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1、Effect of Quantum Confinement on Electrons and Phonons in SemiconductorsWe have studied the Gunn effect as an example of negative differential resistance(NDR).This effect is observed in semiconductors,such as GaAs,whose conduction band structure satisfies a special condition,namely,the existence of

2、higher conduction minima separated from the band edge by about 0.2-0.4eV.As a way of achieving this condition in any semiconductor,Esaki and Tsu proposed in 1970 9.1the fabrication of an artificial periodic structure consisting of alternate layers of two dissimilar semiconductors with layer superlat

3、tice.They suggested that the artificial periodicity would fold the Brillouin zone into smaller Brillouin zones or “mini-zones”and therefore create higher conduction band minima with the requisite energies for Gunn oscillations. iWith the development of sophisticated growth techniques such as molecul

4、ar beam epitaxy（MBE）and metal-organic chemical vapor deposition(MOCVD)discussed in Sect.1.2,it is now possible to fabricate the superlattices(to be abbreviated as SLs)envisioned by Esaki and Tsu9.1.In fact,many other kinds of nanometer scale semiconductor structures(often abbreviated as nanostructur

5、es)have since been grown besides the SLs.A SL is only one example of a planar or two-dimensional nanostructure .Another example is the quantum well (often shortened to QW).These terms were introduced of this chapter is to study the electronic and vibrational properties of these two-dimensional nanos

6、tructures.Structures with even lower dimension than two have also been fabricated successfully and studied. For example,one-dimensional nanostructures are referred to as quantum wires.In the same spirit,nanometer-size crystallites are known as quantum dots.There are so many different kinds of nanost

7、ructures and ways to fabricate them that it is impossible to review them all in this introductory book. In some nanostructures strain may be introduced as a result of lattice mismatch between a substrate and its overlayer,giving rise to a so-called strained-layer superlattice.In this chapter we shal

8、l consider only the best-study nanostructures.Our purpose is to introduce readers to this fast growing field.One reason why nanostructures are of great interest is that their electronic and vibrational properties are modified as a result of their lower dimensions and symmetries.Thus nanostructures p

9、rovide an excellent opportunity for applying the knowledge gained in the previous chapters to understand these new developments in the field of semiconductors physics. Due to limitations of space we shall consider in this chapter only the effects of spatial confinement on the electronic and vibratio

10、nal properties of nanostructures and some related changers in their optical and transport properties.Our main emphasis will be on QWs,since at present they can be fabricated with much higher degrees of precision and perfection than all other structures.We shall start by defining the concept of quant

11、um confinement and discuss its effect on the electrons and phonons in a crystal.This will be followed by a discussion of the interaction between confined electrons and phonons.Finally we shall conclude with a study of a device(known as a resonant tunneling device)based on confined electrons and the

12、quantum Hall effect(QHE)in a two-dimensional electron gas.The latter phenomenon was discovered by Klaus von Klitzing and coworkers in 1980 and its significance marked by the award of the 1985 Nobel Prize in physics to Klitzing for this discovery.Together with the fractional quantum Hall effect it is

13、 probably the most important development in semiconductor physics within the last two decades.Quantum Confinement and Density of StatesIn this book we have so far studied the properties of electrons ,phonons and excitons in either an infinite crystal or one with a periodic boundary condition(the cas

14、es of surface and interface states )In the absence of defects, these particles or excitations are described in terms of Bloch waves,which can propagate freely throughout the crystal.Suppose the crystal is finite and there are now two infinite barriers,separated by a distance L,which can reflect the

15、Bloch waves along the z direction.These waves are then said to be spatially confined.A classical example of waves confined in one dimension by two impenetrable barriers is a vibrating string held fixed at two ends.It is well-known that the normal vibration modes of this string are standing waves who

16、se wavelength takes on the discrete values given by Another classical example is a Fabry-Perot interferometer (which has been mentioned already in Set.7.2.6 in connection with Brillouin scattering).As a result of multiple reflections at the two end mirrors forming the cavity，electromagnetic waves sh

