Molecular fingerprints in the electronic properties of crystalline organic

semiconductors: from experiment to theory

Sergio CiuchiIstituto Sistemi Complessi-CNR and Dipartimento di Fisica Università dellʼAquila

Collaborations

S. Fratini: DMFT

X. Blase, C. Faber: ab-initio GW calculations

R. C. Hatch, H. HöchstARPES measurements

Probe Theory

Outline Introduction

ARPES: the role of disorder and intramolecular vibrations

ARPES: temperature dependence and the role of intermolecular vibrations

Concluding remarks

Outline Introduction

ARPES: the role of disorder and intramolecular vibrations

ARPES: temperature dependence and the role of intermolecular vibrations

Concluding remarks

Organics

GraphenePolyacetylene

Pentacene

Chlorophyll

DNA

Fullerene

Organics

GraphenePolyacetylene

Pentacene

Chlorophyll

DNA

Fullerene

The molecule

HOMO

LUMO

ener

gy

pentacene

rubrene

The crystal

Van der Waals bonding Narrow bands Weak bonding forces large thermal molecular displacement Low symmetry large anisotropy

HOMO

LUMO

ener

gy

[K. Hummer and C. Ambrosch-Draxl, Phys. Rev. B 72, 205205 (2005)]

Band dispersion by ARPES

[Hatch et al. PRL010]

pentacene

Narrow bands of molecular origin seen in pentacene and rubrene Large anisotropy (Wy≈0.4eV Wx < 0.05eV) Temperature dependent broadening e-ph interaction

rubrene

[Yamane et al PRB07]

[Machida et al. PRL010]

Band dispersion by ARPES

[Hatch et al. PRL010]

pentacene

Narrow bands of molecular origin seen in pentacene and rubrene Large anisotropy (Wy≈0.4eV Wx < 0.05eV) Temperature dependent broadening e-ph interaction

rubrene

[Yamane et al PRB07]

[Machida et al. PRL010]

Band dispersion by ARPES

[Hatch et al. PRL010]

pentacene

Narrow bands of molecular origin seen in pentacene and rubrene Large anisotropy (Wy≈0.4eV Wx < 0.05eV) Temperature dependent broadening e-ph interaction - disorder

rubrene

[Yamane et al PRB07]

[Machida et al. PRL010]

! !

!"#$%$&'()*+,-(

,.&/01.2.(34.5"./(6789:;

,/0<<$2=0&(%">(,

?2&/$2@$1(!"#$%$&'

AB#/.2.C(D0%%(µ37"EF"/"G(7A89H;

Carrier mobility

carrier mobility in FETʼs unveils a “power-law” temperature dependence as far as the role of static disorder is reduced by increasing gate voltage/temperature

intrinsic mobility could be explained in term of interaction with intermolecular vibrations [Troisi et. al PRL 96, 086601, (2006), S. Fratini et. al. PRL 103, 266601 (2009)]

M. E. Gershenson et al. Rev. Mod. Phys. 78, 973 (2006)

Outline Introduction

ARPES: the role of disorder and intramolecular vibrations

ARPES: temperature dependence and the role of intermolecular vibrations

Concluding remarks

ARPES Pentacene: Expt. vs DFT

Two inequivalent Pn sites results in two HOMO bands H1,H2 Narrow bands with well-defined dispersion trough the BZ Wexp = 450±15 meV DFT calculation (W=348meV) [H. Yoshida, et.al PRB 77, 235205 (2008)] and GW (W=360meV) [M. L. Tiago, et. al. PRB 67, 115212 (2003)] give smaller BW

Crystalline film on Bi(001) substrateHOMO band dispersion at T=75K

[S. C., R. C. Hatch, H. Höchst, C. Faber, X. Blase, S. Fratini arXiv:1111.2148 (2011) ]

EMV coupling molecule (B3LYP - GW)

correlations (GW) increase the total EMV coupling by 50% w.r.t. standard DFT total intramolecular EMV EP = 69 meV (INDO=72.7meV, B3LYP=45meV) intramolecular polaron e-ph coupling λ = 2 EP / W = 0.4

molecule

0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400 1600

ε P (m

eV)

