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Mesoscopic phenomena in semiconductor nanostructures by quantumdesignFederico Capasso, Jerome Faist, and Carlo Sirtori Citation: J. Math. Phys. 37, 4775 (1996); doi: 10.1063/1.531669 View online: http://dx.doi.org/10.1063/1.531669 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v37/i10 Published by the American Institute of Physics. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Mesoscopic phenomena in semiconductor nanostructuresby quantum design

Federico Capasso, Jerome Faist, and Carlo SirtoriBell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974

~Received 13 June 1996; accepted for publication 17 June 1996!

The energy levels, wave functions, optical matrix elements, and scattering rates ofelectrons can be tailored at will using semiconductor nanostructures as buildingblocks for practically any kind of potential. This allows the design and experimen-tal realization of new artificial materials and devices, with interesting opticaland transport properties arising from quantum confinement, tunneling, and quan-tum coherence on a mesoscopic scale ranging typically from 1 to 100 nm. Thisapproach is illustrated with a number of recent examples based on experimentsand calculations, such as resonant tunneling through double barriers, quantuminterference phenomena in transport and optical absorption, bound states in thecontinuum, quantum well ‘‘pseudomolecules’’ with giant nonlinear optical suscep-tibilities, and quantum cascade lasers. ©1996 American Institute of Physics.@S0022-2488~96!00210-1#

I. INTRODUCTION

Quantum engineering of the electronic energy levels, wave functions and band structure,matrix elements, and scattering rates using ultrathin semiconductor layers1,2 grown by molecularbeam epitaxy~MBE!3 allows one to design and observe quantum phenomena on a mesoscopicscale~typically 1–100 nm!, much larger than the atomic one.4–7 This approach is the basis formodifying and tailoring in unprecedented ways the electronic, transport, and optical properties,which has led in many cases to altogether new materials~materials by design! and useful deviceapplications.2,5–8

Essential to the emergence of this field of research has been MBE.3 This epitaxial growthtechnique allows multilayer heterojunction structures to be grown with atomically abrupt inter-faces and precisely controlled material composition over distances as short as a few nanometers.Such structures include quantum wells. These potential energy wells are formed by sandwiching amaterial such as gallium arsenide~of thickness comparable or smaller than the carrier thermal deBroglie wavelength, which is;25 nm for electrons in gallium arsenide at room temperature!between two wider energy bandgap semiconductors~for example, aluminum gallium arsenide!.The energy spacings of the discrete states of the well arising from quantum confinement dependon the well thickness and depth.

If many quantum wells are grown on top of one another and the barriers are made so thin~typically,5 nm! that tunneling between the coupled wells becomes important, a superlattice isformed and the energy levels broaden into energy bands called minibands separated by minigaps.1

Superlattices are artificial materials with novel optical and transport properties introduced by theartificial periodicity.2,6,7

In this paper we shall review our recent work on mesoscopic quantum phenomena based ontunneling and on electronic transitions between quantized states of the same band~intersubbandtransitions8! in semiconductor nanostructures.

Several of the structures considered in this paper should also appeal to mathematicians andmathematical-physicists since this approach allows one to design, synthesize, and experimentallyinvestigate potentials of significant mathematical interest that cannot be found in nature. We shalllimit ourselves to one-dimensional potentials, i.e., structures based on quantum wells. For struc-

0022-2488/96/37(10)/4775/18/$10.004775J. Math. Phys. 37 (10), October 1996 © 1996 American Institute of Physics


tures based on quantum wires and dots, i.e., relying on quantum confinement along two and threedimensions, respectively, the reader is referred to Ref. 4.

II. RESONANT TUNNELING AND QUANTUM INTERFERENCE IN ELECTRONICTRANSPORT

The resonant tunneling double barrier consists of two potential barriers in series separated bya potential well.9 This well can, of course, have various shapes~Fig. 1!.

The kinetic energy of an electron’s motion perpendicular to the layers is quantized, just as onewould expect for a particle in a box. In the plane of the layers, however, the electron is free, andit behaves semiclassically. As a result, two-dimensional energy subbandsEn(k) are formed:

En~k!5En1\2k2

2m*, ~1!

whereEn is thenth energy level given by the quantization of the perpendicular kinetic energy, andthe second term is the kinetic energy of the electron’s free motion parallel to the layers, with wavenumberk and effective massm* .

The energy levelsEn correspond to a half-integer number of electron de Broglie wavelengthsacross the width of the quantum well. The barriers are thin enough that electrons can tunnelthrough them into and out of the quantum well. This structure is often compared to a Fabry–Perotoptical interferometer: the two barriers play the role of partially transparent mirrors through whichlight is coupled into and out of a resonant cavity.

As we might expect, the transmissivity for electrons through the double barrier shows reso-nant peaks when the perpendicular kinetic energy of the incident electrons equalsEn . At theseresonant energies the transmissivity for a symmetric double barrier reaches 100%, even though thetransmissivity for a single barrier might be less than 1%. This striking resonant enhancement ofelectron transmission is easily understood in terms of constructive interference between multiplyreflected waves. But it cannot be understood within the semiclassical framework, which forbidstunneling through even a single barrier.

This description in terms of Fabry–Perot interferometry is somewhat idealized. In manyrealistic devices the resonant enhancement of the transmission is considerably weakened by scat-

FIG. 1. Energy diagram of resonant tunneling structure under different conditions of applied electric field.~a! Zero field;~b!,~c! the electric field is such that electrons resonantly tunnel through the parabolic portion of the well;~d! the appliedfield is high enough that electrons probe the continuum resonances. The wells are drawn to scale; however, for sake ofclarity only half the number of levels in an energy interval are shown.

