JOURNAL OF CHEMISTRY 571 VOLUME 1, NUMBER 1 DECEMBER 28, 2005

Quantum wells

Meng Guo Department of Chemistry, University of Michigan, 930 N. University Ave., Ann Arbor, MI 48109

(Received November 29, 2005; accepted December 12, 2005)

It is well known that electronic and optical In order to get a good sense of the quantum properties can be altered by using heterostructures well heterostructures, in addition to the electronic and the most widely used heterostructures in and optoelectronic properties of such are quantum wells. When structures, we should begin from semiconductors are fabricated in sufficiently thin some basic concepts. layers, quantum interference effects begin to From transfer theory, the appear prominently in the motion of the electrons, are restricted to a finite region of space, denoted a as a result of the quantum confinement of the “quantum well” or a “.” The carriers in the resulting one-dimensional potential simplest example is the infinitely deep “square” wells. The quantum well, in which a single layer well and is illustrated in Fig. 2. Within this well, of one narrow-gap semiconductor is sandwiched the electron has zero potential energy in the between two layers of a wider-gap material, is region 0

FIG. 2 Energy levels and wave functions for an infinitely deep square

The Schrödinger equation for motion inside the well is identical to that for free space: 22d   h = Ex () (1) 2mdx2

The boundary conditions are FIG. 1 Energy-band profile of a structure containing three quantum wells, showing the confined states in each well.  = 0 for x = 0 (2) The structure consists of GaAs wells of thickness 11, 8, and  = 0 for x = L (3) 5 nm in Al0.4Ga0.6As barrier layers. The gaps in the lines indicating the confined state energies show the locations of nodes of the corresponding wavefunctions.

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We could use real trigonometric functions and 1/2 2  n write  n (xx )=  sin( ) (12) LL  = Akxsin( ) (4) In fact, an infinitely deep well cannot be made, Where but it has become routine to grow structures 1/2 that are close to ideal finite wells. Now we kmE= (2e ) / h (5) consider how Fig. 2 changes if the potential at and me is the mass of the electron. the walls is not infinite. Substitution shows that the function is a solution with (a) 22 h k Ek=   0 () (6) II III 2m I V0 Hence, kL = n, (7) That is: X= -L/2 X= L/2

22kn 2  E ==hh ()2 (8) 22mmLee

Or (b) (c) 222n FIG. 3 (a) Potential energy for a particle in a one-  h dimensional finite rectangular well; (b) The ground-state EE==n 2 (9) 2mLe for this potential; (c) The first excited- state wave function. Where n is a with the value of any positive integer. As we have solved the above wave function and corresponding energy value in It will turn out to be convenient to have the our first problem set, so I will skip some details in origin at the center of the well, so we take the above calculation. The normalization constant, A, can be found from the probability of finding a V(x) = V0 for x < -L/2 (13) particle in the box: 22 V(x) = 0 for -L/2 < x < L/2 (14)  L  n  ()xdx== A sin( xdx ) 1 (10)  0 L V(x) = V0 for L/2 < x (15) This condition means that the particle is localized There are two cases to examine, depending on within the potential well, so that the probability of whether the particle’s energy E is greater than or finding the particle in the potential well is equal smaller than the potential V . to unity. 0 The Schrödinger equation for the barrier region I and III becomes: LLL222 nn2 A  n Asin( x ) dx== A sin( x ) dx [1 cos(2 x )] dx = 1 LL2 L22 000 h   ()x (11) + ()()0EV 0  x= (16) 2m x2 2 Using above equation, we obtain A = . L This is a linear homogenous differential equation with constant coefficients, and there are two Hence, exponential solutions outside the well (say, for x

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> L/2) one increasing to the right, the other derivative, so the energy levels are quantized for decreasing, E

