Electronic Structure And Intraband Health And Social Care Essay

carrier relaxation in hot carrier solar cells

J.P. Trinastic

Department of Physics and Quantum Theory Project,

University of Florida, Gainesville, Florida, 32611, USA

(Dated: March 6, 2013)

Abstract

Third-generation solar cells promise a higher e#ciency compared to

rst generation and second

generation thin

lm cells. In particular, hot carrier solar cells provide a pathway to high e#ciency

without increased device complication if optimal materials can be designed with slow hot carrier

cooling. In this proposal, I discuss the potential for graphene quantum dots and nanoribbons

embedded in a hexagonal boron nitride matrix to act as hot carrier photoabsorbers. Density

functional theory is implemented to calculate the electronic structure of these systems. In addition,

the theory of energy dissipation is discussed that models irreversible energy

system due to coupling with the phonon reservoir. Computational methods are used to calculate

the relaxation rate of an initially photoexcited state via electron-phonon coupling that provides

a measure of hot carrier cooling in the systems of interest. Preliminary results demonstrating

the methodology are presented examining photoexcited carrier relaxation in graphene on a silicon

carbide (3C-SiC) surface.

1

I. INTRODUCTION

A. Third-generation photovoltaics

Global oil production is expected to peak and begin declining around 2020, providing a

rough timeline for a transition from fossil fuels to non-conventional energy sources [1]. Of

the potential options for next-generation sources, e.g., solar, wind, geothermal, biomass,

and nuclear, solar energy is a leading candidate due to the enormous amount of energy

available in solar radiation (4.3x1020 J/hour) compared to other sources [2]. However, recent

calculations indicate that current commercial solar cell e#ciencies, de

ned as maximum

generated power divided by incident power, lead to electricity prices around $0.35 (kW-

hr)??1 compared to $0.02-0.05 (kW-hr)??1 for fossil fuels [2]. Therefore, both improvements

in e#ciency and manufacturing costs are mandatory before solar cells become a commercially

viable renewable energy option.

For a traditional single junction solar cell, the meeting of interfaces of a p-doped and n-

doped material creates a depletion region that is negatively charged in the p-type material

and positively charged in the n-type region. The Fermi level must match across both regions,

leading to a bending of the conduction (CB) and valence (VB) bands within the depletion

region and a resulting electric

eld [3]. Quasi Fermi levels can still be de

ned for the charge

carrier populations in each semiconductor separately. When the p-n junction is connected

to an external circuit, photoexcited electrons and the resulting holes in the depletion region

move in opposite directions under the in

eld. The photocurrent density

(JPC) can be de

ned as

JPC = q

R

EG

b(E)QE(E)dE (I.A.1),

where q is the electron charge, b(E) is the spectral photon

energy, and QE(E) is the quantum e#ciency (QE=1 when each incident photon generates

one electron to the external circuit) [4]. The maximum possible photovoltage for a given

p-n junction is determined by di#erence in the quasi-Fermi levels of the p- and n-regions.

Single junction cells, traditionally made of doped silicon or germanium, presently dom-

inate commercial markets. Known as

rst-generation solar cells, they have a theoretical

limiting e#ency of around 33% based on analyses done by Shockley and Queisser [5]. This

e#ciency limit results from several considerations: 1) blackbody radiation of the solar cell

reduces the total possible energy capture; 2) radiative electron-hole recombination places an

2

upper limit on photoexcited carrier production; 3) all photons with sub-band gap energy

do not contribute to electrical energy conversion; 4) each absorbed photon excites a single

electron; 5) each excited electron (and corresponding hole) relaxes within the CB (VB) to

the band edge prior to extraction; ; and 6) non-radiative recombination is ignored. Non-

radiative recombination can occur due to defects that generate impurity states within the

gap (Shockley-Read-Hall recombination), electron-hole recombination that transfers energy

to another electron (Auger recombination), and surface recombination. Due to the assump-

tion that carriers relax to the band edge (5), the band gap size also plays an important

role, as a very narrow gap leads to high photocurrent but low photovoltage whereas the

reverse is true for a wider gap. For a single junction solar cell under the Shockley-Queisser

assumptions, a 1.3 eV band gap provides the optimal 33% e#ciency [5].

Decades of research into

rst-generation solar cells have optimized their manufacturing

methods and researchers have achieved practical e#encies up to 28% [6], but the high ma-

terial costs of the necessarily pure silicon limits the possibility of continued cost decreases.

As a possible solution, second-generation solar cells that use thin

lms several microme-

ters thick have spurred substantial research interest. These second-generation solar cells are

typically made of copper indium gallum selenide (CIGS), cadium telluride (CdTe), or micro-

morphous silicon [7{9]. Thin

lm solar cells decrease costs by increasing the manufacturing

unit size compared to

rst-generation cells, however e#ciency will likely maximize around

15%, making it di#cult to compete with fossil fuels at comparable electricity costs [10].

As an approach to circumvent the e#ency limits plaguing

rst- and second-generation

solar cells, third-generation cells challenge several of the Shockley-Queisser assumptions

and provide a pathway for solar cells to reach market viability. Third-generation cells are

similar only in that their theroetical models all predict maximum e#ciencies higher than the

Shockley and Queisser limit. Blackbody radiation and radiative carrier recombination will

always bracket e#ciency limits, however each third-generation design attempts to improve

one of the other cell limitations.

