Electron-phonon interactions and Resonance Raman scattering in onedimensional. carbon nanotubes
|
|
- Tyrone Oliver
- 6 years ago
- Views:
Transcription
1 Electron-phonon interactions and Resonance Raman scattering in onedimensional systems: application to carbon nanotubes José Menéndez Giovanni Bussi Elisa Molinari Acknowledgements: M. Canonico, C. Poweleit, J. B. Page, G. B. Adams Supported by the National Science Foundation
2 Infrared absorption and Raman scattering Frequency (cm -1 ) Raman shift (cm -1 )
3 Energy units for optical spectroscopy hw = hck = 2phc l k wave vector; 1/l = wavenumber 1/l µ energy 1 ev = cm cm -1 = 37.2 mev
4 Ï Ô Ì Ô Ó Conservation rules for a Raman process hw = hw + h L S W phonon hk = hk + hq L S phonon BACKSCATTERING W (cm -1 ) 300 Raman K S q phonon L L L K L [ wl - ws]( q) = w( KL) - w( q -KL) = ck -c( q -K ) = 2cK -cq 4p 500 nm Brillouin p 0.5 nm q phonon
5 Phonons in crystals Raman-active
6 What determines Raman intensities? R È S = 1 R Í R Í M «() t 4pcR 2 2 R Í R Î E ( t) = E coswt Ê ˆ M t P e iwt P ei t a() = 1 Á - * w ag + ag E Á 2Â Ë Ê ˆ P P w, u, u= c( f) d P g ab ab g g Á ab Ë P = P ( w, 0) + ab ab  d d f  f f f f g
7 Why a frequency shift? M () t = P(, tt ) E( t ) dt Ú M( w) = P( w, w ) E( w ) dw Ú If P(t,t ) = P(t-t ) fi P(w,w ) = P(w) d(w-w ) fim(w) = P(w) E(w) If a phonon is present: Ptt (, ) Pt ( + Tt, + T) Ê PÊw, w ˆ P( n) w d w w n = ( ) n T p Á - -2 Ë Â Á Ë fi w = w -nw phonon ˆ In QM this is just energy conservation!
8 The carbon nanotube family (n,n) Armchair (n,0) Zigzag (n,m) Chiral C = na + ma h 1 2
9 The electronic structure of graphene V ppp V ppp = 3.1 ev
10 Nanotubes: folding the graphene band structure Ê Á Á K Em( k) = E k 2 Á + mk, ( m = 0,..., N -1); g2dá 1 K Á Ë 2 ˆ (11,8) (15,0) (10,10)
11 Nanotubes: electronic density of states (11,8) (15,0) (10,10)
12 Vibrational modes in nanotubes
13 The radial breathing mode in carbon nanotubes w = 1170 cm R. (armchair) J. Kürti et al. Phys. Rev. B 58, R8869 (1998) w = 1170 cm -1 R (zigzag)
14 Why 1/r dependence? Let us consider an atom of mass M, The radial force is F r = 2F t cos q = F t a C-C /r 0. But F t = K s Da C-C. On the other hand, Da C-C = 2u r cos q = u r a C-C /r 0. Hence F r K a a = Ê s Ë Á ˆ Ê Á r Ë r Therefore C-C C-C 0 0 ˆ ur = K u eff r u r u r F t a C-C u r q K = = M K M Ê a Á Ë r eff s C-C w RBM 0 ˆ r 0 r 0 r0
15 Raman intensities in carbon nanotubes A.M. Rao et al., Science 275, 187 (1997)
16 Resonance Raman excitation profiles hw L Raman shift (cm -1 ) Raman shift (cm -1 ) Raman shift (cm -1 ) Raman shift (cm -1 ) Raman shift (cm -1 ) Intensity (arb. units) Laser Photon Energy (ev)
17 REP for 200 cm -1 RBM E 11 = ev Intensity h = 60 mev Photon energy (ev) M. Canonico et al., Phys. Rev. B 65, (2002)
18 Resonance Raman scattering Light Phonon Light 0 j i 0 Light j j phononi i Light I µ Â 0 0 Ê ˆ Ê ij Áhw -hw -E hw -E Á Ë Á Ë L phonon j L i ˆ 2
19 Raman cross section: quantum theory I The scattering cross section is defined as ds dw = ( radiated power into ) dw incident power per unit area The incident power per unit area is CMP ( ) IP N c Lhw L = n L Other RP IP (R.1) (R.2) where N L = number of incident photons per unit volume and c/n L is the speed of light. The radiated power is RP dw = number of incident photons hw S ( dw ) dt dw = ( NV L ) hw S ( dw ), dt (R.3)
20 Raman cross section: quantum theory II where dw/dt (dw) is the transition probability per unit time from a state with a photon (w L, k L,l L ) to a state with a photon with polarization l S and a wave vector k S within the solid angle dw plus a phonon of frequency W ph and wave vector q. For an arbitrary scattering wave vector k S the transition probability is given, according to Fermi s golden rule, by dw dt CMP Other d hw hw h 2p 2 W fi l S, l L L S  = ( ) - - h k S ( W ) ph (R.4) where W fi is the matrix element of the transition operator. Combining R.4 with R.1-3, we obtain ds wl nv L = Ê dw Ë Á ˆ ÊË ˆ w c S 2 p  Wfi S L d wl w ( l, l ) 2 - S - h k ( dw ) S ( h h hw ) ph (R.5)
21 Raman cross section: quantum theory III The sum over k S can be transformed into an integral: k S 3 Â Ú 2 S 3 kdk S S 3 ( dw ) 2p 2p h c Ú CMP Inserting this into R.5, we finally obtain V V n = = Ê w d ( ) ( ) Ë ˆ ( hw ) Other 2 S S (R.6) ds nl dw = Ê Ë Á w L ˆ w Sn ÊË 4 c S ˆ V Wfi ws S w ÊË 2 ˆ 2 L L p (, l ;, l ) (R.7) 2 h The key quantity that contains resonance effects is the transition matrix element W fi
