Cyclotron Resonance and Faraday Rotation in infrared spectroscopy PHYS 211A Yinming Shao
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Cyclotron Resonance and Faraday Rotation in infrared spectroscopy PHYS 211A Yinming Shao Outline • Cyclotron resonance – Application in Ge: determing effective mass – Experimental detection of cyclotron resonance using FTIR • Faraday Rotation – General expression – Experimental detection • Giant Faraday Rotation in Graphene Cyclotron resonance Apply oscillating in-plane E-field Charges can resonantly absorb energy from E-field B +e -e q G. Dresselhaus et al., Phys. Rev. 98, 368 (1955) Resonance condition π = ππ = ππ΅ ππ Typically changing B field around resonance Using microwaves as AC E-field A word on different masses Resonance condition π = ππ = ππ is the cyclotron mass : ππ΅ ππ S is the k-space area of cyclotron orbits Effective mass: Parabolic bands: Graphene: ππ = π∗ π∗ vanishes ππ exists! Cyclotron Resonance (CR) in Ge In general, effective mass are anisotropic, For Ge, constant energy surfaces near band edge are spheroidal 1. Measuring CR at different field angles π ππ‘ ππ ππ΅ 2. Extract cyclotron mass by ππ = π π G. Dresselhaus et al., Phys. Rev. 98, 368 (1955) Condition to observe cyclotron resonance For 1 T field, Carrier mobility: π = ππ π∗ Ge is the first high purity sample people could obtain in ~1945 Requires π > π2 1 ππ = ππ2 10000 ππ Need high purity samples to see CR!! Organic semiconductors for CR?? Long way… Metals have high conductivities and E-field cannot penetrate sampleο requires special geometry Commercial superconducting magnet ο ~10 T B-field in lab accessible (~late 60s) Resonance condition is easier to realize in THz (1012 Hz) and far-infrared frequencies. (FFT algorithm become popular after ~1965) Use FTIR based transmission to see CR Fourier Transform InfraRed Spectroscopy (FTIR) Transmission set-up Based on a two-beam Michelson Interferometer: 1. Infrared sourceο broad band light source 2. Beam-splitterο divides the beam to two with similar intensity 3. Fixed mirror, moving mirror ο change the optical path difference ο interferogram Fourier transform the πΌ π₯ to get the spectrum πΌ π Fourier Transform InfraRed Spectroscopy (FTIR) • Advantage: 1. Fast: obtain transmittance/reflectance spectrum over a broad frequency range rapidly 2. Simple: moving mirror is the only moving part in the system 3. Sensitive: bright light source; average multiple scans is fast http://mmrc.caltech.edu/FTIR/FTIRintro.pdf CR in graphene from transmittance measurements Transmission data normalized by 0T data ο Cancel out features that are not field dependent Power absorption: It can be shown that the Half Width at Half 1 Maximum is about the scattering rate . π Recall that ππ π = ππ΅, by fitting the cyclotron frequency one get estimates about Carrier mobility. ππ΅ ≈ 0.3 @1π ππ© ≈ π. π @ππ» I. Crassee et al, Nat Phys 7, 48 (2011) Estimate mobility Contact free! Magneto-Optical Faraday Effect First observation (in 1845) of light-magnetism interaction! • Optically active material: ππ ≠ ππ • Field induced circular birefringence π− ≠ π+ • For linearly polarized light, the polarization plane of the transmitted light is rotated Faraday rotation π− ≠ π+ Complex refractive index π = π + ππ π± = π π£± Circular Birefringence left- and right-handed light travel at different speeds in the medium http://cddemo.szialab.org/ General expression of Faraday rotation angle: Single passage approximation Complex transmission: Need Relatively thick sample to suppress multiple reflection Faraday rotation: Detecting Giant Faraday rotation using crossed polarizers • The most straightforward method Rotate the analyzer from 0 to 180 and then Fit the transmitted intensity with πππ 2 (π − ππΉ ) Analyzer Polarizer Combine with FTIR ο Faraday rotation ππΉ (π) at different frequencies π Giant Faraday rotation in graphene (on SiC) Definitely Giant Typical semiconductors (e.g. InSb) comparable rotation but several magnitudes thicker (ππ) 1 atomic layer (~10−10 π) ο > 6 degrees of polarization change! ππ΅ > 0 πππππ‘πππ ππ = = <0 βππππ ππ Negative slope ο hole doping! is maximized around ππ Sign of I. Crassee et al, Nat Phys 7, 48 (2011) Matches the sign of ππ Some modeling based on Drude model Equation of motion: Assuming an harmonically varying field: πΈ = πΈ0 π −πππ‘ and therefore drift velocity Solve v in terms of E and B then compared to Current density π½ = −πππ£ = ππΈ EOM becomes: Dynamical conductivity (magnetic field introduces anisotropy) Explicit form of dynamical conductivity ππ₯π₯ : even function of ππ ππ₯π¦ : odd function of ππ Modeling off-diagonal conductivity Modeling graphene as a two dimensional electron gas, its Faraday rotation angle ππΉ Is proportional to Re(ππ₯π¦ ) up to some positive constant Real part 1. Real part of ππ₯π¦ (ππΉ ) is maximized around cyclotron frequency. Its derivative is maximized at π = ππ 2. Real part of ππ₯π¦ (ππΉ ) changes sign around cyclotron frequency. Its derivative around π = ππ matches the sign of ππ ο gives information about the carrier type! ο Similar to DC Hall effect Giant Faraday rotation in graphene Faraday rotation is enhanced near cyclotron resonance ο Giant ππ = ππ΅ > 0 πππππ‘πππ = <0 βππππ ππ Negative slope ο CR involves hole states (Fermi level in valence band) I. Crassee et al, Nat Phys 7, 48 (2011) Landau Level transitions in MLG (on SiC) Unlike single layer graphene, multilayer graphene are less doped and fall in the quantum regime CRο LL transitions Positive slope indicates The observed LL transition Involves electron like states. Summary • Cyclotron resonance is powerful for determining effective mass in semiconductors and estimate carrier mobility • Faraday Rotation is the optical analogue of Hall effect and is enhanced around cyclotron resonance • FTIR based CR and FR extends traditional measurements to much broader frequency range Thanks for your attention!
