1

Dependence of Optical Spectra on Temperature and

Isotopic MassManuel Cardona

Max-Planck-Institut für FestkörperforschungStuttgart, Federal Republic of Germany

European Union

ICSE-Albany, May 27.2010Querétaro, 18 Agosto. 2010

2

Dielectric function of germaniumobtained from reflectivity spectrumby Kramers-Kronig analysis, at 300KPhilipp and Taft Phys Rev 113 1001 (1959)E1 transitions should be spin-orbit split butnot enough resolution

Pseudopotential band structure of gerrmanium including spin-orbit

interaction (0.2 eV for E1 , E1 + Δ1 )

Fundamentals of Semiconductors, Yu+Cardona

Experimental discoveryof spin-orbit splitting ofE1 gapTauc and AntončikPRL 5 253 1960

Germanium

Spin-orbit splittings have played a very important role in the interpretation of themeasured optical spectra in the visible and eV. Most state of the art codesto perform such calculations do not include them. They also assume that the atomsare at fixed lattice positions and neglect thermal vibrations, including zero-pointquantum mechanical vibrational effects.

Reflectivity + Kramers Kronig Critical points: Band StructureE1 –spin-orbit splitting

3Benedict, Shirley, Bohn, PRB 57, R9385 (1998): Electron-hole quasiparticles plus direct (screened) and exchange (unscreened) exciton interaction

Solid line:Measured,Unfortunately 300K

Calculation lower thanExper. because no exciton interaction

Calculations:No temperature,No zero-point vibrationsNo spin-orbit coupling

E1 E2

E1

E2

E1,E1+Δ1

E2

State of the art: theory versus ab-initio calculations

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Hahn, Seino, Schmidt, FurtmüllerBechstedt, p.stat. sol.(b) 242 2720 (2005)

Exp. at 300 K

Shift 0.2eVIn part 300K E1

silicon

EllipsometryAspnes+Studna300K

Diamond, ellipsometry 300K

1.5 eV

R.Ramírez et al.,Phys Rev B77,045210 (2008); Path Integral Molecular Dynamicscalculation

Diamond Zero point 0.7 eVThe lowest direct gap of diamond

Is lowered by 0.7 eV by e-p interact.(right). The shift in the lower flank ofIm ε (ω) may be due, in part to e-p

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Albrecht et al.PRL80,4150(1998)Cardona et al.PRL83,3970(1999)

Albrecht et al. PRL 80.4510(1998)Cardona, Lastras et al. PRL 83.3970(1999)

State of the art calculation, no phonons

Ellipsometry at 20K, Lautenschlageret al. PRB 36.4821(1987)Ellipsometry at 300K

Zero point?90meV

No excitons

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The electronic properties of crystals are affected by the lattice vibrations throughelectron-phonon interaction. These effects depend strongly on the temperature Tbut do not disappear even at T=0 because of the zero-point vibrations. The T=0 effects are quantum effects which decrease with increasing isotopic mass

Below is shown as an example the temperature dependence of the spectra of theimaginary part ot the dielectric function of Gallium Arsenide for several T ‘s,Measured by ellipsometry.

P.Lautenschlager et al., PRB 35, 9174 (1987)

Photon Energy

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Comparison of calculated with measured imaginary part of dielectric function

of Gallium Arsenide. Calculations performed by averaging 2 for various static

phonon displacements „frozen phonons“ whose amplitude depends on temperature. Lots of computation. No SO splitting, zero-point „by hand“See Ibrahim, Shkrebtii et al, PRB 77, 125218 (2008), theory and experiment.Probably first such calculation! Phonon displacements determined by molecular dynamics. Use supercells.

It is possible, and easier, to calculate the shift with temperature and zero-point vibs. of specific critical point energies. This will be discussed next

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Recent work by A. Marini, PRL 101,106405 (2008)ab initio calculation of imaginary part of dielectricfunction of silicon including electron-phononinteraction and effects of this interaction onexciton effects. Red dotted are experimental dataof Lautenschlager et al., PRB 36, 4821 (1897)

Theory, noexciton

measured

Theory with,exciton

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Three electron-phonon interaction terms must be considered to lowest order in T :

1. Thermal expansion: can be included as a change of the lattice constants with T or with isotopic mass at T=0 : Quasistatic approximation (somewhat trivial)

2. First and second order electron phonon interaction: renormalization of electron propagators:

u u u2

At high T, <u2> kT , independent of M (classical effect)