4776 Capasso, Faist, and Sirtori: Mesoscopic phenomena in nanostructures

J. Math. Phys., Vol. 37, No. 10, October 1996


tering processes because scattering events destroy the phase coherence of multiply reflectedwaves. One way to estimate the role of scattering is to compare the intrinsic energy widthGR ofthe resonance peak with the collisional broadeningGC of the resonance.GR is determined by thedegree of transmission through the individual barriers, that is, by the degree of external couplingto the cavity; it is approximately equal toEnTB , whereTB is the transmissivity of one barrier.GC

is determined by the scattering rate, that is, by the internalQ of the cavity.GC is roughly\/t,wheret is the average time between successive collisions and\ is Planck’s constant divided by2p. If GR is much larger thanGC , the Fabry–Perot description is appropriate. On the other hand,if GC is much larger thanGR , the process can be viewed as sequential rather than coherent: theelectrons tunnel into the well, scatter, and tunnel out the opposite side. Devices usually operatesomewhere in between these extremes. The scattering reduces the peak transmissivity and broad-ens the resonance. The area under the transmissivity curve, however, stays constant.

A very important factor in the operation of resonant tunneling structures is the role of space-charge buildup within the well, which gives rise to an electrostatic potential that shifts the resonantenergy of the wellEn relative to the energy of incident electrons in the emitter. This is analogousto the shift in the resonant frequency of a nonlinear Fabry–Perot interferometer due to light-intensity buildup, which is known to lead to optical bistability. This effect can give rise tointeresting nonlinear oscillations or even chaotic behavior of the charge accumulating in thewell.10,11Other interesting dynamical phenomena associated with charge accumulation in the wellin resonant tunneling structures are discussed in the paper of Presilla and Siostrand of this issue.

In most experiments one measures the current through a resonant tunneling diode when avoltage is applied across the double barrier through the heavily doped contact layers. The appliedbias voltage lowers the resonant energy of the cavity relative to the energy of the incident elec-trons. Once the resonant energy has fallen below the range of incident energies—below theconduction-band edge in the emitter—there is a sharp drop in the current as the applied voltage isincreased further. This negative differential resistance is a useful feature for device applicationssuch as high-frequency oscillators and multistate transistors.5,6,12Oscillators have operated up tofrequencies in excess of 700 GHz.12

One can design and implement by MBE the electronic potential and the wave functions of aresonant tunneling structure in a nearly arbitrary way. This is illustrated by the energy diagram~Fig. 1! of a parabolic well between rectangular barriers under different conditions of appliedelectric fields.13 This structure was grown by MBE. The 44 nm wide well is bound by 3.5 nm thickaluminum arsenide~AlAs! barriers and its chemical composition is varied from Al0.30Ga0.7As ~analloy! at the edges and gallium arsenide~GaAs! at the center. The subscripts indicate the molarfractions of AlAs and GaAs in the alloy. This double barrier is sandwiched between two highconductivity semiconductor layers to allow application of a voltage. Figure 2 shows the measuredcurrent as a function of applied voltage for opposite polarities and the corresponding conductance~i.e., the derivative of the current with respect to voltage!. The latter is plotted to enhance thefeatures corresponding to resonant tunneling through the quantum states.

The overall features of the current voltage curve (I –V) can be interpreted physically bymeans of the energy diagrams of Fig. 1. At zero bias,@Fig. 1~a!#, the first six energy levels of thewell are confined by a parabolic well 225 meV deep, corresponding to the grading fromx50 tox50.30, and their spacing is.35 meV. When the voltage is increased from 0 to 0.3 V the firstfour energy levels probed by resonant tunneling@Fig. 1~b!# remain confined by the parabolicportion of the well, and their spacing is practically independent of the electric field, since it isprimarily controlled by the curvature of the potential. This gives rise to the calculated and ob-served equal spacing of the first four resonances in theI –V characteristic~Fig. 2!. Consider nowthe higher-energy levels confined by the rectangular part of the well~.230 meV! at zero bias.When the voltage is raised above 0.3 V these levels become increasingly confined on the emitterside by the parabolic portion of the well and on the opposite side by a rectangular barrier, thusbecoming progressively more separated, although retaining the nearly equal spacing@Fig. 1~c!#.

4777Capasso, Faist, and Sirtori: Mesoscopic phenomena in nanostructures



This leads to the observed gradual increase in the voltage separation of the resonances as the biasis increased from 0.3 to 1.0 V. Above 1 V the electrons injected from the emitter probe the virtuallevels in the quasicontinuum above the collector barrier@Fig. 1~d!#. These resonances result fromelectron interference effects associated with multiple quantum mechanical reflections at the well–barrier interface for energies above the barrier height. These reflections give rise to the featuresobserved above 1 V in thecurrent–voltage characteristic~Fig. 2!, and must be clearly distin-guished from the ones occurring at lower voltages, which are due to tunneling through twobarriers. It should be noted that in the latter case the reflection from the second barrier is associ-ated with an imaginary wave number in the barrier. In the case of the continuum resonancesshown in Fig. 1~d! instead, the reflections from the second barrier are associated with a real wavenumber since the incident electron energy is greater than the barrier height.

