Theory and Application of Nanomaterialsahrenkiel.sdsmt.edu/.../6_QuantumNanostructures/...(0-D). As...

Outline The Quantum Box Quantum Statistics Density of States

Theory and Application of NanomaterialsLecture 6: Engineered Quantum Nanostructures

S. Smith

SDSMT, Nano SE

FA17: 8/25-12/8/17

S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 1 / 15


Introduction

Quantum dots are semiconductor nano structures with tunable absorption and emissionproperties, which are dictated by choice of composition and tailoring their size in thenanometer scale.

Figure: Quantum dots available from Sigma Aldrich1.

1http://www.sigmaaldrich.com/S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 2 / 15


Outline

1 The Quantum BoxLow-Dimensional NanostructuresSeparation of variablesRelation to low-dimensional nano structures

2 Quantum StatisticsIndistinguishable particlesFermi EnergyFermi Function

3 Density of StatesWavefunction in a periodic solidReciprocal or K-spaceDOS in low-dimensional nano structures



Low-Dimensional Nanostructures

The Quantum Box: Consider a box with walls of infinite potential, this is the 3-Danalogue of the 1-D particle in the box, as shown in the leftmost portion of figure 1:

Figure: The quantum box of size Lx × Ly × Lz . Each plane represents an infinitepotential on the walls of the box. In the limit that each dimensionLi −→∞ (i = x , y , z), we arrive at effective 2D (quantum well), 1D (quantumwire), and 0D (quantum dot) nanostructures.



Solutions to Schrodinger’s equation

Let us solve the 3-dimensional time independent Schrodinger’s equation:

− ~2

2m∇2ψ(r) + V (r)ψ(r) = Eψ(r)

for such a potential. We can employ the method of separation of variables:

ψ(r) = ψx(x)ψy (y)ψz(z)

Inserting this form into the above Schrodinger’s equation and dividing through byψx(x)ψy (y)ψz(z) leads to:

∇2ψx (x)

ψx (x)+∇2ψy (y)

ψy (y)+∇2ψz (z)

ψz (z)+

2m

~2(E − V (r))︸︷︷︸

k2= constantin box

= 0 −→∇2ψx (x)

ψx (x)+∇2ψy (y)

ψy (y)+∇2ψz (z)

ψz (z)+ k2

x + k2y + k2

z = 0

with: k2 = k2x + k2

y + k2z =

2m

~2(E − V (r))



Separation of variables

For each of the above three independent functions ∇2ψi (i)/ψi (i) , i = {x , y , z} toalways sum to a constant, then each of these terms must itself be equal to a constant,which we define as k2

i , i = {x , y , z}. This leads to three constituent equations forψx(x), ψy (y), and ψz(z):

∇2ψx (x) +k2xψx (x) = 0, ∇2ψy (y) +k2

yψy (y) = 0, and ∇2ψz (z) +k2zψz (z) = 0

Solving each of these equations in an identical way as we did for the 1-dimensionalproblem, we find that the total energy of the particle is the sum of the Eigen-energies foreach of these constituent functions ψn(x), ψm(y), and ψl(z):

En,m,l =~2π2

2m(n2

L2x

+m2

L2y

+l2

L2z

)

Where the quantum numbers n, m and l appear in an identical way as for the1-dimensional infinite potential well.



Relation to low-dimensional nano structures

One can see from this equation that as any of the three sides of the box approachesinfinity, the contribution of the term inversely proportional to that dimension to the totalenergy of the particle becomes vanishingly small. As shown in the figure, this leads tothe notion of ”effective” dimensionality, as the particle becomes more and more free tomove, and so is effectively constrained to only a plane (2-D), a line (1-D), or a point(0-D). As shown in the figure, such nanostructures are referred to as quantum wells,quantum wires, and quantum dots, respectively.

Figure: The quantum box of size Lx × Ly × Lz . Each plane represents an infinitepotential on the walls of the box. In the limit that each dimensionLi −→∞ (i = x , y , z), we arrive at effective 2D (quantum well), 1D (quantumwire), and 0D (quantum dot) nanostructures.



Indistinguishability of quantum particles

Based on the principle that quantum particles are identical, a quantum description oftwo or more particles must not be biased towards the identity of any individual particle.This leads to the notion of parity, and ultimately to two classifications for all particles:the so-called Bosons and Fermions, whose wavefuntions have either even or odd parityunder exchange of any two particles. The generalization of this principle to manyparticles leads to the notion of quantum statistics.

Figure: Fermi, Bose and Maxwell distribution functions.



Fermi Energy

The distinguishing characteristic of these statistical descriptions of many particles, andthe most salient point to the discussion of electrons, which are Fermions, is: no twoFermions may occupy the same quantum state. Thus, for electrons, (with twoenergetically equivalent spin states, designated by spin quantum number S = ±1/2), theparticle in the box states would be filled with electrons as shown in figure 2 below:

Figure: Electrons would fill the finite potential well as shown, with only twopossible spin states per energy level. The energy required to add a particle to thesystem is known as the Fermi energy, or Ef .



