthermore, the decentralized nature of wave energy allows for small-scale desalination plants to be established, providing water security for remote and marginalized communities. This can have a transformative impact on the livelihoods and well-being of the people in these areas, empowering them with access to a fundamental resource for life. Despitethe potential of wave energy for desalination, there are also significantchallenges that need to be addressed. One of the primary challenges is the technical complexity of wave energy converters and their integration with desalination systems. Developing efficient and cost-effective technologies that can withstand the harsh marine environment and fluctuating wave conditions is a formidable task. Moreover, the variability of wave energy poses challenges for ensuring a consistent and stable power supply for desalination plants. Research and development efforts are essential to overcome these challenges and optimize the performance of wave-powered desalination systems. Another critical consideration is the environmental impact of wave energy for desalination. While wave energy is renewable and emits no greenhouse gases during operation, the deployment of wave energy converters and desalination plants can have localized environmental effects. These may include disturbances to marine ecosystems, alteration of coastal sediment transport, and potential conflicts with other ocean uses such as fishing and shipping. It is crucial to conduct thorough environmental assessments and engage with stakeholders to ensure that wave-powered desalination projects are implemented in a sustainable and responsible manner. This requires a holistic approach that considers the ecological, social, and cultural dimensions of the marine environment. In conclusion, the power of wave energy for desalination offers a compelling solution to the pressing global challenges of water scarcity and energy sustainability. From technological innovation to socioeconomic empowerment, wave-powered desalination holds great promise for providing clean water to communities in need while mitigating the environmental impact of traditional desalination methods. However, realizing this potential requires concerted efforts to overcome technical, economic, and environmental barriers. By embracing a multidisciplinary approach that integrates engineering, economics, and environmental stewardship, we can harness the power of the wave to create a more water-secure and resilient future for coastal communities and the planet.。

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a r X i v :c o n d -m a t /0205134v 1 [c o n d -m a t .m t r l -s c i ] 7 M a y 2002Energy bands and Wannier-Mott excitons in Zn(P 1−x As x )2and Zn 1−x Cd x P 2crystalsO.A.Yeshchenko,M.M.Biliy,Z.Z.YanchukPhysics Department,Taras Shevchenko National Kyiv University6Akademik Glushkov prosp.,03127Kyiv,UkraineE-mail:yes@mail.univ.kiev.uaAbstractExcitonic absorption,reﬂection and photoluminescence spectra of mixed Zn(P 1−x As x )2crystals over the full range of x (0≤x ≤1)and Zn 1−x Cd x P 2crystals at 0≤x ≤0.05have been studied at low temperatures (1.8K).The decrease of the energy gap in Zn(P 1−x As x )2at the increase of x occurs slightly sublinearly.The rydbergs of excitonic series in this crystals decrease as well,and the dependences Ry (x )for all series are strongly superlinear at small x .In Zn 1−x Cd x P 2crystals the energy gap and rydbergs decrease at the increase of x (at 0≤x ≤0.05)as well.The dependences of E g and Ry on x are considerably stronger in Zn(P 1−x As x )2than in Zn 1−x Cd x P 2.At the increase of x the half-width of excitonic absorption lines increases monotonically in both type crystals that is evidence of the increasing role of ﬂuctuations of crystal potential.1IntroductionIn Ref.