Wavenumber (cm-1)

GWB3LYP

[GW calculation by S. C. et al. arXiv:1111.2148 (2011) ]

EMV coupling molecule (B3LYP - GW)

correlations (GW) increase the total EMV coupling by 50% w.r.t. standard DFT total intramolecular EMV EP = 69 meV (INDO=72.7meV, B3LYP=45meV) intramolecular polaron e-ph coupling λ = 2 EP / W = 0.4

INDO Calculations by [A. Girlando, et al., J. Chem. Phys. 135, 084701 (2011)]

molecule

0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400 1600

ε P (m

eV)

Wavenumber (cm-1)

GWB3LYP

[GW calculation by S. C. et al. arXiv:1111.2148 (2011) ]

EMV coupling crystal (INDO/S)

separation of energy scales between low energy intermolecular modes and high energy intramolecular modes

Intramolecular (Holstein) HARD: ω0≈120-200meV ≈ bandwidth Intermolecular (Periels) SOFT: ω0≈3-20meV << bandwidth

Peierls

INDO Calculations by [A. Girlando, et al., J. Chem. Phys. 135, 084701 (2011)]

Holstein

by courtesy of A. Girlando

DFT+GW+DMFT

Non perturbative approach in both disorder and EMV interaction relevant parameters:

- Intramolecular Holstein EMV coupling λ = 2 EP / W = 0.4 - Einstein model phonon frequency Ω/W=0.5 - Anderson local gaussian disorder variance Δ/W=0÷0.5

DFT

GW

DMFTloop

local gaussian disorder

tight-binding params.

EMV interactionA(k,ω)

[S. C., R. C. Hatch, H. Höchst, C. Faber, X. Blase, S. Fratini arXiv:1111.2148 (2011) ]

ARPES Pentacene: Expt. vs DMFT

DMFT with EMV from GW + Disorder accounts for observed BW Disorder is large (Δ=75±15meV) (single crystal≈10meV,amorphous≈100meV [W. L. Kalb, et al. PRB 81, 155315 (2010)])

ARPES Pentacene: Disorder vs EMV

Disorder causes a uniform increase of the band dispersion, no effect on band splitting Disorder alone is not able to reproduce observed BW EMV coupling with intramolecular modes provide a further enhancement of BW EMV coupling enhances the band splitting

ARPES Pentacene: hallmarks of EMV

Large spectral weight inside the H1/H2 gap H1 phonon overtone shifted and broadened by disorder -2 -1.5 -1

inte

nsi

ty (

arb

. units

)E-EF (eV)

! H1H2

W"

W

d)

!c)

b)

a)

expt

theory

EP>0"=0

EP=0">0

Outline Introduction

ARPES: the role of disorder and intramolecular vibrations

ARPES: temperature dependence and the role of intermolecular vibrations

Concluding remarks

ARPES: finite temperatures

[Hatch et al. PRL010]

pentacene

Narrow bands of molecular origin seen in pentacene and rubrene Large anisotropy (Wy≈0.4eV Wx < 0.05eV) Temperature dependent broadening e-ph interaction - disorder

rubrene

[Yamane et al PRB07]

[Machida et al. PRL010]

ARPES: finite temperatures

Narrow bands of molecular origin seen in pentacene and rubrene Large anisotropy (Wy≈0.4eV Wx < 0.05eV) Temperature dependent broadening e-ph interaction - disorder Temperature band narrowing

[Hatch et al. PRL010]

pentacene

Polarons?

1d model (M-Γ direction)

disorder+dimerization

Intermolecular e-ph interaction: classical phonon approx Intramolecular e-ph interaction: finite T modified DMFT scheme (reduces to MA(0) as T=0)

S. C., F. de Pasquale, S. Fratini, and D. Feinberg, Phys. Rev. B 56, 4494 (1997)M. Berciu, A. S. Mishchenko, and N. Nagaosa, Europhys. Lett., 89, 37 007 (2010)

self-energy matrix of the Holstein single-impurity problem

[S. C. and S. Fratini Phys. Rev. Lett. 106, 166403 (2011) ]