One alternative, the tandem cell, aims to eliminate loss mechanisms due to carrier relax-

ation by stacking cells of increasingly larger band gaps [11]. Since electrons are still assumed

to relax to the band edge prior to extraction, each cell then extracts electrons at a speci

c

band gap energy. The combined cell results in an overall higher e#ciency due to the wider

possible range of photoexcited electron energies. These cells can reach theoretical e#ciences

3

as high as 68% using four-stack cells [12], however the increased amount of materials required

to reach these e#ciencies limits their potential cost reductions.

Intermediate band (IB), multiple exciton generation (MEG), and hot carrier (HC) solar

cells are more promising third-generation designs because device fabrication would be less

costly than tandem cells once optimial materials have been determined. Intermediate band

designs use wide band gap semiconductors with deep level impurities to allow transitions at a

wider range of photon energies. Photoexcitations can then occur from VB to IB, IB to CB, or

VB to CB. If carriers are only extracted from the VB or CB, this increases the photovoltage

due to the large band gap without sacri

cing photocurrrent, leading to e#ciencies as high

as 66% [13]. Multiple exciton generation (MEG) cells rely on the generation of multiple

photoexcited carrier from a single photon (QE > 1 in Eq I.A.1), a mechanism recently

achieved in quantum dots [10, 14] and in graphene [15]. Although both of these technologies

show promise, this proposal will focus on HC solar cells and use computational methods to

assess the electronic structure and carrier relaxation in novel materials to determine their

potential use.

B. Hot carrier solar cells

Hot carrier cells challenge the Shockley-Queisser assumption that carriers must relax to

the band edge prior to extraction to the external circuit [16]. In a traditional solar cell,

photoexcited electrons undergo a series of relaxations to the band edge. Initially, electrons

form a peaked excited distribution in the CB, however elastic electron-electron interactions

broaden the distribution on the femtosecond (fs) timescale. This elastic interaction con-

serves the total energy of the photoexcited carriers, and thus no photoexcitation energy is

lost. However, subsequent inelastic interactions with high-energy optical phonons on a fs to

picosecond (ps) time scale transfer energy from electrons to phonons through lattice vibra-

tions, and the HC distribution shifts to lower energies near the band gap. In this process,

the electrons then lose energy equal to the di#erence between their initial excitation energy

and the conduction band edge, thus decreasing photovoltage and overall cell e#ciency [11].

Hot carrier cells aim to extract electrons prior to this phonon-induced cooling phase by de-

signing materials that either have low electron-phonon coupling or a large phonon band gap

between the acoustic and optical phonon spectrum [17]. Since carrier are extracted prior to

relaxing to the band edge, ideal absorber materials would also exhibit a minimal electronic

4

band gap around 0.5 eV to allow maximum photoabsorbance [18].

The most common HC cooling mechanism in semiconductors is the interaction of an ex-

cited electron with a high energy optical phonon that results in two longitudinal acoustic

(LA) phonons of equal energy and equal and opposite momenta, known as the Klemens

mechanism [19]. If the phononic band gap between the highest LA phonon and the lowest

optical phonon is greater than the optical phonon energy, the Klemens mechanism is in-

hibited and electron cooling signi

cantly reduced. Density functional theory (DFT) studies

have explored possible materials meeting the small electronic and large phononic band gap

requirements, usually a result of the large di#erence between cation and anion masses. As

an example, indium nitride (InN) meets both requirements and has also shown a slower

HC cooling rate compared to gallium arsenide (GaAs) in experiment [20, 21]. However,

most optimal materials require rare elements (e.g., In) that make large scale applications

untenable [17].

As an alternative, recent studies [14, 22{24] have applied quantum dots (QD) as HC

photabsorbers that use more earth-abundant elements. In a traditional semiconductor, a

minimum photoexcitation greater than the material's band gap creates an electron-hole pair,

called an exciton. These carriers, being oppositely charged, feel a weak Coulomb attraction

due to dielectric screening proportional to the material's dielectric constant. The distance

between the electron and hole is a material-dependent quality known as the exciton bohr

radius, de

ned as

re = #h2#

e2 ( 1

me

+ 1

mh

) (I.B.1),

where # is a material's dielectric constant and me and mh are the e#ective masses of

the electron and hole, respectively [25]. If a semiconductoR2s radius R is now reduced to

be comparable to re, the con

nement of the charge carriers leads to discrete energy levels

akin to a particle in a box. Con

nement leads to an increase in the charge carriers2 kinetic

energy, and when the radius is small enough (R # 2re), the kinetic energy contributions

dominate and electron-hole correlation e#ects can be neglected [25]. This discretization

of energy levels provides a robust environment to slow hot carrier cooling due to minimal

overlap between electronic states. In addition, when QDs are embedded in a host matrix

material, the weak coupling of phonons at the interface between the matrix and embedded

QD lead to miniature gaps in the phonon density of states (DOS) that can further prevent

electron-phonon coupling. [17].

