Finite temperature calculations of the electronic and optical ...
Transcript of Finite temperature calculations of the electronic and optical ...
Finite temperature calculations of the
electronic and optical properties of solids and nano-structures
E. Cannuccia
Institut Laue Langevin BP 156 38042 Grenoble,
France
793 K = 68 meV QP gap is 1200 meV
YES ! We do need phonons.
““... unfortunately theorists do ... unfortunately theorists do not even bother to compare not even bother to compare their calculations with low-their calculations with low-
temperature measurements, temperature measurements, using more easily accessible using more easily accessible room temperature spectra."room temperature spectra."
Real life is at finite temperatureReal life is at finite temperature
M. Cardona Solid State Comm. 133, 3 (2005)
⟨u2T ⟩≈ ℏ
4M⟨12NBose T ⟩
The quantistic The quantistic zero-point zero-point
motion effectmotion effect
Outline... Outline...
Finite temperature excitons
Ab-Initio Polarons
Spectral functions and the QP-approximation
Outline... Outline...
Finite temperature excitons
Ab-Initio Polarons
Spectral functions and the QP-approximation
Components of the energy renormalizationComponents of the energy renormalization
Thermal Thermal expansionexpansion
Electron-Phonon interaction
P.B. Allen and M. Cardona Phys. Rev. B 27 4760 (1983)
>>
The Heine-Allen-Cardona Approach (I)The Heine-Allen-Cardona Approach (I)
For a review see M. Cardona, Solid State Commun. 133, 3 (2005).
I
RI
H (R+u)=H (0,e )+H (0, p)
+∑I∇ IV scf ({R})(r )uI+
12∑I∑J
∇ I∇ JV scf ({R})(r)u IuJ
H (0)δH e−p
(1)δH e−p
(2)
Ei(0)=ϵe+ϵpnp
∣Ψi(0) ⟩=∣e ⟩∣np ⟩
∣Ψi ⟩ , E i
δ Ei
Using the Rayleigh-Schrödinger Perturbation Theory, we get the
correction to the energy
δ Ei=⟨Ψi(0)∣δH e− p
(1) ∣Ψi(0)⟩+⟨Ψ i
(0)∣δH e−p(2) ∣Ψi
(0)⟩+⟨Ψi
(0)∣δH e−p(1) ∣Ψi
(1)⟩+...
First order PT Second order PT
V scf ({R})(r)=V ion(R)+V H ({R})(r )+V xc({R})(r )
The Heine-Allen-Cardona Approach (II)The Heine-Allen-Cardona Approach (II)
δ Ei=⟨Ψi(0)∣δH e− p
(2) ∣Ψi(0)⟩+⟨Ψ i
(0)∣δH e− p(1) ∣Ψi
(1)⟩
12∑IJ
⟨e∣⟨np∣∇ IJ2 V scf uI uJ∣np⟩∣e ⟩
We assume that [Ei(0)−H (0)
](−1)
≈(ϵe−H (0,e))(−1)
Phonons as a thermal bath !!!
δ Ee(β)=∑IJ[12⟨∇ IJ
2 V scf ⟩e+⟨∇ IV scf
∇ JV scf
ϵe−H(0, e)⟩e
] ⟨np∣u I uJ∣np⟩β
e
∑IJ⟨e∣⟨np∣∇ IV scf uI
∇ JV scf
Ei(0)−H (0)
uJ∣np ⟩∣e⟩
e→n k
β=1KT
Debye-Waller Fan
The Heine-Allen-Cardona Approach (III)The Heine-Allen-Cardona Approach (III)
δ En k (β)=∑IJ[12⟨∂2V scf
∂RI ∂RJ
⟩+∑m p(En k−Em p)
−1⟨∂V scf
∂RI
∣m p⟩ ⟨m p∣∂V scf
∂RI
⟩]⟨uI uJ ⟩
δ En k(β)=∑IJ{[≈ ]
∑n p
e−βn p ϵp ⟨np∣uI uJ∣np⟩
Z}
Thermal average
Bose Function 1+2B (ϵp)=1+2
eβϵp−1
uI∝∑p(b−p
+ +b p)
Clear dependence on the temperature
Polaron damping neglected
The Heine-Allen-Cardona Approach (IV)The Heine-Allen-Cardona Approach (IV)
The theory satisfies the “translational invariance condition”
∑I⟨n k∣
∂V scf
∂RI
∣m k ' ⟩u I=∑q λgnm k
qλ δk ' ,k−q(b−qλ+ +bqλ)
∑IJ⟨n k∣
∂2V scf
∂RI∂RJ
∣n k ⟩u IuJ=∑q λΛnn ' k
qλ (b−q λ+ +bq λ)(bqλ
+ +b−q λ)
Electron-Phonon Matrix Elements
p→qλ ,ϵp→ωq λk k⃗−q⃗
q
k kq
q
δ En k (β)=∑q λ n '[
∣gn n' kqλ ∣
En k−En ' k+q
−Λnn ' k
q λ
En k−En ' k
](2B (ωq λ)+1)
Generalized Eliashberg Function Eliashberg function allows to visualize which phonon modes contribute to the electronic energy renormalization
Top Valence Band “Bottom” Conduction Band
to the role played by the FAN and DW in order to satisfy the translational invariance of the theory
g2FFAN (ω)=∑q λ[∑n '
gnn ' kq λ N q
−1
ϵn k−ϵn ' k ']δ(ω−ωqλ)
g2FDW (ω)=∑qλ[∑n '
Λn n' kqλ (−2N q)
−1
ϵn k−ϵn ' k]δ(ω−ωq λ)
δ En k (β)=∫dω(g2 FFAN (ω)+g2FDW (ω))(2B(ωq λ)+1)
Temperature dependence
Theory of Polarons: the MBPT perspective
Electron-phonon Self Energy Electron-electron Self Energy
Σ=iGD Σ=i GW
http://arxiv.org/abs/1304.0072E. Cannuccia and A. Marini (submitted PRB)
Many Body
States are not accessible All interactions between electron and phonons are included in the
Electron-Phonon Self Energy Operator
Heine Cardona Allen approach
“Standard” Rayleigh-Schrodinger 2nd order perturbation theory
Solid St. Comm. 133, 3 (2005)
PRB 33, 5501 (1986)PRB 23, 1495 (1981)
∣ϵnk−ϵn ' k−q∣≫ωq λ
STATIC & ADIABATIC
limitω≈ϵnk
Zn k=1
Enk=ϵnk+Zn k(ΣnkFan(ϵnk)+Σnk
DW )
Γnk=ℑΣnk (ϵn k)
Theory of Polarons: from MBPT to HAC approach
Outline... Outline...