[1]we performed the spectroscopic studies of an inﬂuence of substitution of phosphorus by arsenic on the structure of valence and conduction bands,and on the parameters of Wannier-Mott excitons in mixed crystals (solid solutions)of isovalent substitution Zn(P 1−x As x )2at low levels of substitution x ≤0.05.In present work we have proceeded with these studies at 0≤x ≤1.By the other hand we have performed here the similar study (in comparison to Zn(P 1−x As x )2)of another type mixed crystals of isovalent substitution Zn 1−x Cd x P 2at small substitution levels x ≤0.05.As we know,the Zn 1−x Cd x P 2crystals were not studied earlier.Both Zn(P 1−x As x )2and Zn 1−x Cd x P 2belong to the mixed crystals of A II B V type,which are poorly investigated.β-ZnP 2(further ZnP 2)and ZnAs 2crystals are strongly anisotropic direct-gap semiconductors (energygap:1.6026eV for ZnP 2and 1.052eV for ZnAs 2),which are characterized by the same symmetry group C 52h (monoclinic syngony).Besides the symmetry of the lattice,the similarity of ZnP 2and ZnAs 2exists in the structure of energy bands and exciton states,namely,three excitonic series are observed in the absorption spectra of these crystals:dipole allowed C-series at E Z (c )polarization originating from S -states of C-exciton (this series is observed in reﬂection spectra as well),forbidden B-series at E ⊥Z (c )polarization originating from S -states of B-exciton,and partially allowed A-series at E X polarization originating from S -states of A-exciton (see e.g.Refs.[2,3,4]for ZnP 2and Refs.[5,6,7]for ZnAs 2).In the photoluminescence (PL)spectra of these crystals at E Z (c )polarization,a series of lines caused by the radiative transitions from the ground and excited states of allowed C-exciton is observed (see e.g.Ref.[8]for ZnP 2and Ref.[5]for ZnAs 2).Besides this emission series,the so-called B-line is observed in the PL spectra of ZnP 2at E ⊥Z (c ).B-line occurs due to the radiative transitions from the ground state of forbidden B-exciton and corresponds to B 1-line of absorption B-series.In contrast to ZnP 2and ZnAs 2crystals,CdP 2is indirect-gap semiconductor (energy gap:2.155eV:see,e.g.[9]),which is characterized by the diﬀerent symmetry of lattice:symmetry group is D 44for right-rotatingand D 84for left-rotating modiﬁcation (tetragonal syngony).Therefore,the comparison study of an inﬂuence of substitution of P by As in Zn(P 1−x As x )2and of Zn by Cd in Zn 1−x Cd x P 2on parameters of energy bands and excitonic states seems to be rather interesting.The technological operations of growing of Zn(P 1−x As x )2and Zn 1−x Cd x P 2crystals were carried out according to described in Ref.[1].2Zn(P 1−x As x )2crystals over the full range of xIn the present work the low-temperature (1.8K)absorption,reﬂection and photoluminescence spectra of Zn(P 1−x As x )2crystals have been studied at the following levels of substitution of P by As :x =0.01,0.02,0.03,0.05,0.10,0.125,0.40,0.90,0.95.Respective spectra as well as spectra of pure ZnP 2(x =0)and ZnAs 2(x =1)are presented in Fig.1.As one can expect,the Zn(P 1−x As x )2crystals are direct-gap semiconductorsB-and A-series are observed,as in pure crystals.Let us note the doublet structure of an absorption n=1 line of B-series in crystals with x=0.02.Proceeding from intensities and half-widths of the components of this doublet,we have made a conclusion,that narrow high-energy component is n=1line of B-series. An origin of low-energy component,which is missing for crystals with x=0.02,is not clear.One can see from absorption and reﬂection spectra of Zn(P1−x As x)2crystals that at the increase of x(or at the increase of1−x if to go from the ZnAs2side)1the lines corresponding to exciton states with higher n disappear, and the lines with n=1,2broaden.Probably,this fact is a result of”blurring”of the band edges,which takes place owing toﬂuctuations of crystal potential,caused by chaotic distribution of As(P)atoms in sites of lattice at substitution of P(As)atoms.In PL spectra the emission lines of free excitons can be easily separated from the lines of localized excitons(including,most likely,emission of excitons autholocalized on ﬂuctuations of crystal potential)only at rather low substitution levels.At the increase of x the emission lines of both free and localized excitons broaden that is due to theﬂuctuations of crystal potential and correspondingﬂuctuations of the