Experimental Observation and Control of Wave Dispersion Dispersive Effects Analysis and Modeling Kyle McLellan • Electrons in a Lattice • EM wave in a solid • Sound in elastic media Dr. Stephen Remillard, faculty advisor Hope College Department of Physics (A/m) Method-of-Moments simulation using IE3D (Ref 5) yields both scattering parameters and surface current distribution. Condensed Abstract • Dispersion: Deviations from the simple model of constant phase velocity Kronig-Penny potential in the Schrödinger equation • Goal 1: Find a hands-on way to investigate the dispersion of electron waves in a crystal lattice has a transcendental solution: Left Hand Side of Equation 1 cos( k1d1 ) cos( k 2 d 2 ) − k12 − k 22 sin(k1d1 ) sin(k 2 d 2 ) = cos (β (d1 + d 2 )) 2k1k 2 2 R.H.S of E q. 1 1 For Ba bidde n nd 0 ki = ω vi = 2π λi • Analysis: Convert the transmission and reflection coefficients into band structure vi = wave speed • Purpose: Results in an experimental examination of the band theory of solids 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 d1 d2 d1+d2≡Lattice Constant λi=wavelength in region “i” Frequency (GHz) dω ≠ Constant ⇒ Dispersion! dβ L.H.S. d1 + d 2 β = cos −1 dispersionless case: ω 0.7 ω⋅(d 1+d 2)/2πc • Findings: Defects produce engineered states in the band gap K.-P. potential d1=d2=7 mm 2 • Process: Simulate crystals using hand-made transmission lines with periodic variations Wave, β=2π/λ 1 Matlab is used to invert Equation 2 and to calculate the wave number, β. • Goal 2: Introduce defects into the crystal which result in controllable states in the band gap (1) 0.6 Experiment β 0.4 Band Gap 0.3 Transmission line equations from Ref 4. Code Written in MATLAB. (Ref 6) The author uses IE3D to layout the structure and to perform method-ofmoments EM simulation. |LHS|<1 0.5 Transmission & reflection parameters of the dispersive structure are measured with a vector network analyzer. The 400+ lines of Matlab code are used to process raw transmission & reflection data. 0.2 Paper design layout 0.1 π d1 + d 2 β⋅(d1+d 2)/2π Copper tape The dispersive structure is handfabricated using an Exacto knife. Periodic Transmission Lines d1+d2≡Lattice Constant d1 w2 w1 d2 h ε1,eff εs L ε i ,eff ≈ ε S +1 Signal in 2 + ε S −1 2 1 + 12(h / wi ) S11 ≡ Measured reflection coefficient magnitude and phase S21 ≡ Measured transmission coefficient magnitude and phase e j ( α + jβ ) L = Results Periodic variation in εeff produces a periodic impedance mismatch of the wave. ε2,eff 2 2 1 − S112 + S 212 + (1 + S112 − S 21 ) − (2 S11 ) 2 2 S 21 Matlab graphs b. d1 d2 As the size of the gap ‘G’ increases, the state in the gap appears at a continually lower frequency until it reaches the critical point, where the defect jumps back to the far right of the band gap. Distance of defect from the center of the band gap (Delta F) ∆F This transmission line was fabricated using photolithography.(Ref 2) (2) Propagation constant, solved by inverting this equation Attenuation coefficient (Ref 1) Simulated & Measured Transmission, α and β Dispersion Engineering: Impurity States p-Silicon (IV) doped with Al (III) G n-Silicon (IV) doped with As (V) Defect Location From Center Reduced interstitial spacing simulates p-type doping Increased interstitial spacing simulates n-type doping 2.5 Experiment Outline ∆ F > 0 N-type defect state Delta F (GHz)) 1.5 ) 1. Write C-based code to evaluate Equation 1 and to invert Equation 2 2. Design a periodic transmission line using an EM field simulator 3. Fabricate the periodic transmission line 4. Measure the transmission and reflection coefficients vs. frequency 5. Use the computer program to compute β vs. frequency with Eq. 2 6. Plot the dispersion relation in the extended or reduced zone scheme 7. Attempt some “dispersion engineering” with an impurity 2 The dispersion curve generated using the above transmission line is generated three ways: Analytic: Using Equation 1 Simulated S-Parameters: Using Equation 2 and T&R coefficients from EM sim. Measured: Using Equation 2 and T&R coefficients from measurement 1 0.5 0 -0.5 0 1 2 3 4 5 6 7 8 9 10 P type gap state N type gap state Ref 3 -1 ∆ F < 0 P-type defect state -1.5 -2 Gap Size (mm) Acknowledgements References 1. W.R. Eisenstadt and Y. Eo, “S-Parameter Based IC Interconnect Transmission Line Characterization,” IEEE Trans. Components, Hybrids and Manufacturing Technol., 15, no. 4, 483-490 (1992). 2. C. Isaac Angert and S.K. Remillard, "Dispersion in One-Dimensional Photonic Band Gap Periodic Transmission Lines," Microwave and Optical Technology Letters, 51, no. 4, 1010-1013 (2009). 3. E. Yablonovitch, et. al, “Donor and Acceptor Modes in Photonic Band Structure,” Phys. Rev. Lett., 67, no. 24, 3380-3383 (1991). 4. Brian C Wadell, Transmission Line Design Handbook, Artech House, Inc, Norwood, MA, 1991, Page 94. 5. IE3D EM Design System, Zeland Software Inc, Fremont, CA. 6. MATLAB, The MathWorks, Natick, MA. A corporate sponsor of R&D at Hope College Dean of Natural and Applied Science This work was supported by an R&D contract from Mesaplexx, pty ltd., by the National Science Foundation under NSF-REU Grant No. PHY-0452206, and by the Hope College Division of Natural and Applied Science.