Nov 7, 2017 - William Penney. (1909-1991) ..... Tipler, P. A.; Llewellyn, R. A. âModern Physicsâ, 5th Ed. (2008), p. 443. 11 / 07 / 2017 ... HH â Heavy Hole Band.
Dr. Walczak’s Group Meeting 2017
“Intro to Bandstructure Theory of Solids” OBJECTIVES To present a Kronig-Penney model as one of the exactly solvable problems in quantum mechanics. To present plots of E-k diagrams and discuss the concepts of effective mass and density of states.
To explain transport and optical properties of bulk and 2D materials within the banstructure theory.
Dr. Kamil Walczak Department of Chemistry and Physical Sciences, Pace University 1 Pace Plaza, New York City, NY 10038
Dr. Walczak’s Research Group Principal Investigator: Dr. Kamil Walczak Student:
Topics:
Rita Aghjayan
“Thermal Rectifiers”, “Relaxation Processes”
Arthur Luniewski
“Molecular Noise”, “High-Intensity Heat Fluxes”
David Saroka
“Tunneling of Heat”, “Wave-Packet Approach”
Joanna Dyrkacz
“Inelastic Heat Flow”, “Electron-Phonon Effects”
Luke Shapiro
“Thermal Memristors”, “Hysteretic Loops”
Hunter Tonn
“Quantum Interference”, “Tunneling of Heat”
Erica Butts
“Neuronal Networks”, “Memristive Ion Channels”
Omar Tsoutiev
“Thermal Noise”, “Quantum-Interference Devices”
Kamil Walczak
11 / 07 / 2017
Periodic Structure of Crystal Lattice We have to solve the stationary problem of a particle in a one-dimensional potential created by ions in the periodic structure of crystal lattice. (arrangement of ions)
V0 - depth of potential well
V0
2a - width of potential well 2b - distance between wells
V V0
L 2a 2b - periodicity
The regular potential inside the periodic lattice can be simplified by a series of identical one-dimensional potential barriers of rectangular shapes. Kamil Walczak
11 / 07 / 2017
Kronig-Penney Model (1931) Periodic crystal lattice of solids is modeled as ideal quantum-mechanical system that consists of an infinite periodic array of rectangular potential barriers!
Difficulties to include local defects (such as: chemical impurities, dislocations), interactions between electrons and other (quasi)particles!
Ralph Kronig (1904-1995)
The Kronig-Penney model qualitatively describes formation of band structure in solids, but due to unrealistic simplifications, it can not be used for any quantitative analysis!
William Penney (1909-1991)
R.L. Kronig & W.G. Penney, Proc. Roy. Soc. (London) A 130, pp. 499-513 (1931). Kamil Walczak
11 / 07 / 2017
Schrödinger Wave Equation Goal: to find energy levels and the corresponding wavefunctions for an electron in a periodic crystal lattice (solve the eigenvalue problem for energy). Time-independent Schrödinger wave equation:
2 d2 V( x ) ( x ) E ( x ) 2 2m dx V(x) V(x L) The wavefunction of a particle placed in a periodic potential is a product of a periodic Bloch function (that has the same periodicity as the potential) and a plane wave:
(x) u(x) exp ikx Bloch function
u(x) u(x L) Kamil Walczak
Plane wave (free evolution) 11 / 07 / 2017
Bloch’s Theorem (1928)
| ( x ) |2
Charge distribution corresponding to standing waves in 1D crystal.
“When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal… By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation.”