parameters of energy bands.It takes an attention on itself the fact of considerable increase of intensity of PL spectra at the increase of x.Most probably,this eﬀect is due to two following causes.First one consists in the partial suppression of the spatial migration of excitons(due to localization of excitons onﬂuctuations of crystal potential)and the respective decrease of eﬃciency of non-radiative decay of excitons.Second one consists in the fact that the processes of scattering of excitons on the lattice defects favour to more eﬀective radiative decay of excitons.With increase in concentration x,spectral lines shift to the low-energy side,which is caused by the decrease of energy gap.This shift could be easily expected,taking into account the fact,that in ZnAs2 energy gap is0.55eV smaller than in ZnP2.The respective dependence E g(x)is given in Fig.2(a).The dependence is slightly nonlinear(sublinear).It has beenﬁtted by the well-known dependence(see,e.g.Ref.[10])E g(x)=E g1−(E g1−E g2)x+cx(1−x),(1) where E g1and E g2are the energy gaps of ZnP2and ZnAs2respectively,and c is the coeﬃcient of nonlinearity. The coeﬃcient c was obtained to be0.05.But besides the trivial decrease of E g with the increase of x,there is also decrease of the excitonic series rydbergs(see Fig.2(b)).Values of E g and rydbergs were obtained fromﬁtting of excitonic series by simple hydrogenlike dependence:E(n)=E g−Ry/n2.One can see from theﬁgure that the dependences of excitonic rydbergs on x are rather remarkable.These dependences are strongly superlinear at small x(close to ZnP2)and most linear at x→1(close to ZnAs2).Strongest superlinearity takes place at x≤0.05.Let us note that the rydbergs of B-and A-series decrease considerably: at crossing from ZnP2to ZnAs2the rydbergs decrease more than in3times.Meanwhile,the rydberg of A-series decreases suﬃciently less:it decreases in1.4times.Let us note that in ZnP2the n=1state of A-exciton is energy highest,and the n=1state of B-exciton is energy lowest.But,due to the slower decrease of Ry A(x)than ones observed for B-and A-series,at substitution levels x∼0.70the n=1state of A-exciton leaves oﬀto be the energy highest.And in extreme case of ZnAs2an opposite situation takes place: A1-state is energy lowest.The n=1state of allowed C-exciton is energy highest in ZnAs2crystal.There are exist two possible causes of such a fast decrease of excitonic rydbergs at small x.First one is the decrease of exciton reduced mass at the increase of x.This our assumption is based on the data of the exciton reduced masses in ZnP2(µa=0.45m0,µc=0.10m0andµb=0.56m0[11])and ZnAs2(µ⊥bc=0.30m0[5,7])as well as on our previous evaluation of the dependences of m e(x)and m h(x)at small x[1].Here,µa,b,c is the components of the reduced mass of exciton in ZnP2in diﬀerent crystallographic directions,andµ⊥bc is the component ofµin the direction perpendicular to plane(100)(this direction almost coincides with a axis of crystal).Second cause of the fast decrease of rydbergs at small x is the possible fast increase of dielectric constant,as in ZnAs2ǫis suﬃciently higher:average value15[5]versusǫa=9.1,ǫc=9.3andǫb=10.1 in ZnP2.Slower decrease of excitonic rydbergs at higher x can be most likely explained by possible slower decrease of the reduced exciton mass and increase of the dielectric constant at higher x.There were also studied dependences on x of the half-widths of absorption n=1lines of B-and A-series.These dependences are given in Fig.3.2One can seen,that half-widths of B1-and A1-lines increase monotonously with the increase of x.As known,the increase of half-width of exciton lines is the result ofﬂuctuations of crystal potential and respectiveﬂuctuations of energy gap.The theory of inﬂuence of ﬂuctuations of composition x on half-width of exciton absorption lines was developed in Ref.[12],where two extreme cases were considered.First one takes place,if the eﬀective size of area of the crystal potential ﬂuctuation R D=¯h/(2MD)1/2,where M is the total mass of exciton and D(x)=W(x)−W(0)(W(x)isthe half-width of exciton line),is much larger than exciton Bohr radius:R D >>a ex .Such situation,as a rule,takes place,if the eﬀective masses of electron and hole are small and diﬀer slightly:m e ∼m h .In this case D should depend on x asD (x )=0.08α4M 3x 2(1−x )2Na 3ex1/2.