5

An important secondary component of the HC solar cell is an energy selective contact

(ESC) that extracts carriers only at the average energy of the HC distribution. This allows

hot carriers to cool isentropically in the ESC prior to moving to the external circuit, pre-

venting energy loss that destroys the hot carrier e#ect [26]. Initial ideas for such a contact

have been wide band gap semiconductors with narrow conduction and valence bands or QDs

that provide a resonant tunneling mechanism for electron extraction [27].

Recent experiments have demonstrated hot carriers in a variety of QDs, including lead

su

(CdSe) [24]. In addition, hot electron transfer has been shown from lead selenide (PbSe)

nanocrystals to a titanium oxide (TiO2) substrate [28] as well as photovoltages up to 1 volt

using InAs/GaAs QDs [29]. Despite these recent successes, fundamental research into almost

every part of the hot carrier design is needed to reach practical applications, spec

ically in

the identi

cation of earth-abundant materials that demonstrate slow carrier cooling and

hot absorber-ESC tandems that can be economically manufactured. Of particular interest,

studies have synthesized colloidal graphene QDs with tunable band gap and reported slow

carrier cooling on the picosecond time scale [30, 31].

II. STUDY PROPOSAL

Although initial research into hot carriers suggests great potential to improve e#ciency,

fundamental research into new materials and interfaces is required to

nd better methods

to exploit slow carrier cooling while keeping costs low. In this proposal, I will explore the

electronic structure and intraband carrier relaxation in graphene and graphene/hexagonal

boron nitride (h-BN) hybrids as potential hot carrier photoabsorbers. In addition, I will

examine the potential for photoexcited carrier extraction from these materials by examining

carrier relaxation and transfer from these materials to a silicon carbide (SiC) interface, a

common substrate used to create the monolayer graphene/h-BN materials.

Graphene consists of a two-dimensional honeycomb lattice of sp2-bonded carbon atoms.

The linear dispersion of the band structure for low energies near the K-point in the Bril-

louin Zone (BZ) leads to electrons with zero e#ective mass and a host of favorable electronic

properties such as high electron and hole mobility. Due to its high optical transparency

(2.3% of visible light is absorbed [32]), graphene is often used as a transparent conducting

6

electrode (TCE) in devices such as light-emitting diodes (LEDs) and solar cells. However,

recent experiments have found evidence of hot photoexcited carriers in graphene that per-

sist into the picosecond regime, suggesting graphene could be a viable photoabsorber for

hot carrier cells if its optical absorption could be improved [15, 33]. Methods for achiev-

ing this suggest that sandwiching monolayer graphene between highly contrasted dielectric

layers or through chemically doping its surface can theoretically lead to 50-100% optical

absorption [34], however these methods are still largely theoretical.

Another possibility of improving optical absorption is through the use of graphene QDs

or nanoribbons. Graphene is a desirable QD materials because its high dielectric constant

and zero e#ective mass leads to an in

nite exciton bohr radius [35], indicating that quantum

con

nement e#ects could occur at any size and electron-hole correlations are minimal. A new

method of creating graphene QDs has arisen due to the recent production of graphene and

hexagonal boron nitride (h-BN) hybrid monolayer

lms. Hexagonal BN has the same two-

dimensional hexagonal structure as graphene, a similar lattice constant (2.51 #A

vs. 2.47 #A

in

graphene), but has a large band gap of 5.2 eV. Films made from chemical vapor deposition

(CVD) have shown hybrid layers are possible that consist of segregated regions of graphene

or h-BN on the nanometer scale. Of particular interest to the current proposal, the hybrid

monolayers demonstrate higher optical absorbance compared to pristine graphene and a wide

range of band gaps, indicating that this hybrid material could be used as a photoabsorber

in solar cells [36]. Graphene nanoribbons with widths in the nanometer scale have also

been embedded in a h-BN matrix [37]. Given these recent

ndings, this proposal aims to

understand photoexcitd carrier relaxation in graphene QDs and nanoribbons within a h-BN

matrix using

rst-principles calculations.

Once we have examined intraband carrier relaxation in graphene QDs and nanoribbons

in the h-BN matrix, it is important to understand the photoexcited electron transfer at

interfaces that could be used in optoelectronic applications. Many experiments have demon-

strated the simple and cost-e#ective growth of both graphene and h-BN layers on SiC [38{40],

and therefore it is of great importance to understand its potential for carrier transfer and

extraction. Silicon carbide is a wide band gap semiconductor that can occur in a variety of

polymorphs, all of which have tetragonally coordinated Si and C atoms. Each Si or C layer

is geometrically identical, however each polymorph di#ers by shifts of the relative positions

of alternating Si and C layers. Graphene has been epitaxially grown on two polymorphs,

7

4H-SiC (3.0 eV band gap) and 3C-SiC (2.3 eV). Wide gap semiconductors such as these are

ideal for contacts to extract hot carriers at higher energy levels.