 x  x 1/2 2 1/2 1/2 e And e , where (2E V00 )sin[(2 mE ) L /hh ]= 2( V E E ) cos[(2 mE ) L / ] (20) 2  = 2(mV E )/ (17) 0 h Here, we are not going to talk in detail about the derivation of above equation. You can try it if you The wave function outside the well is: are interested.  =+Aexx Be , where A,B are constants. We Let’s go back to the Fig.3, which shows the are assuming here that E < V0, so the particle is wave function for the lowest two energy levels. bound to the well. We shall find this is always The wave function is oscillatory inside the well true for the lowest energy state. Let us try to and dies off exponentially outside the well. It construct the wave function for the energy E turns out that the number of nodes increases by corresponding to this lowest . From one for each higher level. And states with E < V0 the equation with V0 = 0, the wave function inside are bound. the well (let's assume it's symmetric for now) is So far, we have considered only states with 2 EV , the quantity VE becomes proportional to coskx, wherekmE= 2/h . 0 0 0  The wave function and its derivative inside the imaginary. By repeating the same procedures as well must match a sum of exponential terms—the for the case of EV0 are unbound. For the particles in an infinitely deep The only exponential wave function that makes well, all the states are bound. sense is the one for which A is exactly zero. It can Currently, there is great scientific and also be confirmed because the quantity VE0  technological interest in quantization effects in is a real, positive number, and to keep the wave semiconductor structures. The underlying reason function far away from the wall finite, we must is that the optical, electrical and photoredox have A=0. Thus there is only a decreasing wave properties of semiconductors can be tuned and in the wall. Requiring the decreasing wave manipulated in fascinating ways by controlling function, A = 0, means that only a discrete set of dimension, rather than by controlling composition values of k, or E, satisfy the boundary condition alone. Interest in the alloying of two materials to equations above. They are most simply found by reach an objective is due to the fact that: 1) taking A = 0 and dividing one equation by the because the determines the energy of other to give: light emitted and absorbed, which is essential in the /detector area, alloying can create a tan(kL / 2)=  / k (19) desired band gap; 2) it is possible to create a material with a proper lattice constant to match or mismatch with an available substrate. The widely This can be solved graphically by plotting the two studied semiconductor heterostructures are sides as functions of k and find where the curves GaAs/AlAs largely because GaAs and AlAs are intersect. nearly lattice matched so that the alloy can be We can also see from the above equations that grown on GaAs substrate without strain energy only the particular values of E give a wave build-up. Here we will examine the case of function that is continuous and has a continuous GaAlAs as an example [2].

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unit of m/s). In GaAs the effective mass me is 22 * 0.067 m0. Thus  0 ()kE=+c h k /2 m. The sandwich acts like a quantum well because Ec is higher in AlGaAs than in GaAs, and the difference Ec provides the barrier that confines

the electrons. Typically Eevc 0.2  0.3 , a value that is not large. However, we shall approximate it as infinite to find the energy levels in a well of width a. Adapting equation shows that the energy of the bound states, labeled with ne, is

FIG. 4 A schematic of a quantum well formed for the 222 n EGaAs h e (22) electron and holes in a heterostructure  ene  c + *2 2ma The Schrödinger equation for the electron states in the quantum well can be written in a Semiconductors have energy levels in other bands. simple approximation as: The most important of these is the valence band which lies below the conduction band. The top of this is at Ev and the band curves downwards as a 22d   h +=Vx() Ex () (21) function of k, giving  ()kE=  22 k /2 m *', 2mdx*2 vvh which contains another effective mass m*’ *’ Where m* is the effective mass of the electron, (m =0.5m0 in GaAs). The conduction and with a unit of kg. valence bands are separated by an energy called We can obtain exactly the same results as the band gap given by EEEgcv=  . Because Ev discussed above; we just need to replace the mass is at a different level in GaAs well and AlGaAs of a particle by the effective mass of the electron. barriers, there is a quantum well. The energies of We can see that each level of energy is actually a the bound states are subband due to the electron energy in the x-y plane. The subband structure has important 222 n  EGaAs h h (23) consequences for optical and transport properties hnh v 2ma*' 2 of heterostructures. However, the (DOS) of the electronic band will be an important In a pure semiconductor at zero temperature, the manifestation of this subband. Now, we will use valence band is completely filled, and the electron transfer theory to study the quantum [3] conduction band is completely empty. Optical wells . absorption excites an electron from the valence First, we look at the behavior of the electrons. band into the conduction band, and an empty state 22 Free electrons have energy  00()kkm= h /2 (m0 or “hole” will be left in the valence band. is the bulk mass). Electrons in a semiconductor Now we will look at the quantum well [4]. The are in the conduction band, and this fact leads to lowest energy at which absorption can occur is changes in their energy in two ways. First, energy given by the difference in energy eh11 must be measured from the bottom of the band at between the lowest state in the well in the Ec rather than from zero. Second, electrons conduction band and the valence band. behave as though their mass is m*, Absorption can occur at higher energies by using Where the effective mass is defined by other states, and strong transitions occur between analogy with Newton’s second law of motion: corresponding states in the two bands. dv Fm= * (v is the velocity of electron with the dt

