Nanomaterials for Photovoltaics (v11) 11 Quantum Confinement. Quantum confinement (I) The bandgap of a nanoparticle increases as the size decreases.

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1 Quantum Confinement Quantum confinement (I) The bandgap of a nanoparticle increases as the size decreases. A rule of thumb is E g d where d is the particle size. Quantum confinement (II) The Schrodinger equation in -D is d V md E We can write this as d V E md Alternatively d m E V d We define 4 Now m K E 4 U m V,

2 d K U 4 d Quantum confinement (III) Consider a particle in the vicinity of an energy "bo". The potential varies as U, L L U, L Solutions inside the bo must satisfy So d d 4 k ik ik A e B e We define K U k. Outside the bo d 4 K d We will find a discrete set of bound states with E, and continuous set of free states with E. Quantum confinement (IV) First consider the bounds states ( E, K C L K e, K e, ). outside the bo D L If the states are bound, they must be normalizable, so lim( ). The wave function must satisfy the boundary conditions that both the wave funciotn and its derivative are continuous at the bo edges d L continuous and d Quantum confinement (V) Applying the boundary conditions L continuous

3 3 ikl ikl KL Ae Be Ce ikl ikl e e KL ik A B K C e ikl ikl KL Ae Be De ikl ikl e e KL ik A B K D e Let's define e ikl and Kk. The above reduce to two equations A Bi A B AB i AB Quantum confinement (VI) We can evaluate B A AB i B A AB Now define r B A and y. ry yr ry yr This gives r, so r and B A. The solutions can be classified by their parity. Even-parity solutions have AB A A cos k, L Odd-parity solutions have B A B i B sin k, L Quantum confinement (VII) Now we need to find the allowed values of wavenumber k by applying the boundary conditions. For the even (+) solutions KL Acos kl De sin KL ka kl K D e Combining these gives a condition for thee solutions f k ksin kl Kcos kl k sin kl where k U and atn K k We see that the even solutions satisfy kl n, where n. For odd (-) solutions KL Bsin kl De kbcosklk D e KL

4 4 Combining gives f k Ksin kl kcos kl k coskl Now kl n, where n. Using K k k we find k atn k Both types solutions involve conditions on the function atn k Quantum confinement (VIII) The even states satisfy sink and odd states satisfy cosk. Below we plot these functions kl for various values of kl. Solutions are found where the oscillations cross through zero. k k kl Quantum confinement (IX) We can eamine how the bound state energies vary with bo depth and size.

5 5 Quantum confinement (X)

6 6 Quantum confinement (XI) Once we have found the allowed energies for a particular bo, we can plot the wave functions in the vicinty of the bo. We pick AB for simplicity. Then the even solutions are KL cos kl e, L cos k, L KL cos kl e, L The odd solutions are KL sin kl e, L sin k, L KL sin kl e, L Assuming UL, we have the seven bound states shown below o

7 7 Quantum confinement (XII) The even states satisfy k kl tan k It is interesting to consider the limit in which the bo width is zero ( L ), but the depth is infinite ( U ), keeping the product the finite ( UL (finite) ]. Now kl kl KL k LL K L L K L We must have tan klkl k L, so kl L and K K L K L K L k k L k LK L We see that K. So there is eactly one eigenstate with energy 4 K h E m m

Lecture 10: The Schrödinger Equation Lecture 10, p 1

Lecture 10: The Schrödinger Equation Lecture 10, p 1 Overview Probability distributions Schrödinger s Equation Particle in a Bo Matter waves in an infinite square well Quantized energy levels y() U= n=1