Felix Bloch (1905-1983)
F. Bloch, Z. Physik 52, pp. 555-600 (1928). Kamil Walczak
11 / 07 / 2017
Periodicity Due to periodic boundary condition, the following Schrödinger equation is satisfied:
2 d2 V ( x ) ( x L) E ( x L) 2 2m dx Both wavefunctions can differ only by constant and complex factor:
(x L) (x)
(x nL) n (x)
ei
is real
| |n 1
V( x )
a
a
a 2b
x (simplified potential)
V0 Kamil Walczak
11 / 07 / 2017
Specific Solutions (1) in the area a x a
2 d 2 (x) E V0 ( x ) 2 2m dx 2m(E V0 ) q 2 (2) in the area a x a 2b
2 d 2 (x) E ( x ) 2 2m dx Kamil Walczak
2mE 2
and for energy V0 E 0
1 (x) A eiqx B eiqx (wave vector)
and for energy V0 E 0
2 (x) C e x D e x (wave vector) 11 / 07 / 2017
Boundary Conditions Specific boundary conditions for x = a:
1 (x a ) 2 (x a )
A eiqa B eiqa C e a D e a
d1 ( x ) d2 ( x ) dx x a dx x a
iqA eiqa iqB eiqa C e a D e a
Specific boundary conditions for x = a + 2b:
2 (a 2b) 1 (a 2b) ei 1 (x a )
d2 ( x ) d1 ( x ) d1 ( x ) ei dx a 2 b dx a 2 b dx x a
iqA e
C e (a 2b) D e (a 2b) A eiqa B eiqa ei C e (a 2b) D e (a 2b) Kamil Walczak
iqa
involving Bloch’s theorem
iqB eiqa ei 11 / 07 / 2017
Linear Algebra: Special Condition Our specific boundary conditions can be written in the form of homogeneous system of four linear equations for the wavefunction’s coefficients:
eiqa iqa iqe ei ( qa) i ( qa) iqe
e iqa iqe iqa ei ( qa)
e a e a e (a 2b)
iqe i ( qa)
e ( a 2 b )
e a A 0 a e B 0 e ( a 2 b ) C 0 ( a 2 b ) e D 0
From linear algebra we know that non-trivial solution (where all coefficients are equal to zero) can be obtained when matrix determinant is equal to zero:
eiqa iqa iqe det i ( qa) e i ( qa) iqe Kamil Walczak
e iqa iqe iqa ei ( qa)
e a e a e (a 2b)
iqe i ( qa)
e ( a 2 b )
e a e a 0 (a 2b) e (a 2b) e
(*)
11 / 07 / 2017
Further Transformations Extension within the top row of the analyzed fourth-order determinant:
iqe iqa eiqa ei ( qa) iqe i ( qa)
iqe iqa e a ei ( qa) iqe i ( qa)
e a e (a 2b) e ( a 2 b )
iqe iqa ei ( qa) iqe i ( qa)
iqe iqa e a e a e a e (a 2b) e(a 2b) e ( a 2 b ) e iqa ei ( qa) iqe i ( qa) e ( a 2 b ) e ( a 2 b ) e ( a 2 b )
iqe iqa e a e ( a 2 b ) e a ei ( qa) iqe i ( qa) e ( a 2 b )
iqe iqa ei ( qa) iqe i ( qa)
e a e (a 2b) 0 e ( a 2 b )
Extension within the top row of the analyzed third-order determinants (Sarrus): (a 2b) e eiqa iqe iqa (a 2b) e
Kamil Walczak
e (a 2b) e
(a 2b)
e a
ei ( qa) iqe
i ( qa)
e (a 2b) e
(a 2b)
... 11 / 07 / 2017
Further Transformations ... e a
ei ( qa) iqe i ( qa)
e
a
e ( a 2 b ) iqa iqa e ( a 2 b ) e ( a 2 b ) e iqe (a 2b) (a 2b) (a 2b) e e e
ei ( qa) iqe
e (a 2b)
i ( qa)
e
(a 2b)
e
i ( qa) e e a iqe iqa i ( qa) iqe
e a
ei ( qa)
a
iqe i ( qa)
e (a 2b) e ( a 2 b )
ei ( qa)
ei ( qa)
iqe i ( qa)
iqe i ( qa)
iqe
iqa
ei ( qa) iqe
i ( qa)
iqe iqa
ei ( qa)
e (a 2b)
iqe i ( qa)
e ( a 2 b )
i ( qa) e e a iqe iqa i ( qa) iqe
e (a 2b) e
e (a 2b) (a 2b) e
(a 2b)
e
a
e (a 2b) e ( a 2 b )
ei ( qa)
ei ( qa)
iqe i ( qa)
iqe i ( qa)
0
…we can continue analysis by expanding determinants and keep reducing terms… Kamil Walczak
11 / 07 / 2017
Characteristic Function …or we can use “Mathematica” to solve the equation (*)…
eiqa iqe iqa M : i ( qa) e iqe i ( qa)
e iqa iqe iqa ei ( qa)
e a e a e (a 2b)
iqe i ( qa)
e ( a 2 b )
e a a e ; (a 2b) e (a 2b) e
Solve[Det[M] 0, ] Using trigonometric identities to simplify expressions, we obtain the relation:
2 q 2 cosh2b cos2qa sinh 2b sin 2qa cos 2q 1 f (E) 1 Kamil Walczak
(characteristic function) 11 / 07 / 2017
New Variables New representations as inputs in characteristic function:
2mEa 2 2b 2 2i x 2 2mEa 2 2mV0a 2 2qa 2 2i x 2 2 2mEa 2 2 x 2 2 2 q x 2mEa 2mV0a
2mEa 2 2mV0a 2 q 2 x 2 2 x 2mEa