At low T, <u2> M- ½ , zero point vibrations (quantum)

lnln

dEE V

d V , ,

,,[2 (ln 1])q j q j B j

jq

q

VBV

n

12

1

2

; 4M

4M

f

f M

Zero point

<<u2>>

Complications at low T discussed later

Fan termsAntončík terms

Force constant

2 12

[exp( / 1{2 }) 1]4

u kTM

ww

-< > -=< + >h

h

By measuring two isotopes one can determine zero point effect

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Natural Abundance of some Stable Isotopes Carbon 12C 99%; 13C 1%; 14C unstable (dating) ; Nitrogen 14N 99.6% ; 15N 0.4%

Silicon 28Si 92%; 29Si 5%; 30Si 3%

Germanium 70Ge 22%; 72Ge 28%; 73Ge 8%, has nuclear magnetic moment I = 9/2; 74Ge 35%; 76Ge

6%

Tin (gray) 112Sn 1%; 114Sn 0.6%; 115Sn 0.4%; 116Sn 14.3%; 117Sn 7.6%; 118Sn 24.2%

119Sn 24.2%; 120Sn 32.6%; 122Sn 4.7%; 124Sn 5.8%

Copper 63Cu 69.2%; 65Cu 30.8% nuclear magnetic moment

Chlorine 35Cl 75.8%; 37Cl 24.2% “ “ “

Zinc 64Zn 48.6%; 66Zn 27.9%; 67Zn 4.1%; 68Zn 18.8%

Sulfur 32S 95.2%; 33S 0.75%; 34S 4.21%; 36S 0.02%

Information on electron-phonon effects can be obtained either from the temperature dependence or from isotope effects at T=0

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A single crystal of 29Si (5% natural isotopic abundance) grown by VITCON (Germany, Dr.Pohl) from 29Si separated in Russia by EFFECTRON (Moscow, Zelenogorsk)

Because of its nuclear spin, 29Si has been suggested for the realizationof quantum bits [T.D. Ladd et al., PRL 89, 017901 (2002)]

1 cm

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99.95% 28 Si grownfor making two ~ 1kg spheres to to replacethe present Pt-Ir massStandard.

P. Becker et al., Meas. Sci. Tech 20,092002(2009)

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113Cd is a very strong neutron absorber: no INS measurements possible in natural (12% 113Cd) CdS and CdSe crystals (wurtzite structure).One can make crystals with 114Cd or 116Cd instead).

The small GaN crystal (also wurtzite) was used for inelastic x-ray scattering: T.Ruf et al., PRL 86, 906 (2001)

ΔE(Cd)~MCd-1/2

ΔE(Se)~MSe-1/2

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0.95

Linear extrapolation

E0= 60meV

Parks et al. PRB 49,14244 (1994)

2.9meV

Renormalization of the direct gap of Geby electron-phonon interaction

The renormalization E0 can be determined either by the linear extrapolation of E0(T) or from the dependence of the gap on isotopic mass which is proportional to M-1/2

The extrapolation gives 60meV

The isotope shift 2.9x2x(73/6)=70meV

The most recent semi-ab-initio calculation [Olguin et al.SSC 122 575 (2002)] gives 64meV

This renormalization should be taken into account when state of the art ab initio calculations of the gap are comparedwith experimental results.

Can be fitted withsingle oscillator:Einstein model

ΔE(T)= ΔEo[ 1+2(1-exp(ωE /kT)-1]

15A.T.Collins et al., PRL 65,891(1990)

Dependence of the indirect gap of diamond on temperature and isotopic mass

term of average frequency1080 cm-1. The Ramanfrequency of diamond is1300 cm-1. = 370meV

Fit of Eind(T) with a single Bose-Einstein

Linear

extrapolation

Isotope shift=15meV

The T=0 renormalization calculated from the isotope shift using the M-1/2 law is = 15meVx2x(12/1)=360meV,

VERY LARGE! (= 60meV for Ge, 65 meV for Si )

Computational band structure theorist beware!!Tiago, Ismail-Beigi, Louie PRB 69 125212 (2004)Si, Ge and GaAs, GW approximation, no e-p interact.

Electron-phonon interaction calculations byZollner et al.[ PRB 45, 3376 (1992)] usingsemiempirical pseudopotentials give an even larger value: = 600meV !!! STRONG ELECTRON-PHONON INTERACTION:SUPERCONDUCTIVITY?

13C

12C

Δ(M) = AM-1/2= 370 meVΔ(M13)- Δ(M12)=-(1/2)AM-3/2

=-(1/2)• Δ(M)•M= 14.8 meV

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7.1

7.2

7.3

7.4

1000

Classical molecular dynamics

ΔE = 0.7 eV

Measured,Logothetidis et al.PRB 46, 4483 (1992)

Squares ,Quantum (path integral)MD calculations.

