Quantum Dots: Theory, Application, Synthesis Pranjal Vachaspati∗ Massachusetts Institute of Technology (Dated: May 7, 2013) Semiconductor crystals smaller than about 10 nm, known as quantum dots, have properties that differ from large samples, including a bandgap that becomes larger for smaller particles. These properties create several applications for quantum dots, including efficient solar cells and qubits for quantum computing. Various synthesis procedures have been proposed and implemented, including chemical synthesis and lithographic synthesis similar to that used in the semiconductor industry. I. INTRODUCTION Electronic structure of insulators, conductors, and semiconductors Quantum dots were first characterized in 1983 by No electric field Electric field Brus[1] as small semiconductor spheres in a colloidal sus- pension. When the radius of a semiconductor sphere be- comes small, on the order of a few nanometers, the Bohr radii of the charge carriers become larger than the sphere, and their confinement to the sphere causes their energy Insulator to increase. This effect can be modeled in a number of ways. A straightforward method which we present here treats the E quantum dot as a finite spherical potential well and incor- porates both confinement effects and Coulomb attraction Conductor between charge carriers. The easily tunable bandgap of quantum dots makes them especially suited for optical applications, including LEDs and photovoltaic cells. Additionally, their easily described behavior makes them useful as qubits, since Semiconductor k excited electron spins in individual quantum dots can be FIG. 1. When a solid is placed in an electric field, states precisely controlled electronically. with momentum in the direction of the field become more favorable. In an insulator, the valence band is totally filled, so the average momentum does not change. In a conductor, the II. THEORY valence band is only partially filled, so electrons can occupy more energetically favorable states with a net momentum. In A. Bulk Semiconductor Mechanics a semiconductor, the valence band is full, but a band with only a slight energy gap is accessible, so particles can jump into this band and conduct. The energy structure for a solid can be derived from a periodic-potential model. As seen in Figure1, an insula- tor has a ground state that is totally filled, so applying Calculating the precise conduction band shape of a an electric field does not cause a momentum change. In semiconductor is difficult, and using it can be unwieldy. contrast, a conductor has a partially filled ground state, However, for small momenta, the conduction band can so adding an external electric field causes the electrons be approximated by a parabola. This produces the \ef- to shift into states with an overall non-zero momentum. fective mass approximation" : A semiconductor has a ground state (called the va- ¯h2k2 lence band) like that of an insulator, but the first excited E = (1) state is very close to the ground state. Small excitations 2m∗ can lift electrons into the first excited state, known as where m∗ is called the effective mass of the carrier and the conduction band. This allows the excited electrons depends on the curvature of the band. The effective mass to move in the mostly empty conduction band, and the of an electron is typically between 0:1me and me. unexcited electrons to move in the now slightly empty At some finite temperature T , the carrier wave vectors, valence band. The motion of the unexcited electrons can k, are on average described by also be treated as the movement of a positively charged 2 2 hole, which significantly simplifies calculations. 3kBT ¯h k = (2) 2 2m∗ ∗ For a semiconductor with m = me, the characteristic ∗ [email protected] length scale is 1=k, and is on the order of 10 nm. We Quantum Dots: Theory, Application, Synthesis 2 therefore investigate the behavior of semiconductor crys- tals below this size. Allowed values of kin In general, the Hamiltonian of an electron-hole pair in a large semiconductor is given as 2 2 2 ¯h 2 ¯h 2 e H = − rh − re − (3) 2mh 2me jre − rhj where is the dielectric constant. The first two terms represent the kinetic energies of the hole and the electron respectively, and the third term represents the Coulomb attraction between them. B. Small Semiconductor Crystals We next consider a small spherical semiconductor crys- kin tal with radius R. In a large semiconductor, the excitons may be treated as free-particles and the boundary condi- FIG. 2. Equation 14 is solved graphically to find valid electron tions may be ignored, but for sufficiently small crystals, momenta in