Finite temperature excitons
Ab-Initio Polarons
Spectral functions and the QP-approximation
Excitons: the polaronic pictureExcitons: the polaronic picture
The poles of Lare the eigenstates of the Bethe-Salpeter Hamiltonian
HK , K 'el
=(ϵe−ϵh)δK ,K '+(v−W )K ,K '
Quasiparticle energies are real in the optical
range
The BS Hamiltonian is Hermitian
+ -=
L=L0−i L0[v−W ]LQuasihole and quasielectron
K=(e ,h)
Excitons: the polaronic pictureExcitons: the polaronic picture
The poles of Lare the eigenstates of the Bethe-Salpeter Hamiltonian
HK , K 'el
=(ϵe−ϵh)δK ,K '+(v−W )K ,K '
Quasiparticle energies are real in the optical
range
The BS Hamiltonian is Hermitian
+ -=
L=L0−i L0[v−W ]LQuasihole and quasielectron
K=(e ,h)
Excitons: the polaronic pictureExcitons: the polaronic picture
The BS Hamiltonian is NOT Hermitian
=Quasihole and quasielectron
2 ,T ∝∑S T −E T
−1
τλ(T )∝[ℑ(Eλ (T ))]
−1
polaronspolarons
HK ,K '(T )=(Ee(T )−Eh (T ))+i [Γe(T )−Γh(T )]−i (v−W )K , K'
AM, AM, Phys. Rev. Lett.Phys. Rev. Lett. 101101, 106405 (2008), 106405 (2008)
Finite T excitonsFinite T excitons
Bright to dark (and vice versa) transitions
...gradual worsening of optical efficiency
Outline... Outline...
Finite temperature excitons
Ab-Initio Polarons
Spectral functions and the QP-approximation
Quasi particle SF
Green's functions: an (over)simplified pictureGreen's functions: an (over)simplified picture
Real particle SF
Spectral Function
Enk=ϵnk+Zn k(ΣnkFan(ϵnk)+Σnk
DW )
QP approximation
Enk=ϵnk
Enk=ℜ(Enk)+ iℑ(Enk)
Gn k(ω)=1
ω−ϵnk−ℜΣn kFan
(ω)−Σn kDW
−iℑΣn kFan
(ω)
An k(ω)=ℑΣn k
(ω−ϵn k−ℜΣn k)2+(ℑΣnk)
2
An kqp(ω)=
ℑEn k
(ω−ℜEn k)2+(ℑEn k)
2
The case of The case of DiamondDiamond
Logothedis et al. PRB 46, 4483 (1992)
-670 meV
“… a disagreement concerning the energy position of the first direct gap and its origin...”
E. Cannuccia, E. Cannuccia, Phys. Rev. Lett.Phys. Rev. Lett. 107, 255501 (2011) 107, 255501 (2011)
Trans-polyacetylene
C-based nanostructures: polymersC-based nanostructures: polymers
Polyethylene
Zero-Point Motion
√ ⟨u2 ⟩≈0.4 a.u.√ ⟨u2 ⟩≈0.1a.u.√ ⟨uC
2 ⟩≈0.2a.u.
√ ⟨uH2 ⟩≈0.3a.u.
Breakdown of the QP pictureBreakdown of the QP picture
E. Cannuccia, E. Cannuccia, Phys. Rev. Phys. Rev. Lett.Lett. 107107, 255501 (2011), 255501 (2011)
Conclusions... Conclusions...
Finite temperature excitons
Ab-Initio Polarons
Spectral functions and the QP-approximation
HAC approach = Static Perturbation Theory and as an adiabatic and static limit of the MBPT approachEvaluation of the renormalization of the electronic
energy as a function of T
The coupling with the lattice vibrations modifies the state-of-the-art picture of the excitonic states based on a frozen-atom approximation.
Polaronic-induced effect can be HUGE. They can even lead to the breakdown of the electronic
picture.