III. QUANTUM INTERFERENCE IN OPTICAL ABSORPTION

Quantum interference effects in the absorption of atoms and molecules have been knownsince the classic work of Fano.14 The development of MBE and quantum engineering of semi-conductor heterostructures has made possible the observation of new optical absorption phenom-ena. In particular, in this section, we shall focus on intersubband absorption effects.15 Intersubbandtransitions are those where the initial and final quantized states are in the same band, e.g. the

FIG. 2. Current versus applied voltage at 10 K temperature and corresponding differential conductance (dI/dV) for adouble barrier structure~Fig. 1! under opposite voltage polarity conditions. The inset shows the eight resonance on a linearscale. The vertical segments near the horizontal axis indicate the calculated positions of the resonances.




conduction band.8 In an intersubband transition electrons in a lower state characterized by aneigenvalue~energy level! Ei with a dispersion of the type shown in Eq.~1! make a transition to ahigher-energy final state~in the case of optical absorption! characterized by an eigenvalueEf .Conservation of energy and momentum requireEi1\v5Ef , where\v is the photon energy andki5kf . The latter equality comes about because of the negligible momentum of the photoncompared to that of the electron\k. It should be noted that in such a transition the envelopefunction of the electron, determined by the one-dimensional heterostructure potential, changeswhile the rapidly oscillating Bloch function associated with the atomical level periodicity of theunderlying crystal is approximately the same for the final and initial states. The latter approxima-tion is correct under the assumptions that band nonparabolicities15 associated with the effectivemass being energy dependent are small, a condition verified in many absorption experiments.

The matrix element of thei→ f transition is^ i uzu f )&, wherez is the coordinate normal to thelayers. In order to couple to this matrix element the incident radiation must be polarized or at leasthave a component of the polarization normal to the plane of the layers, i.e., alongz. Electrons inthe quantum well structures are introduced during crystal growth by the well-known process ofdoping, i.e., through impurities such as silicon atoms that release an electron. The infrared trans-mission spectrum can then be measured using standard techniques such as Fourier transforminterferometry.

A. Suppression of optical absorption by electric-field-induced quantum interference

Recently the design and demonstration of coupled quantum well structures exhibiting a strik-ing interference effect in the matrix element for intersubband absorption has been reported.16 Thepotential and a specific optical transition~1–3! are designed@Fig. 3~a!# so that under applicationof an appropriate electric field the corresponding matrix element has a null@Fig. 3~b!#.

The sample, grown by MBE, comprises 50 coupled quantum wells. Each period consists oftwo GaAs wells, respectively, 62 and 72 nm thick, separated by a 2 nm Al0.33Ga0.67As barrier. Thecoupled well periods are separated by a 145 nm Al0.33Ga0.67As spacer layer. To supply the electroncharge in the wells an atomically thin layer of silicon dopant~131012/cm2! is inserted in thespacer layers to ensure a symmetric charge transfer. Figure 3~a! shows the energy diagram of thecoupled quantum well structure with no applied voltage. Indicated are the energy levels and themoduli squared of the wave functions. The energy levels and wave functions are computed bysolving Schro¨dinger’s and Poisson’s equations in the envelope function formalism.1,15

To get a better insight into the behavior of the coupled well system as a function of the appliedelectric field, let us first consider the two quantum wells, denoted here as wells a and b, coupledby the barrier in a tight-binding approach.1 In such a model, the calculated wave functionsci

~i51,...,4! of this system are expanded in terms of the eigenfunctionsf1,2a,b of the first two bound

states 1,2 of the two isolated wells. In the tight-binding approximation, the dipole matrix elementz1i5^c1uzuc i& ~i53,4! between the first and the third or fourth state of the coupled well systemcan now be written as the sum of the contribution from the two wells a and b,

z1i5^c1uf1a&^c i uf2

a&z12a 1^c1uf1

b&^c i uf2b&z12

b , ~3!

wherez12a andz12

b are the transition matrix elements computed for the isolated wells. Asc1 is theground state of the system,^c1uf1

a& and ^c1uf1b& have the same sign. On the contrary, since the

second excited statec3 crosses zero twice and is constructed from the antisymmetric wave func-tionsf2

a,b, ^c3uf2a& and^c3uf2

b& have opposite signs. Therefore, if we consider a transition betweenthe first and third state of the coupled well system, the two terms of Eq.~3! have opposite signs.One thus expects large values ofz13 for large absolute values of the electric field, where both wavefunctions are localized in either well a or b@the first or last term of Eq.~3! dominates# and a nullfor some intermediate value of the electric field. At this field the absorption will be suppressed.This behavior is clearly apparent in Fig. 3~b!, where we display the calculateduz13u

2. The absorp-




tion spectra showed very clearly this effect as the applied electric field was varied.16 The experi-mentaluz13u

2 is derived from a measurement of the area under the 1–3 absorption peak~in units ofphoton energy!, and taking the nominal electron sheet density in the wells~531011 cm22!. Theexperimental points in Fig. 3~b! are in good agreement with the calculation.

B. Fano quantum interference

When the excited stateuf& of a quantum system is coupled to a continuumuc& at the sameenergy it broadens due to the finite lifetimet introduced by the coupling to the continuum. Theabsorption spectrum from the ground stateu1& to this excited state will be Lorentzian with a fullwidth at half-maximumG5\/t. However, a peculiar situation arises when the matrix element^1uzuc& from the ground state to the continuum is nonvanishing: the absorption lineshape changesdramatically, becoming asymmetric, and displaying a zero within a fewG from the absorptionpeak. This phenomenon, called Fano interference,14 has been observed in many atomic, molecular,or solid-state systems.