H = −J

i

[1− α(Xi −Xi+1)](c+i ci+1 + c

+i+1ci) +Hlocal+

+Hinter(X)− g

i

(a†i + ai)c+i ci +Hintra(a, a

†)

H0

G(ω) = (ω − Σintra − H0)−1

A(k,ω)

HOMO band including intramolecular phonons and static disorder (1d M-Γ direction) 3

a)

0 !/2a !/a

k

-3

-2

-1

0

1

2

3

"/J

#0

#0

-3

-2

-1

0

1

2

3

-!/2a 0 !/2a

"k/

J

k

HOMO2

HOMO1

b)

T=0.05JT=0.25J

FIG. 1: (a) Spectral function A(k,ω) for a hole in a HOMOband, calculated for J = 0.125eV , λSSH = 0.2, ω0 = 0.05J ,α2H = 0.33, Ω0 = 1.5J , δ = 0.3J , ∆ = 0.2J at T = 0.05J .

(b) Position of the maxima, tracking the renormalized banddispersion εk at low and room temperature (reduced Brillouinzone).

nian Eq. (1) is obtained from Eq. (7) upon averagingover 50000 realizations of disorder variables on a chainof N = 512 sites. The spectral function is A(k,ω) =− 1

π ImG(k,ω), where G(k,ω) = 1N2

i,j

eika(i−j)

Gi,j(ω)is the Green’s function in momentum space.

Results. Fig. 1a shows a photoemission spectrum cal-culated with the above procedure for holes in a one-dimensional HOMO band, ideally representing the di-rection of maximum conduction in an organic crystal,for a representative choice of parameters [slight changesin the parameter values within the range indicated af-ter Eqs. (1)-(6) do not appreciably modify the scenario].To understand the results it is instructive to analyze thedifferent terms in the Hamiltonian separately. We startfrom the interaction with the intra-molecular vibrations,H

(i), that is responsible for the most prominent featuresobserved in the spectrum. As is well known, in the caseof isolated molecules this term gives rise to “shakeoff”satellites of the molecular levels, appearing at multiplesof the vibrational frequency Ω0 [10]. The number of vis-ible satellites is set by the coupling strength, Nvib = α2

H,

and is indicative of the amount of vibrational quanta con-stituting the polaronic deformation.

In a crystalline environment the situation is more com-plex. The molecular picture is only recovered when theelectronic band dispersion is small compared to the vi-bration frequency, W Ω0. This however does not ap-ply to organic crystals, where despite the arguably nar-row bandwidths as compared to inorganic semiconduc-tors, W is still the largest energy scale in the problem.Being W > Ω0, the periodic overtones characteristic ofmolecular spectra are replaced by cuts in the band dis-persion. These features are analogous to the “kinks” thatare commonly observed in the photoemission spectra ofsolids with sizable electron-boson coupling. They adda vibrational fine structure to the spectrum [17] without

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2

(FW

HM

)/J

T/J

HOMO2

HOMO1

a)k=!/4k=!/2

k=0

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2

ba

nd

wid

th/J

T/J

b)

totalHOMO2HOMO1

Therm. exp.

FIG. 2: (a) Temperature dependence of the spectrallinewidths at different points in the BZ, for the same param-eters as in Fig. 1. (b) Widths of the individual subbands andtotal bandwidth. Open circles are obtained including the lat-tice thermal expansion calculated in Ref. [24] (the bandwidthis normalized to the zero-temperature value of J).

much affecting the overall bandwidth [7, 8, 18]. This phe-nomenon is clearly visible in Fig. 1a, where the cosinedispersion starting from the top of the band is cut outby phonon resonances at Ω0 and 2Ω0 (indicated by ar-rows). In the case Ω0 ∼ (1−2)J that is relevant to OSC,the interaction with intra-molecular vibrations effectivelysplits the HOMO dispersion into two main subbands sep-arated by a sizable direct gap, as shown in Fig. 1b. Thegap opens up where the dispersion crosses the first vibra-tional cut, which falls accidentally around k ∼ π/2. Thisresult can explain the large separation between the twoHOMO subbands of pentacene [2, 3], that is one order ofmagnitude larger than that predicted by ab-initio calcu-lations on accounts of the structural non-equivalence inthe unit cell [15, 19].