22 The matrix element W fi I We will use perturbation theory to compute W fi. We assume that the unperturbed hamiltonian H 0 can be written as H 0 = H el + H R + H L, with: A one-electron (band structure) hamiltonian of the form A free radiation term (photons) of the form A harmonic vibrational hamiltonian (phonons): + Hel = E nsc nsc ns CMP k, n,s Other 1 ( 2 ) + HR = a a + H Â k k k Âhw k l k l k l k, l ( 1 ) 2 + = b b + L ÂhW q m q m q m q, m (R.8) (R.9) (R.10)
23 The matrix element W fi II Therefore the quantum states of the system are characterized by three sets of occupation numbers: (R.11) We will be able to simplify this notation considerably because only a few of these occupation numbers change in the scattering process. For example, the initial state consists of the electronic and phonon system in their ground CMPstates and one photon in state Other k L l L. The final state consists of the electronic system in the ground state, one photon in state k S l S, and one phonon in state qm, so that we can express these states as i f m { N } { N } { N },, l nk qm k = 0010,,, ; E = hw L S i L = 010,,, 1; E = hw + hw qm L S f S qm (R.12)
24 The matrix element W fi III The transition between these states is caused by the interaction hamiltonian H int. We will take H int = H el + H er, where H el is the electron-phonon interaction and H er the electron-photon interaction. The electron-phonon interaction can be written as nn + H = M b + b c + c el  kqm qm -qm k -q, ns kn s kqnn ms CMP ( ) Other (R.13) Explicit expressions for M kqm will be discussed below. The electron-radiation hamiltonian arises from the A p coupling term in the interaction of an electron-system with an electromagnetic wave. It has a form similar to R.13: H er e = Ê ˆ Ë m Ê Á Ë 2p 2 Vn w K K 12  kknn ls ˆ nn ( + + e p a + a- ) c -, c l kk Kl Kl k K ns kn s (R.14) where p kk is the matrix element of the momentum operator between the periodic parts of the Bloch wave functions for states nk and n (k-k)
25 The matrix element W fi IV When applied to the specific problem at hand, the notation in these expressions can be simplified. We first limit the summations over electronic bands n, n to a single empty conduction band and a singly fully occupied valence band. We assume parabolic bands. CMP E t k Other 2me + h2 2 E t k - h2 2 2m h
26 The matrix element W fi V We keep the notation c, c + for annihilation/creation operators of electrons in the conduction band, and we introduce v, v + as annihilation/creation operators for electrons in the valence band. We also limit ourselves to the case when a photon of frequency w L is annihilated, a phonon is created, and a photon of frequency w S is created. We then rewrite the electron-phonon hamiltonian as CMP Other ( ) Â el qm kqm k -q, ns kn s kqm k -q, ns kn s kqms + cc vv H = b M c + c + M v + v (R.15) Notice that we have limited ourselves to intraband terms of the form c + c and v + v. We have not included interband terms of the form c + v, etc. We will see later that the interband contribution is negligible.
27 The matrix element W fi VI For the electron-radiation hamiltonian we further assume that the incident and scattered light polarizations are parallel to the same axis z. In nanotubes z must be the axis of the tube. Other directions give zero. We can now write with: H H e CMP Other cv = Ê 12 ˆ Ê 2p ˆ pz a c + v v + c Ë S - S - S m Á Â + 2,,, Ë Vn w + + er kk K k K s ks k K s ks S S ks 12 - er e = Ê ˆ Ë m Ê Á Ë 2p 2 Vn w L L H = H + H + - er er er, ( ) ˆ Â p vc z a c + v v +, c L + L, + + L, ks ( ) s s s s kk K k K k k K k Here we have neglected intraband terms. This is because intraband matrix elements of the momentum operator are very small for small wave vector transfers (by virtue of Green s theorem for periodic functions, see Ashcroft-Mermin, Appendix I). (R.16) (R.17) (R.18)
28 The matrix element W fi VII Since each application of H er can change one photon state (the interaction is linear in the photon creation and annihilation operators) and each application of H el can only change one phonon state, we can only transition from the initial state to the final state by two applications of H er and one application of H el. In other words, we need to go to at least third order in H int : W fi fh H H i = Â int m m int n n int ( E - E ) mn m i ( En - Ei ) CMP Other fh H H i = Â int m m int n n int mn E - hw E hw ( ) ( ) m L n - L (R.19) Notice that the initial state i> contains a filled valence band and an empty conduction band. It is then clear that only the c + v terms in R.17 and R.18 can contribute. (This is of course a manifestation of Pauli s exclusion principle.) Of these, only R.18 can lead to a resonant enhancement by annihilating the incoming laser photon and creating an electronic excitation of equal energy, so that the rightmost denominator in R.19 becomes small.