Fermi Function

In solids there are many states available, on the order of Avagadro’s number, and thusthe Fermi function f (E), defined in the figure, describes the probability that a givenquantum state is occupied. At finite temperature, there is a probability that some statesabove Ef , but within the thermal energy, will be occupied. This means some statesbelow Ef must be empty, as shown in the figure. At absolute zero of temperature, theFermi function is a step-function which terminates at the Fermi energy.

Figure: The Fermi function f (E ) is defined as the probability that a state withenergy E is occupied, and is generally 1 for states below Ef and 0 for statesabove Ef . States within kT of Ef may be empty of filled with some probabilitybetween 0 and 1.



Density of States in a solid

Bloch Functions: We saw that the number of states in the particle in a box problemdepends on the barrier height and the size of the quantum well. Each state was labeledby at least one quantum number (e.g. n, m and l in the quantum box). In solids, thereare Avagodro’s number of nuclei and electrons. Due to the translational symmetry, theappropriate labels are the nearly-continuous range of k-states, as the wave function ofan electron in a solid obeys the following Eigen-value equation:

Hψk(r) = Ekψk(r)

That is, the quantum labels corresponding to the Eignestates and Eigen-energies of theHamiltonian are now the nearly-continuous range of k-vectors: k ∈ { 2nπ

Na} where

−N/2 < n ≤ N/2 and N is the number of nuclei in the crystal (more precisely thenumber of unit cells). The latter is a direct consequence of Bloch’s Theorem, whichposits that the wavefunction in a periodic potential can be expressed as a plane wavestate:

ψk(r) = uk(r)e ik·r

from which the phase equivalence e ik·R = e i(k+G)·R, where e iG·R ≡ 1, means that uniquevalues of k fall between −π

aand π

a.



K-space

The range of k ∈ {−πa, πa} is known as the first Brillouin zone, which we will discuss in more

detail later. While all the “partners” of k formed by adding any G to k satisfy Bloch’s theoremequivalently, the energy of each of these waves is in general not equal:

Ek 6= Ek+G

where G = n 2πax, where n = ± an integer, is known as the reciprocal lattice vector (for a one

dimensional lattice). Thus, for a given value of k, there is an infinite set of “partner” k -vectors, which can be represented graphically by defining what we call k-space.

Figure: Denisty of points in k-space determines the number of quantum states ina solid. All states with energy less than Ef are filled, corresponding to states with|k| ≤ kf .



DOS in low-dimensional nano structuresFor a free electron, since all the energy is kinetic, and has the form:

Ek =~2k2

2m

We can invert this equation to arrive at the value of k which corresponds to the Fermi energy:

kF =

√2mEf

~2

and we can draw a sphere in k-space which encloses all states with energy E ≤ Ef :

Figure: Denisty of points in k-space determines the number of quantum states ina solid. All states with energy less than Ef are filled, corresponding to states with|k| ≤ kf .



DOS in low-dimensional nano structuresWe can thus calculate the total number of states up to the Fermi energy by computing the volume of the spheroid of radius kf

and dividing by(2π)n

an(where in 3 dimensions, n=3):

N(E) = 24/3πk3

f

(2π/a)3= 2

4/3π( 2m~2 Ef )3/2

(2π/a)3=

V

3π2

(2m

~2

)3/2

E3/2

where we multiplied by 2 to account for the two equivalent spin states S = ±1/2. It is often more important to know thenumber of states available at the variable energy E , known as the Density of States:

D(E) =dN(E)

dE=

V

2π2

(2m

~2

)3/2√E for 3 dimensions where n = 3

Thus the density of states for electrons free to move in 3 dimensions in a solid scales as the√E . Similar results can be obtained

for n = 0, 1, 2:

n = 3 bulk N(E) =V

3π2

(2m

~2

)3/2

E3/2 D(E) =V

2π2

(2m

~2

)3/2√E (1)

n = 2 quantum well N(E) =2πk2

f

(2π/a)2D(E) =

A

2π

(2m

~2

)(2)

n = 1 quantum wire N(E) =4kf

2π/a=

2a

π

(2m

~2

)1/2

E1/2 D(E) =a

π

(2m

~2

)1/2 1√E

(3)

n = 0 quantum dot N(E) = 2∑

i diθ(E − Ei ) D(E) = 2∑i

diδ(E − Ei ) (4)



DOS in low-dimensional nano structures

Plotting these expressions may help you to visualize the impact of the density of states on thebehavior of quantum confined nanostructures:

Figure: Number of states at energy E ( Nn(E ) ) and density of states ( Dn(E ) )for quantum confined nano-structures of dimensionality n = 0, 1, and 2,corresponding to quantum dots, quantum wires, and quantum wells, respectively.Bulk density of states (n = 3) shown for comparison.


Date post:	26-Jun-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Theory and Application of Nanomaterialsahrenkiel.sdsmt.edu/.../6_QuantumNanostructures/...(0-D). As...

Documents