Fig. 2a reports the lifetimes for states at differentpoints of the Brillouin zone (BZ). The large difference be-tween the HOMO1 and HOMO2 branches, that was alsoobserved in Ref. [2], is an additional distinctive feature ofthe intra-molecular interaction H

(i): according to the ar-guments given above, the HOMO1 band lies by construc-tion below the threshold for the emission of a vibrationalquantum, |εk − εk=0| < Ω0, and therefore the electroniclifetime there is mostly insensitive to the effects of intra-molecular vibrations [7, 8]. Vibronic scattering processesare instead allowed in the second subband, where theycause a much larger line broadening, with linewidths ofthe order of the electronic transfer rate itself. Since onlythe HOMO1 states near the band edge can be thermallypopulated or doped in a field-effect device, we argue thatscattering from high-frequency vibrations should not playa predominant role in the transport mechanism of OSC.

The effects related to H(i) are essentially temperature

independent, because the considered vibrations cannot

3

a)

0 !/2a !/a

k

-3

-2

-1

0

1

2

3

"/J

#0

#0

-3

-2

-1

0

1

2

3

-!/2a 0 !/2a

"k/

J

k

HOMO2

HOMO1

b)

T=0.05JT=0.25J

FIG. 1: (a) Spectral function A(k,ω) for a hole in a HOMOband, calculated for J = 0.125eV , λSSH = 0.2, ω0 = 0.05J ,α2H = 0.33, Ω0 = 1.5J , δ = 0.3J , ∆ = 0.2J at T = 0.05J .

(b) Position of the maxima, tracking the renormalized banddispersion εk at low and room temperature (reduced Brillouinzone).

nian Eq. (1) is obtained from Eq. (7) upon averagingover 50000 realizations of disorder variables on a chainof N = 512 sites. The spectral function is A(k,ω) =− 1

π ImG(k,ω), where G(k,ω) = 1N2

i,j

eika(i−j)

Gi,j(ω)is the Green’s function in momentum space.

Results. Fig. 1a shows a photoemission spectrum cal-culated with the above procedure for holes in a one-dimensional HOMO band, ideally representing the di-rection of maximum conduction in an organic crystal,for a representative choice of parameters [slight changesin the parameter values within the range indicated af-ter Eqs. (1)-(6) do not appreciably modify the scenario].To understand the results it is instructive to analyze thedifferent terms in the Hamiltonian separately. We startfrom the interaction with the intra-molecular vibrations,H

(i), that is responsible for the most prominent featuresobserved in the spectrum. As is well known, in the caseof isolated molecules this term gives rise to “shakeoff”satellites of the molecular levels, appearing at multiplesof the vibrational frequency Ω0 [10]. The number of vis-ible satellites is set by the coupling strength, Nvib = α2

H,

and is indicative of the amount of vibrational quanta con-stituting the polaronic deformation.

In a crystalline environment the situation is more com-plex. The molecular picture is only recovered when theelectronic band dispersion is small compared to the vi-bration frequency, W Ω0. This however does not ap-ply to organic crystals, where despite the arguably nar-row bandwidths as compared to inorganic semiconduc-tors, W is still the largest energy scale in the problem.Being W > Ω0, the periodic overtones characteristic ofmolecular spectra are replaced by cuts in the band dis-persion. These features are analogous to the “kinks” thatare commonly observed in the photoemission spectra ofsolids with sizable electron-boson coupling. They adda vibrational fine structure to the spectrum [17] without

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2

(FW

HM

)/J

T/J

HOMO2

HOMO1

a)k=!/4k=!/2

k=0

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2

bandw

idth

/J

T/J

b)

totalHOMO2HOMO1

Therm. exp.