Recently, we have reported the observation of Fano interference in a heterostructure, where allthe relevant parameters, i.e., the escape rateG and the matrix element to the continuum^1uzuc&, aretailorable and controlled by design.17

The structures are grown by MBE and consist of ten periods. As shown in Fig. 4, each periodconsists of a GaAs coupled well confined by a high, 40 nm thick Al0.33Ga0.67As barrier on the right

FIG. 3. ~a! Energy diagram of an GaAs/Al0.33Ga0.67As coupled-quantum-well structure used to investigate quantuminterference in optical absorption. Shown are the positions of the calculated energy subbands and the correspondingmodulus squared of the wave functions. We computedE1563 meV,E2580 meV,E35198 meV, andE45250 meV.~b!Square of the transition matrix element~z13!

2 ~right axis!, as derived experimentally from the integrated absorbance belowthe ~1–3! absorption peak~left axis!. The solid line is the calculated value.




and a thick low Al0.165Ga0.835As barrier on the left. The thicknessL5200 nm of theAl0.165Ga0.835As barrier is chosen such that it is much longer than the electron’s coherence lengthlc;20–50 nm, and therefore the states of this region behave as a continuum. Two structures withdifferent quantum well thicknesses were grown. Sample A had the strongest coupling due to therelatively thin 2.0 nm Al0.33Ga0.67As barrier coupling the 5.2 nm left well to the 6.4 nm right well.A doping sheet in the Al0.165Ga0.855As, separated from the quantum wells by a 25 nm spacer layerprovides the 2.531011 cm22 electron sheet density in the coupled well region. Sample B had aweaker coupling due to the thicker 2.5 nm barrier coupling the 5.5 nm left well to the 6.5 nm well.The electron sheet density wasns5531010 cm22 and the spacer layer 50 nm.

In these samples, the individual ground states of the two wells couple through the thinintermediate barrier to form a doublet with a splitting of about 20 meV. The same barrier alsocouples the excited state of the right welluf& with energyEr to the energetically degeneratecontinuum that broadens the stateuf& by G>12 meV for sample A andG>6 meV for sample B.In both cases, we have the conditionG@Gd , where Gd;1–2 meV is the broadening ofuf&~homogenous and inhomogeneous! caused by interface roughness and optical phonon scattering.Since the ground state wave function spans both wells and therefore has a strong dipole couplingto both stateuf& and the continuum stateuc& above the left well, the structure fulfills the require-ments for the observation of Fano interference.14 In a tight-binding picture,1 we are coupling abound-to-bound transition in the right well to a bound-to-continuum in the left well. Note that onewould not observe this interference in a sample having only the right well, since the matrixelement^1uzuc&'0 in this case. The features associated with Fano interference are also clearly

FIG. 4. Energy diagram of a portion of the structure used to study Fano quantum interference in absorption. The thick-nesses and doping used in this self-consistent calculation correspond to sample A~see the text!. The moduli squared of thewave functions of then51 andn52 states are displayed. The modulo squared of the wave function in the continuumuC&is represented as a gray-scale density plot. Points a, b, and c represent the final state energies corresponding to the onsetof the continuum, the zero, and the maximum of the absorption spectrum~see Fig. 5!. The shift between the maximum ofthe absorption and the position of the resonanceEr is a feature characteristic of Fano interference.




apparent in the plot of the modulus squared of the eigenfunctionsuc& of the whole system~con-sisting of the continuum plus the quantum wells! displayed as a gray scale plot in Fig. 4. Aspredicted for Fano interference, the calculated maximum and zero of the absorption~points c andb in Fig. 5! lie above and belowEr , respectively. This occurs because^1uzuf& and^1uzuc& interferewith opposite phaseon the two sides of the resonance.14 The expected energy-dependent phaseshift experienced by the wave functionuc! is evident in Fig. 5 as an abrupt shift of the position ofthe minimum of the modulus squared of the wave functions as the energy crosses the resonantenergy. The absorption for both samples is reported in Fig. 5 along with the calculated spectra.17

The Fano interference is contained in the calculated spectra automatically, since the absorption iscomputed using the wave functionuc&, which is an eigenfunction of the whole system, coupledwell plus continuum. Thus the agreement between the measured and calculated spectra is the proofthat these samples exhibit Fano interference. However, note that these experimental lines cannotbe fitted in satisfactory fashion with the original Fano lineshape14 because the same importantassumptions used to derive that expression~invariance of both the matrix element and the couplingstrength as a function of energy! do not hold in our case. Both spectra also show the qualitativefeatures of the Fano lineshapes with a zero close to the asymmetric absorption peak. The shift~;100 meV! between the absorption peak and the onset of the continuum is another feature thatis specific of these structures exhibiting Fano resonance. An absorption spectrum from a bound-to-continuum single quantum well would peak very close~20 meV! to the onset of the continuum.As expected, the main peak is broader for sample A~50 meV! than for sample B~30 meV! due toits stronger coupling to the continuum.

FIG. 5. Absorption spectra of structures exhibiting Fano quantum interference.~a! The solid curve is the measuredabsorption spectrum for sample A with strong coupling and asymmetry. Points a, b, and c refer to the onset of thecontinuum, the zero, and the maximum of the absorption~see Fig. 4!. The dashed curve is the calculated spectrum.~b! Thesame for sample B. Note the shift between the absorption peak and the onset of the continuum, which is a feature specificto these samples exhibiting Fano interference.