First-principles calculations along with experiment have shown that a single layer of

graphene binds to the SiC surface and does not demonstrate any traditional graphene-like

electronic properties, however a second monolayer recovers the Dirac point and linear disper-

sion, and the third layer demonstrates a 0.3 eV band gap [38, 41, 42]. In addition, graphene

grown on Si-terminated SiC is n-doped [42]. Previous studies have shown photoexcited

electron transfer from graphene to rutile titanium oxide (TiO2). SiC has a similar electron

a#nity to TiO2 [43], but the previous study indicated partial chemical bonding between

the graphene and TiO2 atoms. Since the second graphene layer on the SiC surface appears

to be physisorbed with van der Waals interactions, it will be important to understand the

photoexcited transfer behavior with this di#erent type of bonding to the

rst graphene layer

and SiC bulk. Finally, preliminary studies will be done to assess potential energy selective

contacts (ESC) for carrier extraction by examining the binding energies and band structure

of various QDs on the graphene/h-BN surface. We can determine optimal ESC materials

by matching discrete energy levels in the ESC QD to the average hot carrier energy in the

graphene QD prior to electron-phonon relaxation. This analysis could provide a guide for

experimentalists to test speci

c material combinations.

In review, the current proposal will examine the role of graphene and graphene/h-BN

composites as potential photoabsorbers by: 1) comparing photoexcited carrier relaxation in

pristine graphene to graphene QDs and nanoribbons embedded in a BN matrix; 2) examine

photoexcited electron and hole relaxation in pristine graphene and graphene/h-BN compos-

ites to the 3C-SiC substrate; and 3) determine preliminary candidate materials for ESC for

carrier extraction.

III. METHODS

A. Density Functional Theory

Electronic structure calculations for the graphene, graphene/h-BN, SiC, and combined

systems will be carried out within the density functional theory (DFT) formalism [44]. Be-

ginning from the Born-Oppenheimer approximation, we assume that the electrons readjust

8

instantaneously and adiabatically to any nuclear motion due to the comparably larger nu-

clear mass. Under this approximation, we can uncouple the nuclear and electron wavefunc-

tions and treat the electrons as an interacting electron gas moving within a static potential

from the nuclei. We can write the ground-state energy of a stationary electronic state as (in

atomic units):

[-#Ni

(1/2)r2i

+ #Ni

v(ri) + #N

i<jvee(ri,rj)] = [T + Vnuc + Vee] = E (III.A.1)

where, is the many-body electron wavefunction, N is the number of electrons in the

system, T is the kinetic energy, Vnuc is the external potential exerted on the electrons from

the positive nuclei, Vee is the electron-electron interaction including Coulombic repulsion,

exchange, and correlation, and E is the energy of the many-body state. This Schrodinger

equation can be solved using Hartree-Fock methods, in which a Slater determinant of single

particle antisymmetric wavefunctions is used. This method can take into account the electron

exchange interaction (but approximates correlation), but it is computationally costly and not

tenable for bigger systems. Density functional theory attempts to improve computational

e#ciency and maintain adequate accuracy by using several approximations for the kinetic

energy, electron exchange, and correlation terms.

The goal of most quantum-mechanical many-body theories is to

nd a ground state energy

and corresponding many-body electron wavefunction, from which important observables

such as band structure and density of states can be characterized. Density functional theory

provides a method to do this by using the relationship between electron density, n, and

wavefunction to rewrite the total energy as a functional of n. The electron density can

be written as

n(r) = N

R

...

R

*(r...rN) (r...rN)dr2...drN (III.A.2),

such that if either n or is known, the other quantity can be found. Hohenberg and

Kohn originally proved that if two systems have the same n(r), then the potentials they feel

must only di#er by a constant, which does not a#ect the physics [45]. This important result

demonstrates that the total energy is a unique functional of electron density, and for a given

external potential, we can

nd a unique density that minimizes the energy. We can then

write the expectation value of the energy as a functional of n:

E[n] = T[n] + Vnuc[n] + Vee[n] (III.A.3).

The kinetic energy T[n] and electron-electron Vee[n] functionals are universal, thus theo-

retically we only need to know the nuclear potential for a given system to

nd the electron

9

density and wavefunction. However, T and Vee, contain complicated many-body physics

that cannot be modeled precisely and must be dealt with using approximations. To make

this situation more tractable, Kohn and Sham introduced a method that maps the interact-

ing electron gas onto the non-interacting homogeneous electron gas, a system for which the

kinetic energy, Tnonint is known exactly [46]. We can then write the electron wavefunction as

a determinant of single-particle wavefunctions, #i, known as the Kohn Sham orbitals, each

with energy #i such that

#i(-(1/2)r2 + Vks)#i = #i#i#i (III.A.4), where

Vks = Vnuc + VCoul + Vxc (III.A.5),

VCoul is the electronic Coulomb repulsion and Vxc includes all the many-body exchange

and correlation e#ects. The Kohn-Sham orbitals are the eigenvalues of a Hamiltonian that

includes the kinetic energy of a homogeneous electron gas and a

ctitious e#ective potential;

thus, the Kohn-Sham orbitals themselves are nonphysical. However, by solving Equation

III.A.4, we can construct the electron density as

n(r) = #Ni

j#i(r)j2 (III.A.6),

and based on the Hohenberg-Kohn Theorem discussed above, by minimization we can

nd the unique density for a given external potential Vnuc. By placing all the unknown

many-body physics in Vxc, the Kohn-Sham equations allow us to use approximations for the

exchange and correlation terms and minimize the energy to

nd the ground state electron

density. The ground state energy is then written as

E[n] = Tnonint[n] + Vnuc[n] + VCoul[n] + Vxc[n] = Tnonint[n] +

R

n(r)v(r)d(r) +

R n(r)n(r2)

jr??r2j drdr2

+

R

n(r)vxc(r)d(r) (III.A.7).