New variables:
2mEa 2 x 2 2mV0 a 2 2
b/a
2 q 2 1 q 1 x x 2q 2 q 2 x x Kamil Walczak
11 / 07 / 2017
Characteristic Function: Plot We can use “Mathematica” to define and plot characteristic function:
f [x _, _, _] : Cosh[2i x ]Cos[2i x ]
1 x x Sinh 2i x Sin 2i x 2 x x
6, 16
Procedure in “Mathematica”:
Plot[f [x,6,16],{x,0,1.6}] f (x) allowed bands forbidden bands
x Kamil Walczak
11 / 07 / 2017
Bandstructure Theory of Solids 7, 6
Procedure in “Mathematica”:
Plot[f [x,7,6],{x,0,1.6}] f (x) allowed bands forbidden bands
x Bandstructure theory of solids models the behavior of electrons in solids via the existence of allowed and forbidden energy bands. A solid creates a large number of closely spaced energy levels (molecular orbitals) which are usually overlapped due to broadening effects experienced by them. Bandstructure theory of solids dictates optical and transport properties of solids. Kamil Walczak
11 / 07 / 2017
Formation of Bandstructures (Mesoscopic limit) Discrete electronic structure of atoms/molecules
~ 1020 atoms Periodic arrangement in energy favorable distance
Continuous electronic structure of solids/materials
Physisorption (≈ 0.01 ÷ 0.1 eV) Chemisorption (≈ 0.5 ÷ 0.6 eV)
E
Bonding controls electronic bandstructure of materials and the associated bandgaps!
rC rP Kamil Walczak
r 11 / 07 / 2017
Formation of Bandstructures Z6
Carbon (C) 2
[He] 2s 2p
2
Z 14
Closely-spaced energy levels
~ 1023 eV
Silicon (Si) [ Ne] 3s 2 3p 2 Z 32
Germanium (Ge) [Ar] 3d10 4s 2 4p 2
Splitting of the 2s and 2p states of carbon (diamond), the 3s and 3p states of bulk silicon or the 4s and 4p states of bulk germanium! Tipler, P. A.; Llewellyn, R. A. “Modern Physics”, 5th Ed. (2008), p. 443.
11 / 07 / 2017
Classification of Solids INSULATOR SEMICONDUCTOR
EF
CONDUCTOR
E G 2 eV
EF k BT E G 2 eV
Metal
Semi-metal
Fermi energy is “hypothetical” energy level up to which all accessible energy levels below are occupied by electrons, while energy levels above are empty (at OK). Kamil Walczak
11 / 07 / 2017
Kronig-Penney Model: E-k Diagrams “Extended Zone Scheme”
“Reduced Zone Scheme”
2k 2 E 2m eff
E
E
k 0
L
2 L
3 L
k
L
0
L
Looking at E-k diagrams, we conclude that in certain situations related to solids, “nearly free electron approximation” works quite well! Kamil Walczak
11 / 07 / 2017
Parabolic Band Approximation HH – Heavy Hole Band LH – Light Hole Band SOH – Split-Off Hole Band
Local approximation for energy:
2k 2 E(k ) E 0 2m eff Definition of an effective mass:
E(k ) 2 k 2
m eff
Kamil Walczak
1
2
11 / 07 / 2017
Kronig-Penney Model: Dirac Comb Dirac delta function limit for potential barriers:
V0 ,
a 0,
V0a const
Dirac delta function limit for potential barriers:
2b 0
sinh2b 2b
cosh2b 1
2 q 2 q 2q 2
Under such conditions, the characteristic functions is defined as follows:
cos2qa
2qa sin 2qa cos 2
1 f (E) 1 Kamil Walczak
(characteristic function) 11 / 07 / 2017
Specific Solutions & Boundary Conditions The specific solutions in distinguished areas for positive energy E > 0:
1 (x) A eiqx B eiqx
q 2m(E V0 ) / 2
2 (x) C eikx D eikx
k 2mE / 2
By applying boundary conditions from “slide 9” together with Bloch’s theorem, we obtain the system of four equations for four unknown coefficients:
A eiqa B eiqa C eika D eika iqA eiqa iqB eiqa ikC eika ikD eika
iqA e
C eik (a 2b) D eik (a 2b) A eiqa B eiqa ei ikC eik (a 2b) ikD eik (a 2b) Kamil Walczak
iqa
iqB eiqa ei 11 / 07 / 2017
Characteristic Function Our specific boundary conditions can be written in the form of homogeneous system of four linear equations for the wavefunction’s coefficients:
eiqa iqa iqe ei ( qa) i ( qa) iqe
e iqa iqe iqa ei ( qa)