IV. CONTINUUM RESONANCES: ELECTRON WAVE INTERFERENCE AND BOUNDSTATES ABOVE A POTENTIAL WELL

In the previous section we have seen how confined states of quantum wells are central in anumber of tunneling and optical phenomena. Highly localized states and even bound states canalso be created at energies above the barrier height in a potential well using constructive interfer-ence phenomena.18,19

Consider first a conventional rectangular well@Fig. 6~a!#. At energies greater than the barrierheight one has a continuum of scattering states. For discrete energies corresponding to a semi-integer number of electron wavelengths across the well, one finds transmission resonances. Al-

FIG. 6. Energy diagrams of potentials used to study highly localized states in the continuum.~a! Reference sample. Shownare the ground state of the well~E15204 meV! and the position~dashed line! of the first transmission resonance in thecontinuum~E25560 meV!. ~b! Quantum well cladded by two-period quarter-wave stacks. Shown isuCu2 of the localizedquasibound state~E65560 meV! formed in correspondence to the transmission resonance and the positions of new statescreated at lower energies~E25320 meV,E35322 meV,E45356 meV,E55359 meV!. ~c! In the superlattice limit thel/4stacks behave as Bragg reflectors. The state above the well now becomes a bound state localized by the superlatticeminigap ~5266 meV!. The low-energy miniband extends from 307 to 379 meV.




though at these energies the electron amplitude in the well layer is enhanced, the wave functionsdo not decay exponentially in the barrier, unlike the confined states of the well, but are plane-wave-like. These states can be localized in the well using as barriers stacks of layers of thicknessl/4 each, wherel is the de Broglie wavelength in the layer~at an energy comparable to that of theselected transmission resonance! @Figs. 6~b!–6~c!#. The net effect is that the reflection coefficientof electrons acquires a high value~near unity! for a significant range of energies~typically 0.1–0.2eV! above the barrier height. This high value of the reflectivity is the result of interferencebetween the waves partially reflected by the heterointerfaces of thel/4 stacks, which leads to theformation of a quasibound state above the center well@Fig. 6~b!#. This strongly narrows thetransmission resonance in analogy with a Fabry–Perot optical filter, where sharp optical reso-nances are produced using as high reflectivity mirrors dielectric quarter-wave stacks. The degreeof localization increases with the number of periods; in the structure with just two-period stacks,the wave function is already highly confined@Fig. 6~b!#. In the superlattice limit and at lowtemperatures, to minimize scattering, the stacks behave as Bragg reflectors; a minigap opens up@Fig. 6~c!# and the localized state becomes a bound state at energies greater than the barrier height.The prediction that certain oscillatory potentials support bound states in the continuum, due toquantum interference, was first put forth by von Neumann and Wigner in 1929.20

The reference sample@Fig. 6~a!# had 20 3.2 nm InGaAs doped quantum wells separated by 15nm undoped AlInAs barriers. In the other three structures the 3.2 nm wells, doped to the samelevel were cladded, respectively, by one-period, two-period@Fig. 6~b!#, and six-period@Fig. 6~c!#l/4 stacks consisting of 3.9 nm AlInAs barriers and 1.6 nm GaInAs wells, designed as discussedabove. The phase coherence length in the superlattice structure of Fig. 6~c! is estimated to be;30nm at 10 K.

The room temperature absorption spectra of the reference sample is broad with a long-wavelength cutoff determined by the height of the barrier.18 In the structure with onel/4 periodthe peak is considerably narrower and centered at an energy corresponding to the transitionbetween the ground state of the well and the localized resonant state at the energyE6.

18 As thenumber of quarter-wave stacks is doubled the absorption peak does not shift and considerablynarrows, precisely the behavior expected for an interference filter. In fact, the observed narrowing~16 meV! can be quantitatively explained in terms of the reflectivity increase of thel/4 stacks.18

In the structure with six periods at cryogenic temperatures, the highly localized state becomeseffectively a bound state confined by Bragg reflectors from the superlattice.19 The absorptionspectrum~Fig. 7! shows an isolated peak at 360 meV of width;10 meV corresponding to thetransition from the stateE1 to the stateE2 in Fig. 6~c!.19 It is worth noting that the width of thetransition to the confined state above the well in the two- and six-period structures is identical tothat of the bound-to-bound state transition measured in a conventional 5.5 nm thick GaInAs well

FIG. 7. Absorption spectra at cryogenic temperature for the structure with superlattice Bragg reflectors of Fig. 6~c!~bottom!. The transition to the confined state above the well@E2 in Fig. 6~c!# corresponds to the peak at 360 meV, inexcellent agreement with the calculated value forE22E1 .




with 30 nm thick barriers, thus demonstrating the highly localized nature of the state above thewell.