Since each of the terms depends on the electron density, this equation can be solved self-

consistently with an approximation used for the exchange-correlation term, vxc. This term

includes the di#erence in kinetic energy between the interacting and non-interacting gas as

well as the exchange and correlation electron-electron interactions. Various approximations

have been developed for vxc to account for exchange and correlation e#ects. The local density

approximation (LDA) uses a functional that is proportional to the electron density [44].

This approximation can overestimate binding energies, and more recently a generalized

gradient approximation (GGA) has been developed that depends on both the density and

its gradient [44, 47]. Both methods have been successful in providing accurate ground

state energies, band structure, and other quantum mechanical phenomena, however both

10

underestimate band gaps in semiconducting and insulating materials.

For the current proposal, DFT electronic structure calculations will be carried out us-

ing VASP [48, 49], a code using a plane-wave basis set to construct the Kohn-Sham or-

bitals. Projector augmented-wave (PAW) potentials [50, 51] will be implemented that lin-

early transform the quickly oscillating all-electron wavefunctions of core electrons to smooth

pseudo-wavefunctions to improve computational e#ciency. The PBE generalized gradient

approximation [52] will be used to approximate exchange and correlation e#ects. Forma-

tion energies and band structure will be compared between graphene armchair QDs, zigzag

QDs, and nanoribbons in a h-BN matrix both freestanding and epitaxially on the 3C-SiC

interface.

B. Intraband carrier relaxation

To assess photoexcited intraband carrier relaxation in graphene and graphene/h-BN hy-

brids alone and at the SiC interface, we will employ a rate equation that accounts for

nonadiabatic coupling between electronic states and electron-phonon interactions that me-

diate the relaxation process. The following discussion generally follows the physics of system

relaxation and energy dissipation described in [53].

Since we are primarily concerned with electron and hole relaxation, we consider the

electrons as the relevant system of interest and phonons as the reservoir. In this case, we

can separate the Hamiltonian as

H = HS + HS??R + HR (III.B.1)

where HS is the electronic system Hamiltonian, HR is the reservoir (phonons) Hamilto-

nian, and HS??R represents the coupling between the two systems. Within the density matrix

formalism, we can consider the state of the total system and reservoir at any time t as a

mixture of pure states n(t), given by the density operator as

^#(t) = #npnj n(t)>< n(t)j (III.B.2)

where pn denotes the probability for a given pure state to exist within the mixture and

are normalized such that the probabilities sum to one. By taking the time derivative of ^#(t),

we can write the equation of motion as

@ ^#(t)

@t = - i

#h[H,^#(t)] (III.B.3)

Assuming the total wavefunction is separable into system and reservoir states js> and

jR>, respectively, we can write the density operator matrix elements as

11

#(s,R;s2,R2;t) = <sj<Rj^#(t)jR2>js2> = #npn n(s,R;t) n*(s2,R2;t) (III.B.4).

Since we are not interested in the reservoir dynamics, we integrate over the reservoir

coordinates to de

ne a reduced density matrix #^R(s,s2;t) that only depends on system co-

ordinates. This is the same as taking the trace of the density operator over the reservoir

coordinates (or any basis set in the reservoir space):

#^R(s,s2;t) = #i<Rij#^(t)jRi> = trR(#^(t)) (III.B.5).

Taking the time derivative of #^R(s,s2;t) and using Equation III.B.3 above, we can write

an equation of motion for the reduced density matrix:

@#^R(t)

@t = - i

#h[HS,#^R(t)] - i

#htrR([HS??R, ^#(t)]) (III.B.6).

where we have taken trR([HR, ^#(t)]) = 0 and used the fact that HS is independent of

the trace over reservoir coordinates so the commutator can be expressed with respect to

the reduced density operator. We can rewrite this in the interaction representation where a

given operator A^ is transformed as A^I = Uy

0(t-t0)AU0(t-t0), and the time evolution operator

U0(t-t0) = US(t-t0)UR(t-t0). Then the equation of motion can be rewritten as

@ ^#I

R(t)

@t = - i

#htrR([HI

S??R, ^#I (t)]) (III.B.7).

This equation demonstrates that we only need to calculate the commutation between

the system-reservoir coupling and the density operator to describe the system equation

of motion. To do this, we approximate the commutation using second-order perturbation

theory. We consider a separable HS??R interaction

HS??R = V(s)#(R) (III.B.8),

where V(s) and #(R) are the system and reservoir parts of the interaction. We implement

two projector operators, ^ P and ^Q

, related by ^ P = 1 - ^Q

. The operator ^ P is designed to

separate the density operator ^#(t) into its system and reservoir components,

^ P ^#(t) = ^R

#^R(t) (III.B.9),

where we assume the reservoir component ^R

is in thermal equilibrium and time-

independent. If we apply both projection operators to the equation of motion for ^#I(t)

and take the trace over reservoir coordinates, the result provides a set of coupled equations

in terms of #I(t) and ^Q

^W

I (t):

@#^R

I (t)

@t = - i

#htrR([HI

S??R, ^R

#^R

I + ^Q

^#I ] (III.B.10), and

@ ^Q

^#I

@t = i

#h

^Q

([HI

S??R, ^R

#^R

I + ^Q

^#I ] (III.B.11).