29 The matrix element W fi VIII Maximum resonant enhancement is then obtained if the energy E m is comparable to the energy E n. When the phonon energy is small compared to the separation between the conduction and valence bands, this condition is met when the middle operator causes an intraband transtion. Thus the most resonant contribution to the Raman cross section arises from W fi + - fher m m HeL n n HeR i = CMP Â Other E - E E E mn ( ) ( ) m i n - i (R.20) By inserting R.15, R17, and R18 in R.20 we obtain sums over four indices k,k, k and q. (And also over the spin index s, but this gives trivial factors of 2). There are two types of terms that yield nonvanishing contributions:
30 The matrix element W fi IX CONDUCTION BAND TERMS These terms contain matrix elements of the form + + 0v c c c c + v 0 n ( W ) + 1 S q k - K k k - k k + K k qm where 0> refers to the electronic ground state, and n(w) is the phonon occupation number at temperature T. We assume that there is one incident photon and one scattered photon, so the application of the photon creation/annihilation CMP operators gives just 1. Other These elements are zero except when k = k + K L, k = k + K S, and q = K L - K S. When these conditions are satisfied the matrix element is equal to +1.The last condition is clearly the manifestation of crystal momentum conservation. VALENCE BAND TERMS These terms contain matrix elements of the form + + 0v c v v c + v 0 n ( W ) + 1 S q k - K k k - k k + K k qm These elements are zero except when k = k + K L - K S, k = k + K L, and q = K L - K S. When these conditions are met the matrix element is equal to -1. (Using commutation properties of Fermion creation/annihilation operators.) L L (R.21) (R.22)
31 The matrix element W fi X We thus end up with a single summation over the index k.we now take into account the fact that the relevant light wave vectors are negligible relative to a reciprocal lattice vector in a typical crystal. Thus we can set K L = K S = 0 in all matrix elements and energies. We thus obtain W fi e = Ê ˆ Ë m cv vc Ê 2ph ˆ p p M cc M vv z z m m Á Ê 1 ˆ - Á Â, k, k k0 k0 Ë VnLnS CMP Ë wlws ks ( Ek - hw Other S ) Ek - hws 2 12 where E k is the energy of the intermediate state given by E k We now recognize that the states closest to to k = 0 will make the dominant contribution because they are closest to the singularity in the density of states. We can then replace the momentum and phonon matrix elements for their values at k = 0 and take them out of the sum. We then obtain ( ) ( ) hk hk hk = Et + + = E + ( m ) = m + m 2m 2m 2 m ; with * * e h t e h (R.23) (R.24)
32 W fi  ks where we have defined The matrix element W fi XI e = Ê ˆ Ë m Ê 2ph ˆ Á Ê 1 Á Ë Vn n Ë w w 2 12 L S L S 1 ( Ek - hw E CMP S ) k - ( hw ) cv P p p z = ( ) vc z, 0 z, 0 * S ˆ 2 Pz M cc vv m - Mm ( ) Other. The last factor can be converted into an integral: 1 r EdE ( - ) Ú k E hw E - hw ( E - hw + ih) E - hw + ih  s k ( ) = ( ) ( ) S k S S L (R.25) (R.26)
33 The matrix element W fi XII where we have added a phenomenological broadening parameter that represents the lifetime of the excited states. For a one-dimensional solid: Other where the first factor of 2 takes into account the spin degeneracy. Using R.24 de Hence, using R.27 we obtain h = kdk = m L rkdk = 2 Ê d k (R.27) Ë 2 p ˆ h m 2 2 * * ( ) * È2m E - E Í Î 2 h L m r( E) = Ê * ˆ È Ë ph Í Î2 E - ( ) E t t dk (R.28) (R.29)
34 The matrix element W fi XIII We thus need to calculate the following integral Ú = -12 ( E - Et ) de ( E - hw + ih) ( E - hw + ih) = L ip È Í 1 - Other 1 hw ph Î( hwl - Et -ih) hws - Et -ih These integrals are discussed in the review article by Martin and Falicov. Combining R.30 with R.29, R.25, and R.26, we finally obtain S ( ) e cc vv L m W = ( ) ( )( )( ) * p P - M n fi È Í Î m V n n w w z m L S L S ph hw - E -ih hw - E -ih ( ) L t S t m (R.30) ( )( )( ) ( W ) + 1 ( ) hw 2 12 (R.31) R. M. Martin and L. M. Falicov,"Resonant Raman Scattering", in Light Scattering in Solids, edited by M. Cardona (Springer Verlag, Berlin, 1975), Vol. 8, p. 79. m
35 Raman cross section: quantum theory IV so that the cross section finally becomes ds 1 e = Ê ˆ dw 2 Ë mc h 2 F( w ) L Ê ws S Á Ë w L ˆ Ê n Á Ë n L ˆ L Ê Á Ë W m ˆ 4 P M cc z m M vv 2 * - m m n ( ) ( Wm ) + [ 1] (R.32) with F( w ) = i L Ê Á Ë hw - E -ih hw - E -ih ( ) ( ) L t S t ˆ (R.33)
36 Raman cross section: quantum theory V The measured photon count rate at the detector is proportional to the cross section, but the proportionality factor is usually left undetermined, since the measurement of an absolute Raman cross section requires a careful calibration with a known standard. [Notice that we defined the cross section in terms of power. We can define a photon number cross section which is equal to R.1 times (w L / w S )] If no absolute cross section is measured, only the function F(w L ) is needed to analyze the experimental results, assuming that the approximation of a single conduction and a single Other valence band is valid. This is the case for armchair tubes. From fits of F(w L ) to the experimental data it is possible to determine the transition energy E t and the lifetime of the excited state. In order to study the properties of this function we define Fx ( ) = i Ê Á Ë x -1 -ied x - D -1 -ied ( ) ( ) ˆ (R.34)
37 Raman cross section: quantum theory VI There are two singularities associated with the incoming laser photon or the scattered photon having the same energy as the transition energy. Notice that the imaginary part has the expected asymmetric shape of a one-dimensional density of states, but the real part has the opposite shape, leading to symmetric intensity profiles. Intensity Imaginary part D = 0.1 e = 0.01 Other Real part Real F, Imag F Normalized laser frequency x L 1.15
38 Raman cross section: quantum theory VII A more realistic value for D in carbon nanotubes is D = As the broadening of the electronic states increases, the intensity profile evolves to a single peak Intensity D = 0.01 e = 0.10 Other Intensity D = 0.01 e = Normalized laser frequency x L Intensity D = 0.01 e = Normalized laser frequency x L Normalized laser frequency x L
39 REP for 200 cm -1 RBM E 11 = ev Intensity h = 60 mev Photon energy (ev) M. Canonico et al., Phys. Rev. B 65, (2002)