Other intriguing phenomena arise when, for example, the first continuum resonances of quan-tum wells are at the same energy of the first barrier resonance.21 This can be achieved by suitableadjustment of the layer thicknesses in a superlattice. In this case a very broad miniband is formedthat extends ink space fromp/a to 3p/a ~wherea is the superlattice period!, with a minigapsuppression at 2p/a ~see Fig. 8!. This effect has been confirmed experimentally.21

V. COUPLED QUANTUM WELL PSEUDOMOLECULES WITH GIANT NONLINEAROPTICAL SUSCEPTIBILITIES

Consider an electromagnetic field at frequencyv, propagating through a material. Opticalphenomena such as dispersion, absorption, and stimulated emission are described in Maxwell’sequations by a linear polarizationP(v)5e0x(v)E(v) proportional to the electric field of thewaveE~v! via a coefficientx~v! called susceptibility, wheree0 is the vacuum permittivity. Moregenerally, the polarization contains higher-order but smaller terms at frequencies such as 2v and3v. These nonlinear terms are proportional to powers of the field via nonlinear susceptibilitiessuch asx~2! ~2v! and x~3!~3v!, and are responsible for phenomena such as second harmonicgeneration~SHG! at 2v and third harmonic generation THG 3v. The polarizations for these twophenomena can be written asP(2v)5e0x (2v)

(2) E2(v) andP(3v)5e0x (3v)(3) E3(v). More gener-

ally, when two beams at frequenciesv1 andv2 are present, nonlinear phenomena such as sum ordifference frequency generation atv16v2 are possible.

The structures discussed in this section can be viewed as ‘‘pseudomolecules’’ with giantdipole matrix elements and nearly equally spaced energy levels~Fig. 9!. These characteristics leadto a large enhancement of their nonlinear optical susceptibilities.22 Physically these susceptibilitiesin our structures arise from the interaction of light with the quantized anharmonic oscillations ofelectrons in the potentials of Fig. 10. The latter are grown in the AlInAs/GaInAs system latticematched to InP, and only the thickest well is dopedn-type to provide electrons. The choice of thismaterial system facilitates the tunnel coupling between the layers due to the low effective mass~0.07 m0! of the barrier region and provides a large potential barrier~0.5 eV! essential for con-

FIG. 8. Energy dispersion along the superlattice axis for a structure~inset! where the first barrier resonanceER1B is

degenerate with second well resonanceER1W . The resulting mixing of these states produces a very broad miniband fromp/a

top/3a and suppression of the energy gap atp/2a. The barriers~AlInAs! and the wells~GaInAs! are, respectively, 8.8 and3 nm thick.




fining four states in the three-well structure separated by an energy corresponding to that of thephoton from a carbon dioxide~CO2! laser~'120 meV!. To optically excite the quantized motionnormal to the layer interfaces, one must use light with a component of the polarization normal tothe layers. In our experiments this was done using linearly polarized light in a multipass wave-guide structure wedged at 45°. In our SHG experiment a coherent polarization is created at doublethe frequencyv of the pump wave~a CO2 laser beam! due to the lack of reflection symmetry ofour two-well structure@Fig. 9~a!#. This coherent polarization radiates a wave of frequency 2vcolinear with the pump. The vicinity of the pump photon energy toE22E1 and of 2v to E32E1produces a strong resonant enhancement of the nonlinear susceptibilityx2v

~2! associated with SHG.22

The maximum susceptibility~x2v~2!! corresponds to exact matching, i.e.\v5E22E15E32E2 .

This can be achieved using the large linear Stark effect typical of this structure by applying anelectric field of suitable polarity normal to the layers. In these conditions aux2v

~2!u51027 m/V wasmeasured, approximately 300 times the value ofux2v

~2!u in bulk GaAs atl510 mm.22

The three-well structure@Fig. 9~b!# with the near equal separation of its four energy levels issuitable for triply resonant THG. In this process a pump wave at frequencyv sets up a nonlinearpolarization at 3v that coherently radiates a wave at this frequency.22 The nonlinear susceptibilityx3v

~3! that enters the expression for the polarization is strongly enhanced when the condition

FIG. 9. Energy diagrams of the AlInAs/GaInAs coupled-quantum-well nonlinear optical structures. Shown are the posi-tions of the calculated energy levels and the corresponding moduli squared of the wave functions.~a! The structure usedfor resonant second harmonic generation. The GaInAs wells have thicknesses of 6.4 and 2.8 nm and are separated by a 1.6nm AlInAs barrier.~b! The structure used for triply resonant third harmonic generation. The GaInAs wells have thicknessesof 4.2, 2.0, and 1.8 nm, respectively, and are separated by 1.6 nm AlInAs barriers.




\v.E22E1.E32E2.E42E3 is met. It is important to note that a parabolic well~i.e., a har-monic oscillator potential! is unsuitable for this purpose, since in such a system the electronoscillations are linear~in quantum mechanical terms the transition matrix elements between non-adjacent states are zero, a property of Hermite polynomials!. THG experiments in the structure ofFig. 10~b! have found aux3v

~3!u510214 ~m/V!2 at 300 K.22 At cryogenic temperaturesux3v~3!u is four

times larger. These are the highest third-order susceptibility of any known material system. Thethree-coupled well structure was also used to study multiphoton electron escape from a well underan applied electric field, the analog of multiphoton ionization of an atom.23 In this process elec-trons are photoexcited into the continuum via a CO2 laser using a three photon transition, givingrise to a photocurrent. The cross section for this process is found to be many orders of magnitudelarger than in atoms and molecules.

Consider now the nonlinear optical phenomenon of difference frequency mixing in which twoincident waves at frequenciesv1 andv2 interact in a suitable asymmetric medium to set up apolarization at the difference frequencyv12v2.

24 This polarization is responsible for the genera-tion of radiation atv5v12v2. Quantum nanostructures can be designed to exhibit a very largex~2!~v5v12v2! when the incident photons and their energy difference are resonant with opticaltransitions of the structure. The asymmetric structure of Fig. 3~a! was in fact used for infrared~l.60 mm! difference frequency generation near the energy difference between statesn52 andn51, i.e.,v3.(E22E1)/\.