We can examine the

rst order approximation using Eq III.B.10 if we ignore ^Q

^#I , however

this term only leads to an energy shift and does not introduces relaxation dynamics. We can

12

obtain the second-order perturbation by integrating the

rst-order of approximation (^Q

^#I

! 0) of Eq III.B.11 and plugging it into Eq III.B.10. After simpli

cation, the second-order

equation of motion can be written in the Schrodinger representation as

@#^R(t)

@t = - i

#h[HS + <#>V,#R(t)] - 1

#h2

R t??t0

0 d# (C(# )[V,US(# )V#^R(t - # )Uy

S(# )] - C#(# )[V,US(# )#^R(t

- #)VUy

S(# )]) (III.B.12),

where C(t) = <#(t)#(0)> - <#2> is called the reservoir correlation function. Equation

III.B.12 is known as the Quantum Master Equation. The

rst term reiterates the mean-

eld

contribution from

rst-order perturbation theory. The second term describes energy dissipa-

tion and the inclusion of C(t) introduces memory e#ects that disappear after a characteristic

time #M that depends on the system of interest.

We next rewrite the Quantum Master Equation in an energy representation and apply

several approximations that will allow its use in computational analyses. If we choose

the energy eigenfunctions j a> of HSj a> = Eaj a> as our basis set, we can write #ab(t)

= < aj#^R(t)j b> and Vab = < ajVj b>. In this basis, the equation simpli

es by using

US(# )j a> = e

??iEa#

#h j a> and !ab = Ea??Eb

#h . The equation of motion for the reduced density

matrix elements becomes

@#ab

@t = - i!ab#ab + i

#h#c<#>(Vcb#ac - Vac#cb) + @#ab

@t (diss) (III.B.13), where

@#ab

@t (diss) = -#cd

R t??t0

0 d# ( 1

#h2 C(-#)VbcVcdei!da##ac(t - # ) - 1

#h2 C(#)VacVcd(# )ei!db##db(t - # ) -

[ 1

#h2 C(-#)VcaVbdei!db# + 1

#h2 C(#)VdbVacei!ca# ]#cd(t - # )) (III.B.14).

This last term de

nes the dissipative behavior of the equation of motion and will be our

central concern moving forward in discussing carrier relaxation. For a given system, we

can de

ne a time scale with minimum time step #t to coarse-grain our time axis for the

processes in which we are interested. If #t is much larger than the memory time #M, we are

essentially probing the system at long enough time intervals so that the correlation e#ects

can be neglected. In this case, known as the Markov approximation, we can extend the

integral limit in Equation III.B.14 to in

nity without a#ecting its value and approximate

#ab(t - # ) = #ab(t).

The real parts of each term in the integral of Eq III.B.14 represent the energy dissipation

of the density matrix elements relevant to the discussion of carrier relaxation. We can rewrite

this equation in di#erent notation to eventually de

ne the relaxation rate term for an initial

photoexcited state:

@#ab

@t (diss) = -#cdRab;cd#cd(t) (III.B.15),

13

where

Rab;cd = #ac#eRe

R1

0 d# ei!de# 1

#h2 C(#)VbeVed + #bd#eRe

R1

0 d# ei!ce# 1

#h2 C(#)VaeVec -

Re

R1

0 d# ei!db# 1

#h2 C(#)VcaVbd - Re

R1

0 d# ei!ca# 1

#h2 C(#)VdbVac (III.B.16).

Equation III.B.16 is known as the Red

eld equation, where Rab;cd is the Red

eld ten-

sor [54]. We are interested in electron relaxation, which is represented by population trans-

fer between the eigenstates. This case corresponds to a=b and c=d in the Red

eld tensor

representation, such that the Red

eld tensor reduces to

Raa;cc = 2#ac#eRe

R1

0 d# ei!ae# 1

#h2 C(#)VaeVea - 2Re

R1

0 d# ei!ca# 1

#h2 C(#)VcaVac = #ac#ekae - kca

(III.B.17).

This case represents net population transfer from state j a> to all other eigenstates

(#ekae) and from all other states to j a> (kca). The two transitions rates are related through

the principle of detailed balance such that kab = e#h!ab=kBT kba, however we are only concerned

with the

rst rate describing transitions out of the initial state. If we split the

rst term

of Eq III.B.17 into two integrals and

nity to zero while

taking the complex conjugate of the integrand, we can combine the two into an integral

from negative to positive in

nity. Taking the Fourier transform, the transition rate is

kab = Re 1

#h2VabVbaC(!ab) = Re 1

#h2 jVabj2C(!ab) (III.B.18).

A second case exists when a 6= b, a = c, and b = d, which represents damping rates

for the o#-diagonal matrix elements of #ab(t). These o#-diagonal elements represent phase

di#erences between the energy eigenstates and are referred to as coherences, however they wil

not be the focus of the current proposal. We also assume that there is no coupling between

population transfer and coherence processes, known as the secular approximation. Finally,

we treat the correlation function C(!ab) within the harmonic oscillator approximation such

that the reservoir coupling component is

#(Z) = #lclql (III.B.19),

where ql are harmonic coordinates and cl are the coupling constants. Using this formu-

lation, it can be shown [53] that the reservoir correlation function reduces to

C(!ab) = 2##h[n(!ab)+1][P(#h!ab)] (III.B.20),

where n(!ab) is the Bose-Einstein distribution function and P(!ab) is the phonon density

of states (DOS). Substituting this expression into Eq III.B.18, the rate equation becomes

kab = 2#

#h jVabj2(n(!ab) + 1)P(#h!ab) (III.B.21),

which is the central equation we will be using in the current proposal. Since we are only

14

interested in the diagonal terms of density matrix when considering popuulation transfer,

we can write a simple di#erential equation

@#aa

@t = -#bkab#bb(t) (III.B.22).