eika ike ika eik ( a 2 b )
iqe i ( qa)
ike ik ( a 2 b )
e ika A 0 ika ike B 0 e ik ( a 2 b ) C 0 ik ( a 2 b ) ike D 0
Non-trivial solution exists when matrix determinant is equal to zero, and from this condition we obtain the following relation:
2 q 2 cos2b cos2qa sin 2b sin 2qa cos 2q 1 g(E) 1 Kamil Walczak
(characteristic function) 11 / 07 / 2017
New Variables New representations as inputs in characteristic function:
2mEa 2 2b 2 2 x 2 2mEa 2 2mV0a 2 2qa 2 2 x 2 2 2mEa 2 2 x 2 2 2 q x 2mEa 2mV0a
2mEa 2 2mV0a 2 2 q x 2 2 x 2mEa
New variables:
2mEa 2 x 2 2mV0 a 2 2
b/a
2 q 2 1 q 1 x x 2q 2 q 2 x x Kamil Walczak
11 / 07 / 2017
Characteristic Function: Plot We can use “Mathematica” to define and plot characteristic function:
g[x _, _, _] : Cos[2 x ]Cos[2 x ]
1 x x Sin 2 x Sin 2 x 2 x x
5, 15
Procedure in “Mathematica”:
Plot[g[x,5,15],{x,0,2.1}] g( x ) allowed bands forbidden bands
x Kamil Walczak
11 / 07 / 2017
Ohm’s Physics (1827) The First Ohm’s Law: the electric current I flowing through two points of a conductor is equal to an applied voltage V (potential difference) between those two points:
IG V A
L The Second Ohm’s Law: the electric conductance G is proportional to the cross-sectional area A for the charge flow and inversely proportional to the length of the conductor L:
G
Kamil Walczak
A L
Georg Ohm (1789-1854)
- material-dependent electrical conductivity 11 / 07 / 2017
Conducting Properties of Materials
Insulators: impede the free flow of electrons through the crystal structure. Injected charge usually remains in the initial location (no global charge redistribution is allowed). Semiconductors: thermal energy supports a relatively small amount of electrons (and holes) in their free motion within the sample. As temperature increases, the electrical resistance of the sample decreases. Conductors: contain a lot of electrons moving freely within the sample. Injected charge is quickly distributed across the surface of the object. The electrical resistance increases linearly with temperature. Superconductors: collective behavior of electron pairs which are in quantum state. Those low-temperature metals are characterized by “zero” resistance.
Kamil Walczak
11 / 07 / 2017
Classification of Bulk Materials Semiconductors
Insulators
Conductors
Electrical Conductivity (S/m)
108
1012 Perovskites Cuprates
10 4
Mercury Copper Graphene
100 Germanium
10 4 Boron Silicon
Diamond Air Rubber Glass
Teflon
1024 1020 1016 1012 10 8
The best electrical insulators are teflon and vacuum, while the best conductors are superconductors below critical temperature, graphene, silver, and copper. The most important semiconductor from technological point of view is silicone. Kamil Walczak
11 / 07 / 2017
Density Functional Theory (DFT) The Kohn-Sham single-electron equation:
2 VKS ( r ) n ( x ) E n n ( x ) 2m The K-S potential is the sum of external, Hartree and exchange-correlation terms:
E [n ] q n( r ' ) VKS ( r ) Vext ( r ) VH ( r ) VXC ( r ) Vext ( r ) d 3r ' XC | r r '| n ( r )
2
Since potential is a unique function of electron density, so is any other observable: N 2 n ( r ) | n ( r ) | n 1
with
3 n ( r )d r N
The total energy associated with electrons is a functional of electron density:
Eel [n] TKS[n] U H [n] E XC [n] Problem: LDA-DFT calculations underestimate bandgaps typically by 30-50%! Kamil Walczak
11 / 07 / 2017
Self-Consistent DFT Scheme E [n ] q n( r ' ) VKS ( r ) Vext ( r ) d 3r ' XC | r r '| n ( r ) 2
Guess Initial Density
Create KS Potential
2 out out V ( r ) ( x ) E KS n n (x) n 2m
Solve KS Equations N/2 n ( r ) 2 | nout ( r ) |2 n 1
Obtain New Density
E tot NO Kamil Walczak
Converged?