24 The photons atv1 andv2 are chosen to be near resonance with the~1–3! and ~3–2! transitions, respectively. As the photon energy difference\~v12v2! of the twoincident CO2 lasers was tuned nearE22E1 , the measured far infrared radiation~at l;60 mm!exhibited the typical resonant behavior of this process.24

FIG. 10. ~a! Energy diagram of a quantum cascade laser showing also the moduli squared of the wave functions.~b!Schematic representation of the dispersion of then51, 2, and 3 states parallel to the layers;ki is the corresponding wavenumber. The wavy lines represent the laser transition.




VI. QUANTUM CASCADE LASERS

The quantum cascade~QC! laser is an excellent example of how quantum engineering can beused to design new laser materials and related light sources. It is based on intersubband transitionsbetween excited states of coupled quantum wells and on resonant tunneling as the pumpingmechanism~Fig. 10!. The population inversion between the states of the laser transition is de-signed by tailoring the electron intersubband scattering times. This tailoring of scattering adds animportant dimension to quantum engineering of mesoscopic structures. In QC lasers, unlike allother laser sources, the wavelength is determined by quantum confinement, i.e., by the layerthicknesses of the active region rather than by the chemical properties of the material. As such, itcan be tailored over a very wide range using the same heterostructure material. Since the initialreport of QC lasers in 1994~Ref. 25!, we have demonstrated emission wavelengths in the 4–8.5mm range using AlInAs/GaInAs heterostructures lattice matched to InP.26–29 Figure 10~a! illus-trates the conduction band energy diagram of a portion of the 25-period~active region plusinjector! section of the quantum cascade laser under an applied electric field normal to the layers;105 V/cm corresponding to lasing conditions. The dashed lines are the effective conduction bandedges of the digitally graded electron-injecting regions, where electrons relax their energy beforebeing injected in the next region. These injectors are short period superlattices~Fig. 11!. Thepopulation inversion between the states of the laser transition~n53 and n52 in Fig. 10! isobtained by ensuring, by suitable design, that the scattering time from the upper state~n53! to thelower one~n52! is larger than the lifetime of the latter. At the same time one must reduce asmuch as possible the tunneling escape rate from then53 state to the continuum, since this processin steady state tends to reduce the population of the upper level. Finally, tunneling out of thelowest state~n51! should be fast enough to avoid a buildup of electrons in that subband. Asdescribed below, these requirements are met by a suitable choice of layer thicknesses, number ofquantum wells, and electric field in the active region. More specifically, the latter is designed tohave a laser transition that is ‘‘diagonal in real space’’~Fig. 10! and an energy separation betweenthen52 andn51 states resonant with the optical phonon. Electrons are injected through a 4.5 nmAlInAs barrier into then53 energy level of the active region. The latter includes 0.8 and 3.5 nmthick GaInAs wells separated by a 3.5 nm AlInAs barrier. Note the reduced spatial overlapbetween then53 andn52 states~‘‘diagonal’’ or photon-assisted tunneling transition! and thestrong coupling to an adjacent 2.8 nm GaInAs well through a 3.0 nm AlInAs barrier. Electronsescape from this well through a 3.0 nm AlInAs barrier. The calculated energy differences areE32E25295 meV andE22E1530 meV. The wavy arrow indicates the laser transition. Figure10~b! shows a schematic representation of the dispersion of then51,2, and 3 states parallel to thelayers;ki is the corresponding wave number. The bottom of these subbands correspond to energylevels n51, 2, and 3 indicated in~a!. The wavy arrows indicate that all radiative transitionsoriginating from the electron population~shown as shaded! in then53 state have essentially thesame wavelength. The quasi-Fermi energyEFn corresponding to the population inversion atthreshold~ns51.731011 cm22! is ;8 meV, measured from the bottom of then53 subband. Thestraight arrows represent the intersubband scattering processes by optical phonons.

The tunneling rate through the trapezoidal injection barrier is extremely fast~;0.2 ps!21,ensuring the efficient filling of level 3. The coupled-well region is essentially a four-level lasersystem, where a population inversion is achieved between the two excited statesn53 andn52.The intersubband optical-phonon-limited relaxation time,25 t32, between these states is estimatedto be ;4.3 ps at;105 V/cm; this process is between states of reduced spatial overlap andaccompanied by a large momentum transfer@Fig. 10~b!# associated with the large intersubbandseparation; as such,t32 is relatively long. This ensures population inversion between the two statesbecause the lower of the two empties with a relaxation time estimated around 0.6 ps. Stronginelastic relaxation by means of optical phonons with nearly zero momentum transfer occursbetween the strongly overlapped and closely spacedn52 andn51 subbands@Fig. 11~b!#. Finally,




the tunneling escape time out of then51 state is extremely short~&0.5 ps!, further facilitatingpopulation inversion. The ‘‘diagonal’’ nature of the laser transition increases the escape time intothe continuum from then53 level ~tescp.6 ps!, thus enhancing the injection efficiency. Figure 11is a transmission electron microscope photograph of a cleaved section of the QC laser, showingthree periods of the active region.