40 Trigonal warping in carbon nanotubes I The figure shows the 1D density of states for several metallic nanotubes of approximately the same diameter, showing the splitting of singularities (increasing from zero in armchair tubes to a maximum value in zig-zag tubes) due to trigonal warping in the band structure of graphene. The associated optical transitions are very close to each other and therefore one has to include two conduction bands and two valence bands in the calculation of Raman cross sections. Other From R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Phys. Rev. B 61, 2981 (2000)
41 Trigonal warping in carbon nanotubes II Because the two transitions can interfere, it is critical to know the sign and magnitude of the prefactor cc P M - M vv m z ( )( ) 2 * 12 m m in the expression for the transition matrix element. The effective mass can be obtained by fitting a parabolic band to the calculated energy bands. The matrix element of p z can be easily obtained from pseudopotential calculations of the band structure. Other For the electron-phonon matrix elements, we can derive a simple expression for the particular case m = RBM, the radial breathing mode. If we limit ourselves to this mode, the electron phonon interaction is H el H = d where d RBM is the normal coordinate for the RBM and H is the electronic hamiltonian. The derivative means that the atomic positions are displaced by along the phonon mode eigenvector. RBM d RBM (R.36) (R.37)
42 Trigonal warping in carbon nanotubes III The normal coordinate can be written in terms of the mode eigenvectors c and the atomic displacements u as d  * f c lka f M k u lka = ( ) ( ) ( ) lka where l is the cell index, k the basis index, a a cartesian index, and M(k) the mass of atom k within the unit cell. For a periodic solid, Other 1 c( lka qm) = ka NM el ( q me ) c k iqr ( lk ) (R.38) (R.39) where N c is the number of unit cells and e is a unit vector normalized to unity:  e * ( lka qm ) e ( lka q m ) = d qq d mm ka (R.40)
43 Trigonal warping in carbon nanotubes IV If we give all atoms a displacement Du( lka ) = DR e( ka RBM ), (R.41) where R is the nanotube radius, and we assume q RBM = 0, the corresponding change in the normal coordinate is Dd = N M D Other R RBMc where we have used R.38, 39, 40 and the fact that the atomic mass is the same for all atoms The electron-phonon hamiltonian thus becomes H el = 1 NM c H R d i h RBM = 2NMw c RBM H ( R b - b + ) RBMRBM (R.42) (R.43)
44 Trigonal warping in carbon nanotubes V We therefore need matrix elements of the form where we have invoked the Hellman-Feynman theorem. By comparing R37 with R.15, and using R.41, we can write cc vv M - M = -i RBM RBM = i H n R n nk k k = ER h 2NMw c h 2NMw c RBM RBM Ê Ec Ë R Et R Other Ev - R Thus the relevant quantity for the calculation of Raman cross-sections is the derivative of the transition energy relative to the radius of the nanotube. ˆ (R.44) (R.45)
45 Trigonal warping in carbon nanotubes VII E f 2.0 Energy (ev) ev 1.99 ev Other E f Energy (ev) G Z G Z -3.0
46 Trigonal warping in carbon nanotubes VIII Autovalori a G Transizioni a G Energia (ev) ev 1.99 ev 1.99 ev Other 1.62 ev Energia (ev) Raggio del nanotubo Raggio del nanotubo (A)
47 Trigonal warping in carbon nanotubes IX Squared momentum matrix element (1/Å) Square root of effective mass Other ev 1.99 ev ev 1.99 ev
48 Trigonal warping in carbon nanotubes X We can use this to compute the corresponding Raman cross section. When we reverse the sign of the radial derivative for one of the transitions, we obtain a different profile, indicating that interference effects cannot be neglected. Intensity E 2 -E 1 = 0.37 ev h = 0.07 ev Intensity Other E 2 -E 1 = 0.37 ev h = 0.07 ev de 2 /dr reversed Laser photon energy (ev) Laser photon energy (ev)
49 REP for 188 cm -1 RBM Intensity (12,6) (15,3); (16,1) (11,8) Photon energy (ev)
50 CONCLUSIONS Interference effects are important for the understanding of the Raman cross section in carbon nanotubes. The shape of the Raman cross section as a function of the excitation energy may provide an optical tool to identify the (n,m) values. It is unclear to what extend the inclusion of excitonic effects may affect the simple calculations presented here.
Summary lecture VII. Boltzmann scattering equation reads in second-order Born-Markov approximation
Summary lecture VII Boltzmann scattering equation reads in second-order Born-Markov approximation and describes time- and momentum-resolved electron scattering dynamics in non-equilibrium Markov approximation
More informationFig. 1: Raman spectra of graphite and graphene. N indicates the number of layers of graphene. Ref. [1]
Vibrational Properties of Graphene and Nanotubes: The Radial Breathing and High Energy Modes Presented for the Selected Topics Seminar by Pierce Munnelly 09/06/11 Supervised by Sebastian Heeg Abstract
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationnano.tul.cz Inovace a rozvoj studia nanomateriálů na TUL
Inovace a rozvoj studia nanomateriálů na TUL nano.tul.cz Tyto materiály byly vytvořeny v rámci projektu ESF OP VK: Inovace a rozvoj studia nanomateriálů na Technické univerzitě v Liberci Units for the
More informationFrom Graphene to Nanotubes
From Graphene to Nanotubes Zone Folding and Quantum Confinement at the Example of the Electronic Band Structure Christian Krumnow christian.krumnow@fu-berlin.de Freie Universität Berlin June 6, Zone folding
More information5 Problems Chapter 5: Electrons Subject to a Periodic Potential Band Theory of Solids
E n = :75, so E cont = E E n = :75 = :479. Using E =!, :479 = m e k z =! j e j m e k z! k z = r :479 je j m e = :55 9 (44) (v g ) z = @! @k z = m e k z = m e :55 9 = :95 5 m/s. 4.. A ev electron is to
More informationQUANTUM WELLS, WIRES AND DOTS
QUANTUM WELLS, WIRES AND DOTS Theoretical and Computational Physics of Semiconductor Nanostructures Second Edition Paul Harrison The University of Leeds, UK /Cf}\WILEY~ ^INTERSCIENCE JOHN WILEY & SONS,
More information[ ( )] + ρ VIII. NONLINEAR OPTICS -- QUANTUM PICTURE: 45 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 88
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 88 VIII. NONLINEAR OPTICS -- QUANTUM PICTURE: 45 A QUANTUM MECHANICAL VIEW OF THE BASICS OF N ONLINEAR OPTICS 46 In what follows we draw on
More informationOptical Properties of Lattice Vibrations
Optical Properties of Lattice Vibrations For a collection of classical charged Simple Harmonic Oscillators, the dielectric function is given by: Where N i is the number of oscillators with frequency ω
More informationC. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.
Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is
More informationELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES
ELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES D. RACOLTA, C. ANDRONACHE, D. TODORAN, R. TODORAN Technical University of Cluj Napoca, North University Center of
More informationSemiconductor Physics and Devices Chapter 3.
Introduction to the Quantum Theory of Solids We applied quantum mechanics and Schrödinger s equation to determine the behavior of electrons in a potential. Important findings Semiconductor Physics and
More informationRefering to Fig. 1 the lattice vectors can be written as: ~a 2 = a 0. We start with the following Ansatz for the wavefunction:
1 INTRODUCTION 1 Bandstructure of Graphene and Carbon Nanotubes: An Exercise in Condensed Matter Physics developed by Christian Schönenberger, April 1 Introduction This is an example for the application
More information2 Symmetry. 2.1 Structure of carbon nanotubes
2 Symmetry Carbon nanotubes are hollow cylinders of graphite sheets. They can be viewed as single molecules, regarding their small size ( nm in diameter and µm length), or as quasi-one dimensional crystals
More informationElectron Interactions and Nanotube Fluorescence Spectroscopy C.L. Kane & E.J. Mele
Electron Interactions and Nanotube Fluorescence Spectroscopy C.L. Kane & E.J. Mele Large radius theory of optical transitions in semiconducting nanotubes derived from low energy theory of graphene Phys.
More informationCONTENTS. vii. CHAPTER 2 Operators 15
CHAPTER 1 Why Quantum Mechanics? 1 1.1 Newtonian Mechanics and Classical Electromagnetism 1 (a) Newtonian Mechanics 1 (b) Electromagnetism 2 1.2 Black Body Radiation 3 1.3 The Heat Capacity of Solids and
More informationPHYSICS OF SEMICONDUCTORS AND THEIR HETEROSTRUCTURES
PHYSICS OF SEMICONDUCTORS AND THEIR HETEROSTRUCTURES Jasprit Singh University of Michigan McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico Milan Montreal
More informationchiral m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Three major categories of nanotube structures can be identified based on the values of m and n
zigzag armchair Three major categories of nanotube structures can be identified based on the values of m and n m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Nature 391, 59, (1998) chiral J. Tersoff,
More informationSummary lecture II. Graphene exhibits a remarkable linear and gapless band structure
Summary lecture II Bloch theorem: eigen functions of an electron in a perfectly periodic potential have the shape of plane waves modulated with a Bloch factor that possess the periodicity of the potential
More informationOptical Lattices. Chapter Polarization
Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose- Hubbard model used to describe ultracold atoms in optical lattices. We show how the AC-Stark
More informationElectrons in Crystals. Chris J. Pickard
Electrons in Crystals Chris J. Pickard Electrons in Crystals The electrons in a crystal experience a potential with the periodicity of the Bravais lattice: U(r + R) = U(r) The scale of the periodicity
More informationi~ ti = H 0 ti. (24.1) i = 0i of energy E 0 at time t 0, then the state at afuturetimedi ers from the initial state by a phase factor
Chapter 24 Fermi s Golden Rule 24.1 Introduction In this chapter, we derive a very useful result for estimating transition rates between quantum states due to time-dependent perturbation. The results will
More informationGraphene and Carbon Nanotubes
Graphene and Carbon Nanotubes 1 atom thick films of graphite atomic chicken wire Novoselov et al - Science 306, 666 (004) 100μm Geim s group at Manchester Novoselov et al - Nature 438, 197 (005) Kim-Stormer
More informationM.S. Dresselhaus G. Dresselhaus A. Jorio. Group Theory. Application to the Physics of Condensed Matter. With 131 Figures and 219 Tables.
M.S. Dresselhaus G. Dresselhaus A. Jorio Group Theory Application to the Physics of Condensed Matter With 131 Figures and 219 Tables 4) Springer Contents Part I Basic Mathematics 1 Basic Mathematical Background:
More informationHarald Ibach Hans Lüth SOLID-STATE PHYSICS. An Introduction to Theory and Experiment
Harald Ibach Hans Lüth SOLID-STATE PHYSICS An Introduction to Theory and Experiment With 230 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Contents
More informationReview of Optical Properties of Materials
Review of Optical Properties of Materials Review of optics Absorption in semiconductors: qualitative discussion Derivation of Optical Absorption Coefficient in Direct Semiconductors Photons When dealing
More informationINTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM
INTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM I. The interaction of electromagnetic fields with matter. The Lagrangian for the charge q in electromagnetic potentials V
More informationTwo-phonon Raman scattering in graphene for laser excitation beyond the π-plasmon energy
Journal of Physics: Conference Series PAPER OPEN ACCESS Two-phonon Raman scattering in graphene for laser excitation beyond the π-plasmon energy To cite this article: Valentin N Popov 2016 J. Phys.: Conf.