17、ow maxima and minima in transmission through the interferometer at discrete wavelengths.If the space inside the cavity is filled with air,the condition for constructive interference is given by (9.1).At a transmission minimum the wave can be considered as “confined”inside the interferometer.=2L/n, n

18、=1,2,3 .(9.1) For a free particle with effective mass confined in a crystal by impenetrable barriers(i.e.,infinite potential energy)in the z direction,the allowed wavevectors of the Bloch waves are given by =2/=n/L, n=1,2,3 (9.2)And its ground state energy is increased by the amount E relative to th

19、e unconfined case: (9.3)This increase in energy is referred to as the confinement energy of the particle.It is a consequence of the uncertainty principle in quantum mechanics. When the particle is confined within a distance L in space(along the z direction in this case)the uncertainty in the z compo

20、nent of its momentum increases by an amount of the order of /L.The corresponding increase in the particles kinetic energy is then given by(9.3).Hence this effect is known also as quantum confinement.In addition to increasing the minimum energy of the particle,confinement also causes its excited stat

21、e energies to become quantized.We shall show later that for an infinite one-dimensional”square well”potential the excited state energies are given by n ,where n=1,2,3as in (9.2).It is important to make a distinction between confinement by barriers and localization via scattering with imperfections。F

22、ree carriers in semiconductors are scattered by phonons and defects within an average scattering time introduced in Sect,5.2.We can define their mean free pathas the product of their average velocity and .Such scattering can also decrease the uncertainty in a particles position and hence increase it

23、s momentum uncertainty.This results in an uncertainty in its energy of an amount given by (9.3)with .This effect is typically associated with defects or disorder in solids and is not the same as the quantum confinement effects of interest in this chapter,One way to distinguish between these two case

24、s is to examine the wavevector along the confinement direction.The wavevector of particle confined in a quantum well,without scattering,is discrete as it corresponds to a standing wave, and is given by (9.2).Scattering at defects dephases a wave so that its amplitude decays exponentially within the

25、mean free path .The Fourier transform of such a damped wave involves ,which is not discrete but has a Lorentzian distribution with a width equal to 1/.Tiong et al.【9.2a】have proposed a model to estimate or phonons from the frequency shift and broadening of optical phonons localized by defects introd

26、uced by ion implantation. Most excitations have a finite lifetime.Optical phonons,for instance, decay via interactions with other phonons (through anharmonicity)or defects.As a result,their energies have an imaginary part represented by the damping constant. The effect ofis to introduce a width to t

27、he energy levels.Therefore ,in order to see confinement effects it is necessary for the confinement energy to be at least .Equivalently,this translates,via(9.3),into a maximum value in L for observing confinement effects. In other words, when L is too large the excitation will decay before even reac

28、hing the barrier.Since the confinement energy is inversely proportional to ,it is more difficult to observe quantum confinement effects in heavier particles.Typically the sample has to be cooled to low temperature(so as to decrease )in order to observe a small confinement energy. The confinement beh

29、avior of excitons is different from both electrons and phonons since they consist of an electron plus a hole separated from each other by a Bohr radius .When L is much larger than ,the exciton can move between the barriers like a free particle with total mass M (equal to the sum of the electron and

30、hole masses).The maximum value of L for confinement is determined by the exciton mean free path.When L is smaller than ,the exciton properties are modified by the confinement of its constituent electron and hole. For example,the exciton binding energy will be increased since the electron and hole ar

31、e forced to be closer to each other.In the limit of a two-dimensional exciton,the binding energy is increased by a factor of four relative to the three-dimensional case .Sometimes the confinement potential (if assumed to be infinite)has a larger effect on the two convenient to regard the heavier of

32、the two particles as being trapped inside the potential well(since its wavefunction will be more confined in the center of the well) while the other particle is attracted to it via the Coulomb interaction.Similarly,we expect donor and acceptor binding energies to be enhanced when such impurities are

33、 confined to a distance smaller than their Bohr radii. In addition to changing the energies of excitations, confinement also modifies their density of states(DOS).We have already considered the effect of dimensionality on DOS in the vicinity of critical point. In general, reducing the dimensionality