Ramírez et al.PRB 73,2452022006

For diamond the volume effectis negligible:0.03eV at T=0compared with 0.7 eV total

Lowest direct gapof diamond (E‘

o),Γ25‘ Γ15

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Superconductivity in boron-doped (~2x1021 cm-3)diamond films produced by CVD

Y.Takano et al., APL 85, 2851 (2004); Ekimov E.A. et al. Nature 424, 542(2004)

More recent Tc =11.4 K

Y. Takano et al.

Diamond and rel.mat.

16, 911 (2007)

Because of strong electron-phonon interaction, superconductivity possible

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Recent calculations:Giustino, Yates, Souza, Cohen, Louie U.C. Berkeley PRL 98, 047005 (2007)

Use millions of points in k-space and either C-B virtual crystalor a supercell with boron in the center. Include boron vibrations

Find for Nh~ 2x1021 cm -3 Tc= 4K

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Why is the electron-phonon interaction so muchstronger in diamond than in germanium or silicon?

The large e-p interaction with the top of the valence band seems to be a property of the elements of the second row ofthe periodic table ( carbon, nitrogen, oxygen…). We shall see next that that the e-p interaction, as obtained from the dependence of the gap on T and on M, is also large in GaN a in ZnO and, as will be seen later, in SiC.

Handwaving argument:

C, N, O.. have no p-electrons in the core and the p valence electrons, as the atoms vibrate, can get much closer to the core than in cases where p-electronsare present in the core: germanium, silicon, GaAs….

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Carbon contributionSi contrib.

TotalRenorm.

Temp. Dep. of gap: SiC, path integral molecular dynamics

Hernández, Herrero, Ramírez, Cardona, PRB 77 045210 2008

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Superconductivity in boron-doped siliconE. Bustarret et al., Nature 444,465 (2006)

1. (5x1021 holes/cm3)2. (2.8x1021 „ )

Silicon Carbide

Zhi-an Ren, J. Akimitsu et al., J.Phys. Soc. Jap.76, 103710ß (2007)

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Specific heat of diamond vs.T: Einstein, Nernst-Lindemann, Debye

Debye, TD =1830 KAnn. Phys. 39,789 (1912)

Einstein, TE =1450 Kaccording to Debye

Einstein, TE =1300 KAnn.Phys. 22, 180 (1907)

Nernst-LindemannZ.Elektrochemie 817 (1911)

Fre

quen

cy,T

hz

Reduced wavevector

Nernst-Lindemann

experimental data

So far we used fits with 1 Einstein oscillator, Sometimes two Einstein oscillators better

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Walter Nernst F.A. Lindemann,Viscount Cherwell

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170meVGaN

Temperature and isotope effects in binary compounds: GaN

The zero-point vibrations of different atoms contribute differently to T=0 renormalizations: Different atoms show different isotope effects, especially if their M’s are very different:GaN

So far we have used single oscillator B-E fits (Einstein model of Cv). For binaries with very different M’s two oscillators are needed (Nernst-Lindemann Cv). The Debye model may be necessary at very low temperatures, as it is for Cv

0 2 [2 ( / ) 1

2 [2 ( / ) 1

( ) ]

]

Ga B GaGa

N B NN

dEM n kT

dM

dEM n kT

d

E T E

M

dE/dMN=4.2meVdE/dMGa=0.4meV

N =790cm-1

Ga =350cm -1

For diamonddE/dMC= 7meV/atom!,why C and N so large? No p- electrons in core?

The temperature shift gives directly the sumof anion and cation contributions. The isotope effects are more informative since they give the separate contributions

Manjon et al.Europ.phys.j. B40,4553(2004)

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Exc

itoni

c ba

nd g

ap v

s. M

zn +

Mo

(MZn + Mo) (amu)

Zinc 29meV

Oxygen 135 meV

Contribution of zinc and oxygen to the zero-point renormalization of the ZnO gap

For GaN it is:Gallium 56 meV, nitrogen 203 meV

For diamond it is 360meV= 180 +180 meV

RB Capaz… and SG Louie PRB 94, 036801 (2005) use twoOscillators (breething and squashing modes) to fit T-dependence of gaps of nanotubes

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Anion and cation contributions to the zero-point renormalization andthe temperature dependence of gaps can be very different. For CuCl,CuBr, and CuI they even have opposite signs! Unusual effects appear.