More informationLecture 6 Photons, electrons and other quanta. EECS Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku
Lecture 6 Photons, electrons and other quanta EECS 598-002 Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku From classical to quantum theory In the beginning of the 20 th century, experiments
More informationNeutron and x-ray spectroscopy
Neutron and x-ray spectroscopy B. Keimer Max-Planck-Institute for Solid State Research outline 1. self-contained introduction neutron scattering and spectroscopy x-ray scattering and spectroscopy 2. application
More informationReview of Semiconductor Physics
Solid-state physics Review of Semiconductor Physics The daunting task of solid state physics Quantum mechanics gives us the fundamental equation The equation is only analytically solvable for a handful
More informationElectronic and Optoelectronic Properties of Semiconductor Structures
Electronic and Optoelectronic Properties of Semiconductor Structures Jasprit Singh University of Michigan, Ann Arbor CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE INTRODUCTION xiii xiv 1.1 SURVEY OF ADVANCES
More informationA. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017.
A. F. J. Levi 1 Engineering Quantum Mechanics. Fall 2017. TTh 9.00 a.m. 10.50 a.m., VHE 210. Web site: http://alevi.usc.edu Web site: http://classes.usc.edu/term-20173/classes/ee EE539: Abstract and Prerequisites
More informationPhonons (Classical theory)
Phonons (Classical theory) (Read Kittel ch. 4) Classical theory. Consider propagation of elastic waves in cubic crystal, along [00], [0], or [] directions. Entire plane vibrates in phase in these directions
More informationLecture contents. A few concepts from Quantum Mechanics. Tight-binding model Solid state physics review
Lecture contents A few concepts from Quantum Mechanics Particle in a well Two wells: QM perturbation theory Many wells (atoms) BAND formation Tight-binding model Solid state physics review Approximations
More informationNeutron Scattering 1
Neutron Scattering 1 Cross Section in 7 easy steps 1. Scattering Probability (TDPT) 2. Adiabatic Switching of Potential 3. Scattering matrix (integral over time) 4. Transition matrix (correlation of events)
More informationMatter-Radiation Interaction
Matter-Radiation Interaction The purpose: 1) To give a description of the process of interaction in terms of the electronic structure of the system (atoms, molecules, solids, liquid or amorphous samples).
More informationBasic Semiconductor Physics
Chihiro Hamaguchi Basic Semiconductor Physics With 177 Figures and 25 Tables Springer 1. Energy Band Structures of Semiconductors 1 1.1 Free-Electron Model 1 1.2 Bloch Theorem 3 1.3 Nearly Free Electron
More informationChapter 12: Semiconductors
Chapter 12: Semiconductors Bardeen & Shottky January 30, 2017 Contents 1 Band Structure 4 2 Charge Carrier Density in Intrinsic Semiconductors. 6 3 Doping of Semiconductors 12 4 Carrier Densities in Doped
More informationMolecular spectroscopy
Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering
More informationPhysics 606, Quantum Mechanics, Final Exam NAME ( ) ( ) + V ( x). ( ) and p( t) be the corresponding operators in ( ) and x( t) : ( ) / dt =...
Physics 606, Quantum Mechanics, Final Exam NAME Please show all your work. (You are graded on your work, with partial credit where it is deserved.) All problems are, of course, nonrelativistic. 1. Consider
More informationQuantum Physics in the Nanoworld
Hans Lüth Quantum Physics in the Nanoworld Schrödinger's Cat and the Dwarfs 4) Springer Contents 1 Introduction 1 1.1 General and Historical Remarks 1 1.2 Importance for Science and Technology 3 1.3 Philosophical
More informationGroup Theory and Its Applications in Physics
T. Inui Y Tanabe Y. Onodera Group Theory and Its Applications in Physics With 72 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Contents Sections marked with
More informationLattice Vibrations. Chris J. Pickard. ω (cm -1 ) 200 W L Γ X W K K W
Lattice Vibrations Chris J. Pickard 500 400 300 ω (cm -1 ) 200 100 L K W X 0 W L Γ X W K The Breakdown of the Static Lattice Model The free electron model was refined by introducing a crystalline external
More informationSemi-Classical Theory of Radiative Transitions
Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13 Atomic Structure (recap) Time-dependent
More informationPhysics with Neutrons I, WS 2015/2016. Lecture 11, MLZ is a cooperation between:
Physics with Neutrons I, WS 2015/2016 Lecture 11, 11.1.2016 MLZ is a cooperation between: Organization Exam (after winter term) Registration: via TUM-Online between 16.11.2015 15.1.2015 Email: sebastian.muehlbauer@frm2.tum.de
More informationRPA in infinite systems
RPA in infinite systems Translational invariance leads to conservation of the total momentum, in other words excited states with different total momentum don t mix So polarization propagator diagonal in
More informationQuantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationNotes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering
More informationvan Quantum tot Molecuul
10 HC10: Molecular and vibrational spectroscopy van Quantum tot Molecuul Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ j.rojo@vu.nl Molecular and Vibrational Spectroscopy Based
More informationLecture 3: Optical Properties of Insulators, Semiconductors, and Metals. 5 nm
Metals Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals 5 nm Course Info Next Week (Sept. 5 and 7) no classes First H/W is due Sept. 1 The Previous Lecture Origin frequency dependence
More informationMany-Body Problems and Quantum Field Theory
Philippe A. Martin Francois Rothen Many-Body Problems and Quantum Field Theory An Introduction Translated by Steven Goldfarb, Andrew Jordan and Samuel Leach Second Edition With 102 Figures, 7 Tables and
More informationQuantum Condensed Matter Physics Lecture 9
Quantum Condensed Matter Physics Lecture 9 David Ritchie QCMP Lent/Easter 2018 http://www.sp.phy.cam.ac.uk/drp2/home 9.1 Quantum Condensed Matter Physics 1. Classical and Semi-classical models for electrons
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationC2: Band structure. Carl-Olof Almbladh, Rikard Nelander, and Jonas Nyvold Pedersen Department of Physics, Lund University.