Equations III.B.21 and III.B.22 provide an intuitive interpretation of how intraband

phonon-asisted relaxation is calculated. For two given states, the nonadiabatic system-

coupling term jVabj2 provides a measure of state overlap that will increase the likelihood

of such a transition occurring. However, this is weighted by the phonon density of states,

such that there must be an available phonon equal to the transition energy !ab for the given

transition to occur. If we consider an initial photoexcitation as a pure state in the density

matrix formalism, then we can

x our initial condition by setting the matrix element cor-

responding to the photoexcited state equal to one and the rest to 0. We can then solve Eq

III.B.22 with this initial condition to determine the carrier relaxation to all other states for

a given photoexcited state..

We will use Eq III.B.21 to compute relaxation rates for the systems discussed within this

proposal. Therefore, we need to compute jVabj2 and P(!ab) using computational methods.

Referring to Eq III.B.18, we determine Vab, the system coupling component, by calculat-

ing the nonadiabatic coupling between electronic states. This coupling is calculated by

performing quantum molecular dynamics (MD) of the system using the Verlet velocity al-

gorithm [55] in VASP and calculating the overlap of eigenstates at adjacent time steps. For

two normalized eigenstates j a> and j b>, the coupling matrix element can be written as

jVabj2 = #Ni

#h2

N j< ia

(t)j @

@t j ib

(t)>j2 = #Ni#h2

N j< ia

(t)j ib

(t+#t)?? ib

(t)

#t >j2 (III.B.23)

where #t is the simulation time step and N is the total number of time steps. The system

is annealed to 300K using an NVT ensemble and then simulated within the microcanonical

ensemble until the matrix elements jVabj2 converge to stable values. The phonon DOS will

be calculated using the frozen phonon method in VASP, in which each atom is displaced

by 0.015 #A

and the resulting forces on all other atoms are calculated to calculate the force

constants and subsequent frequencies. Gaussian broadening of the phonon DOS is used when

calculating photoexcited relaxation, which provides a rough representation of the existence

of rarer multiphonon processes that can assist relaxation at higher transition energies.

Using this method, we will compare photoexcited carrier relaxation in pristine graphene

to that in graphene QDs and nanoribbons embedded in the h-BN matrix. We de

ne pho-

toexcited states as those with the largest oscillator strengths between normalized occupied

15

and unoccupied states, de

ned as 1

me#h!ab

< ajpj b>2, where me is the electron mass and p

is the momentum operator. Starting from this pure photoexcited state, we then solve the

di#erential equation and measure the time taken for the excited electron and correspond-

ing hole to relax to the conduction and valence band edges, respectively. After analyzing

relaxation for freestanding monolayers, we will examine the same structures at the 3C-SiC

interface to understand carrier transfer mechanisms.

IV. PRELIMINARY RESULTS

The preliminary results demonstrate the feasibility of the proposed study of electronic

structure and intraband carrier relaxation in graphene-SiC structures. In particular, we

report initial results looking at photoexcited carrier relaxation in pristine graphene at the

C-terminated 3C-SiC interface (Figure 1).

FIG. 1. Interface between the C-terminated, 3C silicon carbide polymorph and two graphene layers.

Small, brown sphere and large, blue spheres represent C and Si, respectively. The Si-terminated

interface has a similar structure with a 2 #A

bond length between surface Si and graphene C atoms.

As shown in previous studies [42], the

rst epitaxial graphene layer binds to the SiC

surface 1.65 #A

above the bulk C atom. The second graphene layer is 3.35 #A

above the

rst and recovers pristine graphene band structure, including the Dirac point at the K

point (Figure 2(a)). Unbonded SiC carbon atoms at the surface lead to a

the Fermi level. For Si-terminated 3C-SiC surfaces, a similar physical structure results,

however the bond length between the Si surface atom and the graphene C atom extends to

2 #A

. The second graphene layer again recovers its traditional band structure, however it is

now n-doped (Figure 2(b)). This di#erence in electronic structure could have a signi

cant

impact on dominant transition dipole moments for photoexcitation and subsequent carrier

16

relaxation, which will be further explored. At this time, I report carrier relaxation results

only for the C-terminated SiC-graphene interface that leaves the second graphene layer

undoped.

FIG. 2. Band structure for C- and Si-terminated graphene-SiC interface along high-symmetry

k-point paths. a) C-terminated SiC; b) Si-terminated SiC. Note that the

near the Dirac point arises due to dangling bonds at the SiC surface.