The 25-period active region is sandwiched between thick AlInAs cladding layers to providean optical waveguide parallel to the layers. The optical cavity is obtained by cleaving 0.5–3 mmlong bars normal to the layers. The crystalline cleavage planes serve as mirrors. With the designof Fig. 10 laser action was obtained in pulsed mode atl54.3mm with several tens of mW of peakpowers and up to;100 K operating temperatures, but with relatively high thresholds. In thisdesign, the width of the luminescence transition is relatively broad@full width at half-maximum~FWHM!522 meV# since a diagonal transition is more sensitive to interface roughness associatedwith thickness fluctuations~; one atomic layer! in the plane of the layers. As a consequence, thepeak gain is reduced. To circumvent this problem we designed the structure of Fig. 12, whereelectrons make a vertical radiative transition essentially in the same well.26 This reduces consid-erably the width of the gain spectrum~FWHM'10 meV!, and therefore the laser threshold currentdensity. To prevent electron escape in the continuum, which is greatly reduced in the case of thediagonal transition, the superlattice of the digitally graded injector is now designed as an effectiveBragg reflector for electrons in the higher excited state and to simultaneously ensure swift electronescape from the lower states via a miniband facing of the latter~Fig. 12!.26 The active regionconsists of 4.5 mm InGaAs quantum well coupled to a 3.6 nm well by a 2.8 nm AlInAs barrier.Tunneling injection from the superlattice into the active region is through a 6.5 nm AlInAs barrierand electrons escape out of then51 state through a 3.0 nm AlInAs barrier. As in the otherstructure, the lower state of the laser transition is separated by an optical phonon~'30 meV! fromthe n51 state. The calculated relaxation time ist21.0.6 ps, which is considerably less than thatbetween then53 andn52 state~1.8 ps!, thus creating population inversion condition betweenthese energy levels. Electrons can, in turn, tunnel out of then51 state in a subpicosecond time toprevent electron buildup.

Dramatic performance improvements have been obtained with vertical transition QC lasers.26

The threshold current density is considerably reduced~; a factor of 2! leading to higher operating

FIG. 11. Transmission electron micrograph of a portion of the cleaved cross section of the quantum cascade laser of Fig.10. Three periods of the 25 stage structure are shown. The superlattice period of the digitally graded regions is 3 nm andthe duty cycle of the AlInAs barrier layers varies from 40% to 77% top-to-bottom, creating a compositionally gradedpseudoquarternary alloy in these regions. This is used for injecting electrons from top-to-bottom into the 0.8 nm GaInAswell. The wells and barriers of the digitally graded regions are dopedn-type to 1.531017 cm23 to avoid space-chargebuildup under injection, while the other layers are undoped.




FIG. 12. The schematic energy diagram of a portion of the Ga0.47In0.53As/Al0.48In0.52As quantum cascade laser with verticaltransition under positive bias condition and an electric field of 8.53104 V/cm. The dashed lines are the effective conduc-tion band edges of the 20.8 nm thick superlattice electron injector. As shown, this superlattice is also designed as to createa minigap that blocks electron escape from level 3. The wavy line indicates the transition responsible for laser action. Themoduli squared of the relevant wave functions are shown.

FIG. 13. Continuous optical output power from a single facet versus injection current for various temperatures of aquantum cascade laser~Ref. 30!. ~a! Sample D-2122, with a device 2.9 mm long and 7mm wide.~b! Sample D-2160, witha device 3 mm long and 9mm wide. Single mode high resolution spectra are shown in the inset at various temperatures.The lasers operated in pulsed mode up to room temperature and above~320 K!.




temperatures. In addition, the peak optical power is also greatly enhanced and the lasers canoperate in continuous wave.27,28

More recently, the design of vertical transition QC lasers has been further improved by addinga thin quantum well between the graded injector layer and the double-well active region.29 Thisincreases the tunneling injection efficiency. The above features, together with the substitution ofthe AlInAs cladding layers with InP regions of much higher thermal conductivity, has led to theroom-temperature high peak power~;200 mW! pulsed operation of QC lasers atl55.2 mm.30

Continuous wave single mode operation has also been achieved up to 140 K~Fig. 13!. These arethe first semiconductor lasers operating at room temperature in the mid-infrared. Their overallperformance makes them excellent candidates for many applications such as environmental sens-ing and pollution monitoring in the 3–5mm and 8–13mm atmospheric windows.

VII. CONCLUSIONS

This review has highlighted the range of interesting transport and optical mesoscopic phe-nomena made possible by wave function engineering in semiconductor nanostructures grown byMBE. By controlling the phase of the electronic envelope function states and of their transitionmatrix elements, a number of interesting quantum interference effects have been observed indouble-barrier transport and optical absorption. This quantum engineering approach has also led tothe design and demonstration of new materials with giant nonlinear optical coefficients. Finally,not only the electronic states but also the scattering rates can be tailored. This adds a newdimension to quantum design and has allowed us to demonstrate new light sources~quantumcascade lasers!, where the population inversion is designed rather than determined by relaxationtimes intrinsic to the laser material.

ACKNOWLEDGMENTS

We are in debt to A. Y. Cho, A. C. Gossard, L. Pfeiffer, S. Sen, D. L. Sivco, K. West, J. N.Baillargeon, S. N. G. Chu, and A. L. Hutchinson for their collaboration in this work. One of us~F.C.! is grateful to Professor Giovanni Jona-Lasinio for the invitation to contribute to this journalissue, for past collaborations, and for having brought him in touch with the community of math-ematical physicists interested in the problems discussed in this issue.

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1-mesoscopic phenomena in semiconductor nanostructures by quantum design.pdf

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multiply reected waves

digitally graded regions

reduced spatial overlap

nato asi series

perpendicular kinetic energy

quantum cascade lasers

quantum cascade laser

transition matrix elements