C2: Band structure Carl-Olof Almbladh, Rikard Nelander, and Jonas Nyvold Pedersen Department of Physics, Lund University December 2005 1 Introduction When you buy a diamond for your girl/boy friend, what
More informationLecture 1 - Electrons, Photons and Phonons. September 4, 2002
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2002 Lecture 1-1 Lecture 1 - Electrons, Photons and Phonons Contents: September 4, 2002 1. Electronic structure of semiconductors 2. Electron statistics
More informationOptical spectra of single-wall carbon nanotube bundles
PHYSICAL REVIEW B VOLUME 6, NUMBER 19 15 NOVEMBER 000-I Optical spectra of single-wall carbon nanotube bundles M. F. Lin Department of Physics, National Cheng Kung University, Tainan, Taiwan 701, The Republic
More information( ) /, so that we can ignore all
Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)
More informationManifestation of Structure of Electron Bands in Double-Resonant Raman Spectra of Single-Walled Carbon Nanotubes
Stubrov et al. Nanoscale Research Letters (2016) 11:2 DOI 10.1186/s11671-015-1213-8 NANO EXPRESS Manifestation of Structure of Electron Bands in Double-Resonant Raman Spectra of Single-Walled Carbon Nanotubes
More informationEE 223 Applied Quantum Mechanics 2 Winter 2016
EE 223 Applied Quantum Mechanics 2 Winter 2016 Syllabus and Textbook references Version as of 12/29/15 subject to revisions and changes All the in-class sessions, paper problem sets and assignments, and
More informationA few principles of classical and quantum mechanics
A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system
More informationElectron energy loss spectroscopy (EELS)
Electron energy loss spectroscopy (EELS) Phil Hasnip Condensed Matter Dynamics Group Department of Physics, University of York, U.K. http://www-users.york.ac.uk/~pjh503 Many slides courtesy of Jonathan
More informationLecture 4: Basic elements of band theory
Phys 769 Selected Topics in Condensed Matter Physics Summer 010 Lecture 4: Basic elements of band theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 Introduction Most matter, in particular most insulating
More informationS. Bellucci, A. Sindona, D. Mencarelli, L. Pierantoni Electrical conductivity of graphene: a timedependent density functional theory study
S. Bellucci, A. Sindona, D. Mencarelli, L. Pierantoni Electrical conductivity of graphene: a timedependent density functional theory study INFN Laboratori Nazionali Frascati (LNF), Italy Univ. Calabria,
More informationOBE solutions in the rotating frame
OBE solutions in the rotating frame The light interaction with the 2-level system is VV iiiiii = μμ EE, where μμ is the dipole moment μμ 11 = 0 and μμ 22 = 0 because of parity. Therefore, light does not
More informationPHYSICS 4750 Physics of Modern Materials Chapter 5: The Band Theory of Solids
PHYSICS 4750 Physics of Modern Materials Chapter 5: The Band Theory of Solids 1. Introduction We have seen that when the electrons in two hydrogen atoms interact, their energy levels will split, i.e.,
More informationMOLECULAR SPECTROSCOPY
MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective
More information16. GAUGE THEORY AND THE CREATION OF PHOTONS
6. GAUGE THEORY AD THE CREATIO OF PHOTOS In the previous chapter the existence of a gauge theory allowed the electromagnetic field to be described in an invariant manner. Although the existence of this
More informationAtomic cross sections
Chapter 12 Atomic cross sections The probability that an absorber (atom of a given species in a given excitation state and ionziation level) will interact with an incident photon of wavelength λ is quantified
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationTheory of Rayleigh scattering from metallic carbon nanotubes
PHYSICAL REVIEW B 77, 5 8 Theory of Rayleigh scattering from metallic carbon nanotubes Ermin Malić,, * Matthias Hirtschulz, Frank Milde, Yang Wu, Janina Maultzsch, Tony F. Heinz, Andreas Knorr, and Stephanie
More informationNiS - An unusual self-doped, nearly compensated antiferromagnetic metal [Supplemental Material]
NiS - An unusual self-doped, nearly compensated antiferromagnetic metal [Supplemental Material] S. K. Panda, I. dasgupta, E. Şaşıoğlu, S. Blügel, and D. D. Sarma Partial DOS, Orbital projected band structure
More informationarxiv:cond-mat/ v1 22 Aug 1994
Submitted to Phys. Rev. B Bound on the Group Velocity of an Electron in a One-Dimensional Periodic Potential arxiv:cond-mat/9408067v 22 Aug 994 Michael R. Geller and Giovanni Vignale Institute for Theoretical
More informationOptical & Transport Properties of Carbon Nanotubes II
Optical & Transport Properties of Carbon Nanotubes II Duncan J. Mowbray Nano-Bio Spectroscopy Group European Theoretical Spectroscopy Facility (ETSF) Donostia International Physics Center (DIPC) Universidad
More informationChapter 4: Summary. Solve lattice vibration equation of one atom/unitcellcase Consider a set of ions M separated by a distance a,
Chapter 4: Summary Solve lattice vibration equation of one atom/unitcellcase case. Consider a set of ions M separated by a distance a, R na for integral n. Let u( na) be the displacement. Assuming only
More informationSupplementary Figures
Supplementary Figures 8 6 Energy (ev 4 2 2 4 Γ M K Γ Supplementary Figure : Energy bands of antimonene along a high-symmetry path in the Brillouin zone, including spin-orbit coupling effects. Empty circles
More informationSpin orbit interaction in graphene monolayers & carbon nanotubes
Spin orbit interaction in graphene monolayers & carbon nanotubes Reinhold Egger Institut für Theoretische Physik, Düsseldorf Alessandro De Martino Andreas Schulz, Artur Hütten MPI Dresden, 25.10.2011 Overview
More informationWe already came across a form of indistinguishably in the canonical partition function: V N Q =
Bosons en fermions Indistinguishability We already came across a form of indistinguishably in the canonical partition function: for distinguishable particles Q = Λ 3N βe p r, r 2,..., r N ))dτ dτ 2...