34、 “enhances” the singularity in the DOS at a critical point. For instance, on reducing the dimension from three in bulk samples to two in a QW, the electronic DOS at the bandgap changes from a threshold depending on photon energy as (-)to a step function .Since the transition the transition probabili

35、ties calculated using the Fermi Golden Rule involve the density of final states, confinement can have an important impact on the dynamics of scattering processes in semiconductor devices.For example, it has been demonstrated that laser diodes fabricated from QWs have higher efficiency and smaller th

36、reshold current than corresponding bulk laser diodes.It has been predicted that quantum dot lasers (zero-dimensional)should have even smaller threshold currents.In addition,their lasing frequencies will be much less sensitive to temperature change. In this book we shall not consider these effects of

37、 confinement on devices.Interested readers should refer to other books specializing on this topic. 9.2 Quantum Confinement of Electrons and Holes As an illustration of how electrons are confined in semiconductors and how to calculate their properties, we shall consider the case of single QW. Its str

38、ucture is a sandwich consisting of a thin layer(thickness L) of a semiconductor material (denoted by A)between two layers of another semiconductor B (of equal thicknesses L).The direction perpendicular to these layers will be referred to as the z axis.There are more complex structures ,consisting of

39、 several repeating units of the form B/A/B/A/B/A/B/A.(where LL),which are known as multiple quantum wells or MQWs .Superlattices and MQW are similar in construction except that the well separations in a MQW are large enough to prevent electrons from tunneling from one well to another .The barrier wi

40、dth L in a SL is thin enough for electrons tunnel through so that the electrons see the alternating layers as a periodic potential in addition to the crystal potential. We shall assume that the bandgap of the well A()is smaller than that of the barriers B() in a single QW.Owing to this bandgap diffe

41、rence, the conduction and valence band edges of A and B do not align with each other.The difference between their band edges is known as the band offset and has already been introduced in Sect.5.3.This band offset produces the potential responsible for confining the carriers in one layer only. Thus

42、the control and understanding of this band offset is crucial in the fabrication of quantum confinement devices. While our understanding of what determines the band offset of two dissimilar semiconductors is still not perfect, great progress has been made in the fabrication techniques to control the

43、shape of the bandgap discontinuity .For example ,in the well-studied GsAs(=A)/GaAlAs (=B) system the interfaces between A and B have been shown by high resolution transmission electron microscopy to be as narrow as one monolayer.Extensive comparisons between experimental results and theoretical calc

44、ulations have also shown that the band edge discontinuities can be rather abrupt,making a simple square well a good approximation for the confinement potential in most QWs.As a result we shall not discuss further the various theories proposed to explain the band offset,instead we shall assume that i

45、ts value is known from experiment .Dingle et al .9.6 have defined a factor Q equal to the ratio between the conduction band offset () and the bandgap difference ()as a way to characterize the band offset.For example ,in the technologically important GaAs/GaA/As and InGaAs/InP QWs the values of Q hav

46、e been determined to be 0.6 and 0.3,respectively. Fig.9.1Semiconductor Materials for Quantum Well and Superlattices Although a square confinement potential is not the only kind existing in nanostructures, it is nevertheless the most common one .The achievement of a sharp interface imposes very strin

47、gent requirements on the growth conditions, such as purity of the source materials,substrate temperature and many others too numerous to list here.However ,ultimately the quality of the interface between two dissimilar materials A and B,known as a heterojunction, is determined by their chemical and

48、physical properties.Of these perhaps the most important is the difference in their lattice constants.When these are nearly the same,it is easy for all the atoms of A to be aligned perfectly with those of B.This lattice alignment is known as pseudomorphic growth and is highly desirable for achieving

49、high-quality heterojunctions. There are only a few such lattice-matched systems.Figure9.2 plots the low temperature energy bandgaps of a number of semiconductors with the diamond and zinc-blende structures versus their lattice constants.The shaded vertical regions show the groups of semiconductors with similar lattice constants.Materials within the same shaded region but having different bandgaps can