(3)The experimental dependences D (x )for absorption B 1-and A 1-lines of Zn(P 1−x As x )2crystals are presentedin Fig. 3.Likely to data presented in our work [1]the experimental points are badly ﬁtted both by function (2)and (3).In ZnP 2and ZnAs 2crystals the eﬀective size of area of crystal potential ﬂuctuation is about some tens of angstrom,and the Bohr radii of B-and A-excitons in these crystals are the following:a B =16˚A ,a A =29.5˚A in ZnP 2,and a B =34˚A ,a A =25.7˚A in ZnAs 2,i.e.R D ∼a ex .Therefore,the extreme conditions R D >>a ex and R D <<a ex are not fulﬁlled,and the intermediate case takes place which is,nevertheless,more close to case of Eq.(2).Therefore,since the intermediate case takes place,the experimental dependences can be ﬁtted by the functionD (x )=(1−c )D 1(x )+cD 2(x ),(4)which is the superposition of function D 1(x )of type (2)and D 2(x )of type (3),c is the weighting factor.One can see from Fig.3that for the mixed crystals close to ZnP 2,i.e.at x →0,the experimental dependence of the half-width of B 1-line on x is ﬁtted by function (2)rather well,and the contribution of the function (3)is rather small (c =0.05).The situation is quite diﬀerent for the mixed crystals close to ZnAs 2,i.e.at x →1,as the contribution of the function (3)is considerably larger (c =0.20).It is rather simple to understand,as the Bohr radius of B-exciton in ZnAs 2is about two times larger than in ZnP 2.So,in ZnP 2the condition R D >>a ex is fulﬁlled better than in ZnAs 2.For the dependence of the half-width of A 1-lines on x we have the quite opposite situation.For the mixed crystals close to ZnP 2,the experimental dependence of the half-width of A 1-line on x deviates considerably from function (2),and the contribution of the function (3)is rather large (c =0.29).For the mixed crystals close to ZnAs 2,the contribution of the function (3)to the ﬁtting function is quite small (c =0.08).The Bohr radius of A-exciton,as we have noted above,almost does not diﬀer in ZnP 2and ZnAs 2.So,as the contribution of function (3)at x →1is smaller than at x →0,the condition R D >>a ex is fulﬁlled better for A-exciton in the crystals close to ZnAs 2.Thus,we can make a conclusion of larger eﬀective size of area of the crystal potential ﬂuctuation in crystals close to ZnAs 2,than in ones close to ZnP 2.3Zn(P 1−x As x )2and Zn 1−x Cd x P 2at small x :comparisonThe comparison studies of the low-temperature absorption,reﬂection and photoluminescence spectra of Zn(P 1−x As x )2and Zn 1−x Cd x P 2crystals have been performed at small levels of substitution of Zn by Cd :x ≤0.05.Respective spectra of Zn(P 1−x As x )2are presented in Fig.1,and the spectra of Zn 1−x Cd x P 2–in Fig. 4.In spite of the diﬀerence of the symmetries of lattice and structure of energy bands of ZnP 2and CdP 2crystals (monoclinic and tetragonal,direct-and indirect-gap respectively:see Sect.1),at small x Zn 1−x Cd x P 2remain the direct-gap crystals with monoclinic lattice.One can see from the ﬁgure,that in the Zn 1−x Cd x P 2crystals the same excitonic C-,B-and A-series are observed,as well as in pure ZnP 2and mixed Zn(P 1−x As x )2crystals.One can see from absorption and reﬂection spectra that,likely to Zn(P 1−x As x )2,already at rather small x only the excitonic states with n =1,2are observed clearly in the spectra of Zn 1−x Cd x P 2crystals.The higher components of exciton spectra disappear,and the lines with n =1,2broaden.In PL spectra,at the increase of x the emission lines of both free and localized excitons broaden.As noted above (see Sect.2),all these eﬀects are due to the ﬂuctuations of crystal potential and corresponding ﬂuctuations of the parameters of energy bands.Likely to Zn(P 1−x As x )2,an eﬀect of considerable increase of intensity of PL spectra at the increase of x takes place in Zn 1−x Cd x P 2crystals too.An explanation of such an eﬀect is given as well in Sect.2.In spite of the fact