More information12 Carbon Nanotubes - script
12 Carbon Nanotubes - script Carola Meyer Peter Günberg Insitute (PGI-6), and JARA-FIT Jülich Aachen Research Alliance - Future Information Technology Research Center Jülich, 52425 Jülich, Germany 12.2
More informationChem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.
Chem 44 Review for Exam Hydrogenic atoms: The Coulomb energy between two point charges Ze and e: V r Ze r Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative
More informationPhysics 221A Fall 1996 Notes 19 The Stark Effect in Hydrogen and Alkali Atoms
Physics 221A Fall 1996 Notes 19 The Stark Effect in Hydrogen and Alkali Atoms In these notes we will consider the Stark effect in hydrogen and alkali atoms as a physically interesting example of bound
More informationLecture 0. NC State University
Chemistry 736 Lecture 0 Overview NC State University Overview of Spectroscopy Electronic states and energies Transitions between states Absorption and emission Electronic spectroscopy Instrumentation Concepts
More informationIntroduction to Sources: Radiative Processes and Population Inversion in Atoms, Molecules, and Semiconductors Atoms and Molecules
OPTI 500 DEF, Spring 2012, Lecture 2 Introduction to Sources: Radiative Processes and Population Inversion in Atoms, Molecules, and Semiconductors Atoms and Molecules Energy Levels Every atom or molecule
More informationMethoden moderner Röntgenphysik I + II: Struktur und Dynamik kondensierter Materie
I + II: Struktur und Dynamik kondensierter Materie Vorlesung zum Haupt/Masterstudiengang Physik SS 2009 G. Grübel, M. Martins, E. Weckert, W. Wurth 1 Trends in Spectroscopy 23.4. 28.4. 30.4. 5.4. Wolfgang
More informationLecture 10. Transition probabilities and photoelectric cross sections
Lecture 10 Transition probabilities and photoelectric cross sections TRANSITION PROBABILITIES AND PHOTOELECTRIC CROSS SECTIONS Cross section = = Transition probability per unit time of exciting a single
More informationSupplementary Information
Ultrafast Dynamics of Defect-Assisted Electron-Hole Recombination in Monolayer MoS Haining Wang, Changjian Zhang, and Farhan Rana School of Electrical and Computer Engineering, Cornell University, Ithaca,
More informationLow Bias Transport in Graphene: An Introduction
Lecture Notes on Low Bias Transport in Graphene: An Introduction Dionisis Berdebes, Tony Low, and Mark Lundstrom Network for Computational Nanotechnology Birck Nanotechnology Center Purdue University West
More informationElectron-phonon interaction in ultrasmall-radius carbon nanotubes
Electron-phonon interaction in ultrasmall-radius carbon nanotubes Ryan Barnett, Eugene Demler, and Efthimios Kaxiras Department of Physics, Harvard University, Cambridge, Massachusetts 2138, USA Received
More informationUltraviolet-Visible and Infrared Spectrophotometry
Ultraviolet-Visible and Infrared Spectrophotometry Ahmad Aqel Ifseisi Assistant Professor of Analytical Chemistry College of Science, Department of Chemistry King Saud University P.O. Box 2455 Riyadh 11451
More informationCarbon nanotubes and Graphene
16 October, 2008 Solid State Physics Seminar Main points 1 History and discovery of Graphene and Carbon nanotubes 2 Tight-binding approximation Dynamics of electrons near the Dirac-points 3 Properties
More informationCHAPTER 9 FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS
CHAPTER 9 FUNDAMENTAL OPTICAL PROPERTIES OF SOLIDS Alan Miller Department of Physics and Astronomy Uni ersity of St. Andrews St. Andrews, Scotland and Center for Research and Education in Optics and Lasers
More informationLecture notes for QFT I (662)
Preprint typeset in JHEP style - PAPER VERSION Lecture notes for QFT I (66) Martin Kruczenski Department of Physics, Purdue University, 55 Northwestern Avenue, W. Lafayette, IN 47907-036. E-mail: markru@purdue.edu
More informationQUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer
Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer Contents 1. Historical and Experimental
More information1 Quantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ
More informationSolid State Physics Byungwoo Park Department of Materials Science and Engineering Seoul National University
Solid State Physics Byungwoo Park Department of Materials Science and Engineering Seoul National University http://bp.snu.ac.kr Types of Crystal Binding Kittel, Solid State Physics (Chap. 3) Solid State
More informationDark pulses for resonant two-photon transitions
PHYSICAL REVIEW A 74, 023408 2006 Dark pulses for resonant two-photon transitions P. Panek and A. Becker Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden,
More informationChem 442 Review of Spectroscopy
Chem 44 Review of Spectroscopy General spectroscopy Wavelength (nm), frequency (s -1 ), wavenumber (cm -1 ) Frequency (s -1 ): n= c l Wavenumbers (cm -1 ): n =1 l Chart of photon energies and spectroscopies
More informationIntermediate valence in Yb Intermetallic compounds
Intermediate valence in Yb Intermetallic compounds Jon Lawrence University of California, Irvine This talk concerns rare earth intermediate valence (IV) metals, with a primary focus on certain Yb-based
More information