YES
3 TKS[n] Vext ( r )n( r )d r
U H [n] E XC [n] 11 / 07 / 2017
Fixing Bandgap Problem: GWA-DFT Exchange-correlation term:
Experimental Bandgap [eV]
8
VXC ( r )n ( r )
3 ( r , r ' )n ( r ' )d r '
6
4
Self-energy term:
( r , r ' ) iG ( r , r ' )W( r , r ' )
2 Screened Coulomb interaction: (red) LDA-DFT (blue) GWA-DFT
0 0
6 2 4 Theoretical Bandgap [eV]
Schilfgaarde et al., Phys. Rev. Lett. 96, 226402 (2006).
8
VC ( r , r ' ' ) 3 W( r , r ' ) d r ' ' ( r ' ' , r ' )
Dyson equation for propagator:
G G 0 G 0G 11 / 07 / 2017
E-k Diagram & DOS: Bulk Silicon
Kamil Walczak
11 / 07 / 2017
Bandstructure of Bulk Silicon: Details Temperature-dependent indirect bandgap (temperature in K):
4.37 104 T 2 E G 1.17 T 636 Six equivalent valleys at conduction bandedge:
Kamil Walczak
11 / 07 / 2017
Effective Mass / Bare Mass
Effective Mass Vs. Bandgap
0.2
0.1
0.0 0
1
2 Bandgap [eV]
3
Concluding Remark: in general, the larger the bandgap in the semiconductor, the higher the value of the effective mass associated with charge carriers! Kamil Walczak
11 / 07 / 2017
E-k Diagram & DOS: Bulk Copper
Kamil Walczak
11 / 07 / 2017
Spin-DOS: Copper, Cobalt, Iron
Kamil Walczak
11 / 07 / 2017
E-k Diagram & DOS: Graphene (2D)
Dirac cone
Kamil Walczak
11 / 07 / 2017
Hybridization & E-k Diagram: Graphene (2D)
sp2 hybridization
conduction
Relativistic Wave Equation:
valence
F k E(k ) Dirac cone
Kamil Walczak
11 / 07 / 2017
Bandstructures of 2D Materials
Energy [eV]
Silicene
Germanene
10 8 6 4 2 0
2 4 6 8 10
M K k-points
P. Miro et. al. Chem. Soc. Rev. 43, pp. 6537-6554 (2014).
M K k-points
11 / 07 / 2017
Bandstructure Calculations Among the standard methods used to calculate bandstructure of solids are: Kronig-Penney Model with nearly free electron approximation, where interactions between electrons are completely neglected and Bloch’s theorem is applied. Density Functional Theory as a first-principles method addressing electron-electron interaction problem via exchange-correlation term in functional of electron density. Theory of Propagators in which many-particle effects (correlations) are included via self-consistent computation of Green’s functions satisfying Dyson equation. Tight-Binding Approach is based on the assumption that electrons in solid are highly localized near atomic sites and LCAO with Wannier wave functions is applicable. The “k dot p” Method which is based on perturbation theory formulated in terms of energies and wave functions exactly at the Gamma point (where k = 0). Kamil Walczak
11 / 07 / 2017
Closing Remarks Bandstructure theory is based on a few assumptions that system is homogeneous and impurity-free, macroscopically large (composed of 1023 atoms or more), while electrons do not interact dynamically with other electrons, lattice vibrations, etc. Electronic structures of solids show bands of allowed energies, being separated by zones of forbidden energies as a consequence of periodic structure of solids! Bandstructure of a solid dictates its optical and transport properties! Due to applicability of Bloch’s theorem, the unbound electrons in solids can be treated as nearly free particles which do not contribute to resistance (there is no scattering on periodic potential created by crystal ions)!
Impurities and interfaces modify bulk bandstructures (surface and dopant states, local charges and band bending), while conventional bandstructure theory is not applicable to strongly correlated systems (Mott insulators, Hubbard model)! Bandstructure theory is only an approximation to the quantum state of a solid! Kamil Walczak
11 / 07 / 2017