that the CdP 2crystal has larger energy gap than the ZnP 2,likely to Zn(P 1−x As x )2there is a decrease of the energy gap of Zn 1−x Cd x P 2crystals at the increase of x (see Fig.5(a)).But theZn(P1−x As x)2and Zn1−x Cd x P2crystals have the similarity in the dependences of the rydbergs of excitons on x as well(see Fig.5(b)).The rydbergs of all the three A-,B-,and C-series decrease at the increase of x.But,also likely to the dependence E g(x),the excitonic rydbergs in Zn1−x Cd x P2decrease considerably slower than in Zn(P1−x As x)2.Considerably weaker dependences of the energy gap and the rydbergs on xin Zn1−x Cd x P2are,most probably,the results of the weaker changes in parameters of the energy bands,in particular in eﬀective masses of electrons and holes,and in dielectric constant.But,at higher x one can expect the more interesting and stronger changes in the parameters of energy bands and excitonic states with regard to the fact that CdP2and ZnP2have the diﬀerent symmetry of lattice,and that CdP2is indirect-gap semiconductor.The dependences of half-widths of absorption B1-and A1-lines on x have been also studied in Zn1−x Cd x P2 crystals.The results of comparison of such dependences with ones observed for Zn(P1−x As x)2are presentedin Fig.6.One can see that D(x)dependences for both types of the mixed crystals are similar for excitonB1-line.This dependences are described rather well by the function of Eq.(2)type,contributions of the function(3)are quite small for both type crystals:the weighting factor is c=0.05for Zn(P1−x As x)2andc=0.035for Zn1−x Cd x P2.It is an evidence that in both cases the eﬀective size of area of the crystal potentialﬂuctuation R D is rather large comparing to the exciton Bohr radius a ex.Likely to Zn(P1−x As x)2,in Zn1−x Cd x P2the dependence of half-width of A1-line on x deviates strongly from the function(2),the contribution of the function(3)is large.As we assumed above,it is the result of the fact that A-exciton has considerably lower binding energy comparing to B-exciton,and larger Bohr radius correspondingly.So,the condition R D>>a ex is not fulﬁlled,and the dependence D(x)is not described by the function(2).At the same time,there are two remarkable features of D(x)dependence for A1-line in Zn1−x Cd x P2.First one is the considerable deviations of the experimental points from theﬁtting function.It is quite remarkable as such deviations are very small for the D(x)dependences for A1-line in Zn(P1−x As x)2and for B1-line in the mixed crystals of both types.Second one is the fact that the contribution of the function(3)is determinative in theﬁtting function:the weighting factor of the function(3)is c=0.89.For Zn(P1−x As x)2the contributionof the function(3)in theﬁtting function is considerably smaller:c=0.29.At present moment,both above features of D(x)dependence for A1-line in Zn1−x Cd x P2are obscure.Thus,a comparison of the energy bands and exciton parameters behaviour versus x reveals as similar tendencies and diﬀerent ones.4ConclusionsIn conclusion,we have obtained the following results.We have found that,likely to”pure”ZnP2crystal, both in Zn(P1−x As x)2and Zn1−x Cd x P2mixed crystals the same three excitonic hydrogenlike series(C,B, and A)are observed.At the increase of x in the range of small x(0≤x≤0.05)the decrease of the energy gap and exciton rydbergs takes place.The dependences of E g and Ry on x are considerably stronger in Zn(P1−x As x)2than in Zn1−x Cd x P2.Besides the comparison study of these crystals at small x,the Zn(P1−x As x)2crystals have been studied over the full range of x:0≤x≤1.We have obtained the following results for Zn(P1−x As x)2.At the increaseof x the energy gap decreases slightly sublinearly.The exciton rydbergs decrease as well.The dependences Ry(x)are strongly superlinear at small x(close to ZnP2)and most linear at x→1(close to ZnAs2).At the crossing from ZnP2to ZnAs2,the rydbergs of B-and C-series decrease more than in3times.Meanwhile, the rydberg of A-series decreases suﬃciently less:in1.4times.At the increase of x the half-width of excitonic absorption lines increases monotonically both in Zn(P1−x As x)2 and Zn1−x Cd x P2crystals that is evidence of the increasing role ofﬂuctuations of crystal potential. References[1]O.A.Yeshchenko,M.M.Biliy,Z.Z.Yanchuk,Physica B307,(2001)231.