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 semiconductors are quantum wells. When semiconductor structures, we should begin from semiconductors are fabricated in sufficiently thin some basic concepts. layers, quantum interference effects begin to From electron transfer theory, the electrons 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 “particle in a box.” 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<x<L, and infinitely high potential illustrated in Fig.1 [1]. In such a quantum well barriers prevent it from straying beyond this heterostructure, the confinement can change the region. Note that L is the full width of the well optical adsorption from the smooth function, as in here. a bulk material, to a series of steps. The confinement can also increase the binding energy of excitons, resulting in exceptionally clear excitonic resonances at room temperature in quantum well heterostructures. Thus, quantum well hetero-structures are key components of many electronic and optoelectronic devices, because they can increase the strength of electro- optical interactions by confining the carriers to small regions. FIG. 2 Energy levels and wave functions for an infinitely deep square potential well 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. 11 JOURNAL OF CHEMISTRY 571 VOLUME 1, NUMBER 1 DECEMBER 28, 2005 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 wave function for this potential; (c) The first excited- state wave function. Where n is a quantum number 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 12 JOURNAL OF CHEMISTRY 571 VOLUME 1, NUMBER 1 DECEMBER 28, 2005 > L/2) one increasing to the right, the other derivative, so the energy levels are quantized for decreasing, E<V0. The particular values of E are given by: 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 bound state. 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 E<V . For E>V , 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 E<V0, however, this time, we wave function in the wall—at x = L/2, so cannot set A as zero. With this additional constant, the energy E need not be restricted to obtain cos(kL / 2) =+ AeLL/2 Be /2 properly behaved wave functions, because A and (18) B can match each other. Therefore, all energies kkLAeBesin( / 2) = LL/2 /2 above V0 will be allowed and states with E>V0 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 band gap determines the energy of other to give: light emitted and absorbed, which is essential in the laser/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]. 13 JOURNAL OF CHEMISTRY 571 VOLUME 1, NUMBER 1 DECEMBER 28, 2005 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.