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If the wells were really infinitely deep there When the active region is made of a material with would be an infinite series of lines with a narrower gap than that of the material through frequencies given by equation: which radiation is emitted, loss of light in passive regions of the structure is reduced, and the loss is 222 222 222 GaAshhhnnn GaAs GaAs 11 lowered by carrier recombination in emitters. DwEnenhnc=  =+()()* 2 E v *' 2=+ E g () * + *' 22ma m a 2 m m These advantages were first actualized in AlGaAs (24) heterostructure LEDs, in which a nearly 100%

internal quantum efficiency of radiative The barriers in the semiconductor are finite, and recombination was achieved. absorption occurs in the AlGaAs barriers for all AlGaAs The , where quantum well frequencies where hwE> g and it is heterostructures are always employed, acts the AlGaAs same way as LEDs. However, the laser structure impossible for adsorption for hwE< g . So there is a discontinuity in the absorption spectrum. is designed to create an optical “cavity” which And there are two discrete lines produced by can guide the photons generated. An important transitions between states in the quantum well parameter of the laser cavity is the optical lying between these two frequencies, confinement factor , which gives the fraction of the optical wave in the active region. This EwEAlGaAs GaAs . gg<

Applications: Light Emitters

Heterostructures have multiple advantages over homo-p-n-junctions, regarding the parameters of light emitting diodes (LEDs) and laser diodes (LDs). The LEDs and LDs are both very important technology devices with FIG. 5 Forbidden gap width in AlGaAs/GaAs application in displays and communications, heterostructure LEDs of three types and schematic of light including optical communications, optical reflection and ''re-emission'' processes. memories, optical logic and lighting. The advances in these areas are coming from low- Heterostructure Solar Cells and Photo- dimensional systems (quantum wells for example) detectors and new materials. The LED is essentially a forward biased p-n diode. Electrons and holes are AIIIBV heterostructure solar cells can be used injected as minority carriers across the diode to produce solar batteries with higher efficiency junction and they recombine either by radiative and radiation hardness. An important advantage recombination or non-radiative recombination.

15 JOURNAL OF CHEMISTRY 571 VOLUME 1, NUMBER 1 DECEMBER 28, 2005 of heterostructure photoelectric converters is their ability to efficiently convert strongly concentrated (up to 1000–2000-fold) sunlight, which opens up possibilities of reducing significantly (in proportion to the extent of concentration) the area and cost of solar cells and, as a result, making ''solar'' electric power cheaper. MOCVD-grown heterostructure solar cells with built-in Bragg mirrors are of particular interest and AlGaAs/GaAs heterostructure solar cells are widely used in space due to their higher efficiency and improved radiation hardness. AIIIBV heterostructure materials and their solid solutions are widely used now in new types of , GaAs/AlGaAs-based for the spectral region 0.3–0.9 μm range, InP/InGaAsP- based for the 1.0-1.6 μm range, and the GaInSbAs system for longer wavelengths (2.0–6.0 μm). It was shown that the use of in photodetectors not only enable the control over their spectral sensitivity region, but also makes it possible to increase the quantum efficiency and speed of photodetectors by bringing into coincidence the regions of light absorption and separation of nonequilibrium carriers, to reduce dark currents, and, in some cases, to control impact ionization processes. We can expect the further exploring of the optical and magnetic properties the quantum well heterostructure could bring to us.

[1] R. Dingle, W. Wiegmann, and C. H. Henry, Phys. Rev. Lett. 33, 827 (1974). [2] Jasprit Singh, Electronic and Optoelectronic properties of semiconductor structures, Cambridge University Press 2003, UK. [3] R.J. Dwayne Miller, George L. Mclendon, Authur J. Nozik, Wolfgang Schmickler, Frank Willig, Surface Electron Transfer Processes, VCH publishers, 1995, U.S.A. [4] John H. Davies, The physics of low- dimensional semiconductors- An introduction, Cambridge University Press, 1998, New York, U.S.A.

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