[2]I.S.Gorban,M.M.Biliy,I.M.Dmitruk,O.A.Yeshchenko,Phys.Stat.Sol.(B)207,(1998)171.[3]A.B.Pevtsov,S.A.Permogorov,A.V.Selkin,N.N.Syrbu,A.G.Umanets,Fiz.Tekhn.Poluprovodn.16, (1982)642.[4]I.S.Gorban,M.M.Biliy,V.A.Borbat,V.O.Gubanov,I.M.Dmitruk,Z.Z.Yanchuk,Dopovidi AN URSR: Series A No.4,(1988)48.[5]A.V.Mudryi,V.M.Trukhan,A.I.Patuk,I.A.Shakin,S.F.Marenkin,Fiz.Tekhn.Poluprovodn.31, (1997)1029.[6]M.M.Biliy,I.S.Gorban,I.M.Dmitruk,O.A.Yeshchenko,Z.Z.Yanchuk,Ukr.Fiz.Zhurn.44,(1999)863.[7]V.A.Morozova,V.S.Vavilov,S.F.Marenkin,O.G.Koshelev,M.V.Chukichev,erd.Tela40,(1998), 877.[8]I.S.Gorban,M.M.Biliy,I.M.Dmitruk,O.A.Yeshchenko,Phys.Stat.Sol.(B)191,(1995)337.[9]I.S.Gorban,V.A.Gubanov,M.V.Chukichev,Z.Z.Yanchuk,Fiz.Tekhn.Poluprovodn.19,(1985)1312.[10]A.N.Pikhtin,Fiz.Tekhn.Poluprovodn.11,(1977)425.[11]J.Engbring,D.Fr¨o lich,R.Schepe,S.Spitzer,O.Arimoto,K.Nakamura,Phys.Stat.Sol.(B)196, (1996)461.[12]N.N.Ablyazov,M.E.Raikh,A.L.Efros,erd.Tela25,(1983)353.1.501.531.56 1.591.501.531.56 1.591.441.471.501.531.56Photon energy (eV)Photon energy (eV)Photon energy (eV)A b s o r p t i o n k d (a r b . u n .)Photon energy (eV)Photon energy (eV)Photon energy (eV)Figure 1:Optical spectra of Zn(P 1−x As x )2crystals at temperature 1.8K.(a)Absorption spectra.Obser-vation conditions:q ⊥(110),E ⊥Z (c )-for crystals with any x except x =0.05;q ⊥(100),E ⊥Z (c )-for crystals with x =0.05.(b)Reﬂection spectra.Observation conditions:q ⊥(100),E Z (c ).(c)Photoluminescence spectra.Observation conditions:q ⊥(100).E n e r g y g a p (e V )xxR y d b e r g s (m e V )Figure 2:(a)Dependence of energy gap on x for Zn(P 1−x As x )2crystals.Solid line represents ﬁtting of the experimental points by expression (1),dotted one is the line connecting points of two extreme cases:ZnP 2(x =0)and ZnAs 2(x =1).(b)Dependences of excitonic rydbergs on x for Zn(P 1−x As x )2.Connected open squares -experimental dependence for B-series,connected solid up triangles -the same for C-series,connected open diamonds -the same for A-series.Dashed line connects two extreme cases (mentioned above in this caption)for B-series,dotted one -for C-series,dashed-dotted one -for A-series.xD (x ) (m e V )Figure 3:Dependences of half-widths of excitonic absorption B 1-and A 1-lines in Zn(P 1−x As x )2crystalson x .Open circles and squares -experimental dependences for B 1-and A 1-lines in crystals close to ZnP 2(x ≤0.125:bottom x-axis);solid circles and squares -the same for B 1-and A 1-lines in crystals close to ZnAs 2(x ≥0.90:top x-axis).Solid lines represent the ﬁtting of experimental points by expression (4)for crystals with x ≤0.125;dashed lines -the same for crystals with x ≥0.90.P L I n t e n s i t y (a r b . u n .)Photon energy (eV)Photon energy (eV)Photon energy (eV)Figure 4:Optical spectra of Zn 1−x Cd x P 2crystals at temperature 1.8K.(a)Absorption spectra.Observation conditions:q ⊥(110),E ⊥Z (c ).(b)Reﬂection spectra.Observation conditions:q ⊥(100),E Z (c ).(c)Photoluminescence spectra.Observation conditions:q ⊥(100).xE n e r g y g a p (eV )xR y d b e r g s (m e V )Figure 5:(a)Dependences of energy gap on x for Zn(P 1−x As x )2(connected open circles)and Zn 1−x Cd x P 2crystals (connected open squares).(b)Dependences of excitonic rydbergs on x for Zn(P 1−x As x )2and Zn 1−x Cd x P 2.Connected open circles,squares,and up triangles -the dependences Ry (x )for B-,C-,and A-series of Zn(P 1−x As x )2crystals respectively;connected solid circles,squares,and up triangles -the de-pendences Ry (x )for B-,C-,and A-series of Zn 1−x Cd x P 2crystals respectively.D (x ) (m e V )xFigure 6:Dependences of half-widths of excitonic absorption B 1-and A 1-lines on x :open circles and squares respectively –in Zn(P 1−x As x )2,and solid circles and squares respectively –in Zn 1−x Cd x P 2crystals.Solid lines represent the ﬁtting of experimental points by expression (4)for Zn(P 1−x As x )2crystals;dashed lines -the same for Zn 1−x Cd x P 2crystals.11。