the finite spherical well potential. the Hamiltonian in Eq.3 must be modified to take into account the fact that the semiconductor acts as a poten- tial well. The modified Hamiltonian is which simplifies to ¯h2 ¯h2 e2 2 @ @2 2m(E − V (r)) H = − r2 − r2 − + V + V (4) h e e h + = − 2 (r) (7) 2mh 2me 4πj~re − ~rhj r @r @r2 ¯h where Ve and Vh are zero inside the sphere and some For clarity, we write positive constant V , that depends on the medium the 0 s r ∗ crystal is in, outside the sphere. The dielectric constant 2m0(E − V0) 2m E for the crystal is given by . For an infinite well, the con- kout = ; kin = (8) ¯h2 ¯h2 finement energy disregarding the Coulomb term is given h¯2π2n2 by 2mR2 as in a square well, and the electrostatic en- The solutions to Equation7 are Bessel functions. Inside e2 the sphere, the wavefunction must be finite at the origin ergy disregarding the confinement term is of order R . As will be shown, the infinite square well approximation and therefore has the form significantly overestimates the band gap for small quan- sin k r tum dots, so we must consider a finite spherical square (r) = A in ; r < R (9) well instead. kinr The inverse quadratic dependence on R in the confine- Outside, the wavefunction must converge to zero at large ment energy compared to the inverse linear dependence r, so the state has the form in the electrostatic energy means that in the small R limit, the confinement term dominates. Therefore, we e−koutr first calculate the confinement energies and wavefunc- (r) = −B ; r > R (10) k r tions for a finite well, and treat the Coulomb energy as out a perturbation on top of that. The probability current is preserved by [2]: We are calculating the properties of the first excited state, which is spherically symmetric and has no angular 1 d (R) 1 d (R) in = out (11) momentum. Thus, the wavefunction has no dependence ∗ m in(R) dr m0 out(R) dr on the spherical angles. The unperturbed confinement Hamiltonian for an electron or hole is given by Substituting Equations9 and 10 for in and out gives 2 ¯h 2 1 1 1 1 + koutR H = − r + V (r) (5) ∗ − + kin cot(kinR) = − (12) 2m m R m0 R Schr¨odinger'sequation gives Following Equation8, kout is written in terms of kin: ¯h2 1 @ @ (r) r 2 m0 2 V0 − r + V (r) (r) = E (r) (6) kout = k − (13) 2m r2 @r @r m∗ in ¯h2m∗ Quantum Dots: Theory, Application, Synthesis 3 Equation 13 is then substituted into Equation 12: Successive approximations to quantum dot bandgap s ∗ ∗ 2 ∗ 12 m m 2 2 V0R m kinR cot(kinR) = 1 − − R kin + 2 Infinite well m0 m0 ¯h 10 Finite well (14) Experimental Equation 14 can be solved numerically (Figure2) to find 8 valid values of kin and therefore the wavefunctions and energy spectrum of the confined electron and hole. (eV) 6 We can account for the effect of electron-hole interac- E tions by applying perturbation theory. The perturbation ∆ 4 to the Hamiltonian is given by 2 e2 δH = (15) 4πjre − rhj 0 0 1 2 3 4 5 so the first-order perturbation to the energy is given by Radius (nm) e2 δE = h e0; e1j j e0; e1i (16) 4πj~re − ~rhj FIG. 3. The infinite square well approximation diverges sig- nificantly from the finite square well approximation and ex- perimentally observed energies for small quantum dots. Ex- 2 Z 2 2 perimental data from [3]. e h(~rh) e(~re) 2 2 δE = sin θh sin θer r drdθdφ 4π p 2 2 h e re + rh − 2~re · ~rh (17) This integral does not have a simple analytic solution, and must be evaluated numerically. In the large V limit, 0 Effect of Coulomb perturbation 1:8e2 2.5 δE ≈ : (18) 4πR Finite well Finite well w/Coulomb This perturbation becomes less significant as the dots 2 Experimental become larger and the confinement potential V0 becomes weaker. 1.5 (eV) E 1 C. Comparison of Experimental and Theoretical ∆ Results 0.5 To validate this theory, we compare successive levels of approximation to experimental results. For cadmium sul- 0 fide dots in an oleic acid solution, the confinement poten- 0 1 2 3 4 5 tial V0 is 5 eV, the effective electron mass is 0:18me, and Radius (nm) the effective hole mass is 0:54me. The infinite square well approximation (Figure3) closely follows the finite square FIG. 4. The Coulomb perturbation is small for CdS, since well approximation and experimental results for dots the confinement potential (V0 = 1:25eV) is relatively small. larger than about 2 nm, but diverges sharply for larger It is roughly ten percent the size of the confinement energy. dots, overestimating the bandgap change for 0:5 nm dots Experimental data from [3]. by a factor of five. The unperturbed finite well approximation (Figure4) works well for all CdS dot sizes for which experimen- tal data was available.