FIG. 2: (a) Temperature dependence of the spectrallinewidths at different points in the BZ, for the same param-eters as in Fig. 1. (b) Widths of the individual subbands andtotal bandwidth. Open circles are obtained including the lat-tice thermal expansion calculated in Ref. [24] (the bandwidthis normalized to the zero-temperature value of J).

much affecting the overall bandwidth [7, 8, 18]. This phe-nomenon is clearly visible in Fig. 1a, where the cosinedispersion starting from the top of the band is cut outby phonon resonances at Ω0 and 2Ω0 (indicated by ar-rows). In the case Ω0 ∼ (1−2)J that is relevant to OSC,the interaction with intra-molecular vibrations effectivelysplits the HOMO dispersion into two main subbands sep-arated by a sizable direct gap, as shown in Fig. 1b. Thegap opens up where the dispersion crosses the first vibra-tional cut, which falls accidentally around k ∼ π/2. Thisresult can explain the large separation between the twoHOMO subbands of pentacene [2, 3], that is one order ofmagnitude larger than that predicted by ab-initio calcu-lations on accounts of the structural non-equivalence inthe unit cell [15, 19].

Fig. 2a reports the lifetimes for states at differentpoints of the Brillouin zone (BZ). The large difference be-tween the HOMO1 and HOMO2 branches, that was alsoobserved in Ref. [2], is an additional distinctive feature ofthe intra-molecular interaction H

(i): according to the ar-guments given above, the HOMO1 band lies by construc-tion below the threshold for the emission of a vibrationalquantum, |εk − εk=0| < Ω0, and therefore the electroniclifetime there is mostly insensitive to the effects of intra-molecular vibrations [7, 8]. Vibronic scattering processesare instead allowed in the second subband, where theycause a much larger line broadening, with linewidths ofthe order of the electronic transfer rate itself. Since onlythe HOMO1 states near the band edge can be thermallypopulated or doped in a field-effect device, we argue thatscattering from high-frequency vibrations should not playa predominant role in the transport mechanism of OSC.

The effects related to H(i) are essentially temperature

independent, because the considered vibrations cannot

Observed HOMO splitting in Pn is due to a intramolecular phonon resonance Intramolecular e-ph interaction → high energy features in ARPES (HOMO2) Intermolecular e-ph interaction → low energy damping (HOMO1) The “Band narrowing” is due to large thermal crystal expansion [M. Masino et al.,

Macromol. Symp. 212, 375 (2004)]

Outline Introduction

ARPES: the role of disorder and intramolecular vibrations

ARPES: temperature dependence and the role of intermolecular vibrations

Concluding remarks

Increasing the doping: superconductivity

Superconductivity found in Alkali-doped picene Tc=7-18K Relevance of local e-e repulsion [G.Giovannetti M.Capone Phys. Rev. B 83, 134508 (2011)] Picene vs C60 (anisotropic vs isotropic) Intermolecular e-ph interaction → resistivity Intramolecular e-ph → pairing Relevance of intercalant induced modes

[Mitsuashi et. al. Nature, 464, 76 (2010)]

0 500 1000 1500 2000 2500 3000

! (cm-1)

0

0.0005

0.001

0.0015

0.002

"2 F(

!)/!

Picene (C22H14)

[A.Subedi L.Boeri PRB 84, 020508(R) (2011)]

[M. Casula et. al Phys. Rev. Lett. 107, 137006 (2011)]

Conclusions The proper inclusion of the interaction with molecular vibrations and disorder, beyond electronic band theory calculations, provides a remarkably accurate description of the experimental photoemission spectra in pentacene

Coexistence of dispersive features characteristic of the extended band regime, together with multi-phonon shakeoff resonances reminiscent of the molecular spectra, provides solid spectroscopic evidence for the widespread idea that organic materials are located in a crossover region with both band and molecular characters

The total HOMO width exhibits a moderate increase with temperature that is entirely caused by the coupling to low-frequency intermolecular vibrations

Despite the relatively high value of the intramolecular EMV coupling via GW calculations (λ = 0.4) polarons are not formed in large molecule organics. The observed band narrowing is of thermodynamic origin

References[S. C., R. C. Hatch, H. Höchst, C. Faber, X. Blase, S. Fratini arXiv:1111.2148 (2011) ][S. C. and S. Fratini Phys. Rev. Lett. 106, 166403 (2011) ]

[S.C. S. Fratini D. Mayou. Phys. Rev. B.83, 081202(R) (2011)][S. Fratini and S.C. Phys. Rev. Lett. 103, 266601 (2009)]