BranchCu-like

branch Cl-like

65CuCl63CuCl

65CuCl

Cu37ClCu35Cl

dE0/dMCu= -0.076meV/amu

dE0/dMCl=+0.036meV/amu

The isotope data have been crucial for understandingthe strange behavior of E0 ( T) in copper halides. CuGaS2 ?

A.Göbel, PRB 57, 15183 (1998)

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Periodic Table ( Section)

Mg Al Si P S Cl

Cu Zn Ga Ge As Se Br (3d-core electrons)

Ag Cd In Sn Sb Te I (4d-core electrons)

-10 eV

-20 eV

-30 eV

-40 eV

-50 eV

Cor

e d-

elec

tron

s

Cu and Ag d-core electrons hybridize withvalence p-electrons

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Low temperature behavior of E0 (T): Debye Model

The average energy of the phonons is:

0

1

2[ ( / ) ] ( )phonon B dn kT N d

<E

The contribution of electron-phonon interaction to the gap is obtained by replacing [nB(+ 1/2)] by a coupling constant Ce-p multiplied by < u2>. Since < u2> (M ) -1 we find for the gap renormalization at T0:

0

1[ ( / ) ] ( )

2e p

d

Cn kT N d

This equation differs from that above by a factor of -2 in the integrand. However:Ce-p must vanish for ω 0 ( 0 phonons are a uniform translation) and Ce-p must be proportional to 2 . Hence in the low T limit behaves like <Ephonon> T4

T4 for T 0

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Sharp indirect edge luminescence in isotopically pure Si

The T4 dependence of the gap of a semiconductor at low T( T<10K) was first observed for 28Si. The reason is that the shift is smaller than the widthsfor natural Si. Karaiskaj et al., PRL 89, 016401(2002)

Width 10eV, slit limited

The luminescence peaks of isotopically pure 28Si are much sharper than thoseof natural Si (92% 28Si, 5% 29Si, 3% 30Si). The reason is that within an excitonicradius Rex fluctuations of the isotope composition are sampled giving a fluctuation ofthe gap which depends on isotopic composition. Excitons bound to neutral boron.

Rex

28Si

29Si

30Si

28

30

0.15GHz=

0.08GHz=0.3eV

0.6eV

T4

Very high resolution(< 0.2 eV) excitation spectroscopy measurementswere performed last year by M.Thewalt et al. on pure 28Si. They confirmed, for the first time, the T4 dependence of the gap shift.

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( T in Kelvin)

(F

in G

Hz)

Cardona,Meier,Thewalt,PRL 92,196403(2004)

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Ultrahigh resolution of photoluminescence excitation spectrum of excitons bound to neutralphosphorus in isotopically pure Si (99.991% 28Si). Width of exciting laser: 0.3neV. Observedwidths: 150neV(inhomogeneous still, should be ~ 5 neV).

Exciton boundto neutralP

Neutral donor

These lines are split by half the hyperfine splitting = 486/2 = 243 neV

M.L.W. Thewalt et al. JAP 101 081724 (2007)

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90 meV, confirmed by isotope measurements, 28Si vs. 30 Si

P. Lautenschlager et al., P.R B31, 2163 (1985)L.F. Lastras Martínez et al. PRB 6112946 (2000)

Germanium

Germanium

E1,

E1 +

Δ1 (e

V)

Rönnow, Lastras et al. Eur. Phys. J. 5, 29 (1998)

90 meV

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CONCLUSIONS1. State of the art calculations of dielectric function of semiconductors is in good shape but (most) do not includeelectron-phonon interaction (Temperature dependence)nor spin-orbit coupling

2. Measurements of dependence on temperature and isotopic mass are beginning to be compared with state of the art ab initio calculations 3. Particularly interesting are the isotope effects in binary or multi-nary materials ( ZnO, CuGaSe2)4. Dependence on mass at T~ 0 is strong for atoms of the second row of the Periodic Table ( C, N, O ). It seems related to the appearance of superconductivity in B-doped diamond

5. Superconductivity has been seen in heavy B-doped Si, SiC It should be observable in heavy p-type GaN, ZnO… p-type InN:R.E.Jones,E.E. Haller et al., PRL 96,12505 (2006) ?

6. Ultra- sharp edge emission spectra for isotopically pure materials: hyperfine splittings

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Is it possible to measure or calculate the temperature dependence of the absolute band energy (not the gap difference)? Absolute with respect to what? Infinity? A deep core level?

Measurements by M. Beye et al: N. J. Phys. In press.

Calculations: Lautenschlager et al PRB 36 4821 (1987)

Cardona and Gopalan, Progress in electron properties of solids, 1989 p.51, Kluwer