For the present results, we only examine photoexcitation and carrier relaxation at the

K-point. Further developments in the code will allow for averaging over k-points to provide

a more representative picture of carrier relaxation. However, the smallest required photoex-

cited energies occur at the K-point and provide an important characterization of electron

relaxation. Initial photoexcited states were chosen as those states with the largest oscillator

strengths with the initial state at the Dirac point. Several possible photoexcitations are

chosen to describe qualitative di#erences in the photoexcited states based on their localiza-

tion either on the graphene or SiC. The

rst photoexcited state (PE1) has the majority of

its charge density residing on the second graphene layer (Figure 3(a)), the second (PE2) is

mixed between the graphene and SiC (Figure 3(b)), the third (PE3) is largely in SiC (Fig-

ure 3(c)), and the last (PE4) largely resides on the

rst graphene layer (Figure 3(d)). By

comparing relaxation rates between these examples, we can study the e#ect of localization

on relaxation and carrier extraction.

We next calculate the carrier relaxation for each chosen PE state. As shown in Figure

4, we can visualize the relaxation in both real (top row) and energy (bottom row) for each

photoexcitation. In real space, the intitial and subsequent states are projected along the

z-axis of the structure to visualize localization at each layer over time. Figures 4(a) and 4(b)

show similar relaxation patterns since both PE states (PE1 and PE2) are localized at least

17

FIG. 3. Charge density isosurfaces (yellow) and cross-sections (red) for sampled photoexcited

states (PE) overlaid on the graphene-SiC structure. a) PE localized to graphene layers (PE1); b)

PE spread across graphene and SiC (PE2); c) PE localized to SiC (PE3); d) PE localized to

rst

graphene layer (PE4).

partially on the second graphene layer. As seen in the energy space, these photoexcitations

broaden to include other available bands in the femtosecond time range, which corresponds to

the renormalization of the hot carrier distribution [11] and has been experimentally identi

ed

previously in graphene [56]. This hot carrier distribution persists into the picosecond range,

at which point they relax to the SiC surface and the dangling bond impurity state. These

results indicate the potetntial for hot carrier extraction prior to phonon-initiated carrier

cooling. Photoexcited states localized either on the SiC or

rst graphene layer (Figures

4(c)-(d)) do not demonstrate a similar hot carrier distribution across the femtosecond time

scale and relax more quickly to the SiC surface.

FIG. 4. Photoexcited carrier relaxation within real (top row) and energy (bottom row) space for

each of the chosen photoexcited states. The graphene-SiC structure on the right aligns with the

z-axis of the graphs in the

rst row to help visualize the electrion relaxation pathway. a) PE1; b)

PE2; c) PE3; and d) PE4.

18

Across all PE states, the photoexcited electron relaxes to the second graphene layer before

relaxing to the impurity state, indicating that 3C-SiC does not act as an electron acceptor

at this interface. However, the graphene hot carrier distribution does persist long enough

that, if an ESC material were adsorbed to the graphene surface, these hot carriers may be

extracted prior to phonon cooling. This situation requires further investigation and will be

included in the proposed study. In addition, current analyses are being done to determine

photoexcited hole behavior at the interface.

V. CONCLUSIONS AND FUTURE WORK

In conclusion, the proposed study aims to understand photoexcited electron and hole relax-

ation in pristine graphene as well as graphene QDs and nanoribbons embedded in a hexgonal

boron nitride matrix. The study will use computational methods based on energy dissipation

theory to model electron-phonon coupling and to calculate relaxation rates for photoexcited

carriers. Photoexcited carrier relaxation will be examined on freestanding graphene QD/h-

BN surfaces and subsequently at the SiC interface to determine photoexcited charge transfer

mechanisms. Once this interface is well understood, energy selective contact (ESC) materi-

als will be tested at the graphene surface. Preliminary results suggest that SiC does not act

as an electron acceptor, and therefore electron accepting materials will be initially studied

as ESC candidates.

It is important to note that the current computational methodology does not take into

account all relaxation methods and does not explicitly account for multiphonon processes.

As discussed in the proposal, electron-hole correlation e#ects should be minimal in graphene

QDs, however an inclusion of this interaction could con

rm this and extend the methodol-

ogy to QDs where this coupling is important. Other future work in this area could include

accounting for explicit multiphonon processes, electron-phonon interactions that change the

electron's momentum, and extending this methodology to include photoexcitation and relax-

ation at multiple k-points. In addition, the inclusion of other electron-electron interactions

such as impact ionization would provide a more complete analysis of electron relaxation and

allow the study of multiple exciton generation within the same framework.

Finally, the graphene-SiC interface is a prime candidate for intermediate band (IB) solar

cell design when SiC is doped with boron, aluminum, or nickel to create deep donor and

19

acceptor levels. Therefore, future research will include examining the electronic structure

of the doped SiC-graphene interface and calculating electron relaxation and charge transfer.

Non-radiative recombination via the impurity states is an important limiting relaxation

process in IB cells, and therefore we will use DFT calculations to understand the non-

radiative transition energies using various dopants to determine optimal doping materials.

Acknowledgments

I am extremely grateful to Dr. Hai-Ping Cheng for her guidance and support in the devel-

opment of this proposal and to Dr. Dmitri Kilin for his guidance in using the nonadiabatic

coupling code. I would also like to thank The Quantum Theory Project, University of

Florida High Performance Computing (UFHPC), and The National Energy Research Scien-

ti

c Computing Center (NERSC) for computational resources.

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