Les Houches School of Physics Lecture 1: Piezoelectricity & mechanical resonators

Transcription

1 Les Houches School of Physics Lecture 1: Piezoelectricity & mechanical resonators Institute for Molecular Engineering Andrew Cleland August 2015 Ecole de Physique Les Houches

2 πιέζειν ήλεκτρον Piezoelectricity 1880: Paul-Jacques Curie (age 24) and Pierre Curie (age 21) discover that some materials under stress generate electric voltages. They term this effect piezoelectricity. piezin to squeeze or press electron amber from tree resin (an early source of static electric charge via friction) 1881: Following a theoretical prediction based on symmetry by the mathematician Gabriel Lippman, they discover that conversely electric voltages cause stress. The Curie brothers then build the piezoelectric quartz electrometer, an electromechanical transducer for measuring very small electric currents, later used by Pierre & Marie Curie for studies in radioactivity (NP 1906). Piezoelectric quartz electrometer Little is done for the next 30 years...

3 Langevin Sandwich Transducer Paul Langevin ( ): Student of Pierre Curie s, invents a metal-piezoelectric-metal sandwich now called a Langevin sandwich transducer. Uses this device as a source and receiver of acoustic signals in water, later called sonar (1916). This is the second practical application of piezoelectricity (after electrometer). metal piezoelectric metal Langevin s British patent 145,691 (1918) for sending & receiving sound waves for detecting submarines, icebergs, or other echo-reflecting objects Langevin sandwich transducers, for ultrasonic cleaners. Found on indiamart.com

4 Sonar Ernest Rutherford and his student Robert William Boyle (not the chemist!) worked in parallel with Langevin to develop piezoelectric acoustic sensors. Ernest Rutherford ( ) Robert William Boyle ( ) Robert Boyle s failed attempt to replicate Rutherford s early experiments with detecting sound using piezoelectric quartz. Boyle then worked with Langevin to develop practical ultrasonic sonar; 1917 the UK developed ASDIC, with prototypes tested during WWI.

5 Langevin sandwich transducer metal piezoelectric metal Displacement ½ wave resonance neutral plane Stress-free top and bottom surface Metal properties differ (somewhat) from piezo properties Operates at ½ wave resonant frequency ( 2 ) Voltage generates stress; stress generates voltage

6 Stress / (N/m 2 ) Stress, strain, and all that Strain Δ (dimensionless) Hooke s law relates stress and strain: F Area A length L F Local displacement, : vector displacement of solid from rest point Stress ΔL/2 ΔL/2 stress Young s modulus strain Strain are elastic moduli (N/m 2 )

7 Stress, strain, and all that

8 Stress, strain, and all that Write symmetric strain tensor as a six-component vector: = Six-vector form of Hooke s law: sixvector stress 6 by 6 elastic modulus tensor Isotropic solid has two independent elastic moduli: and 0 0 sixvector strain 2 is the shear modulus

9 Stress, strain, and all that Lamé constants 2 Dynamic equation for an isotropic solid: Differential cube or Young s modulus and the Poisson ratio: 3 2 Net force along : 2 Force original location Newton s law: Newton s law for stress: isotropic solid

10 Mechanical Loss Many different models for mechanical loss/dissipation Specific models for specific type of loss mechanism Some examples: Two-level states Thermoelastic loss Phonon-phonon Phonon-electron Hooke s law for an elastic solid: Zener s empirical model for an anelastic solid: constant strain relaxation time relaxed modulus Microplastic deformation Elastic hysteresis... Harmonic motion: 1 1 constant stress relaxation time

11 Zener s Inelastic Solid Model Harmonic stress & strain: 1 1 Static/equilibrium: (relaxed modulus Hooke s law) High frequency/adiabatic: Intermediate frequencies: Strain & stress out of phase Mechanical loss 1 log 1 (unrelaxed modulus Hooke s law) Complex elastic modulus: 1 Frequency dependent loss (similar to Drude model): 2 1

12 Piezoelectricity Piezoelectricity couples mechanical deformation and electric fields Electric displacement & field: or Stress and strain: or Piezoelectric material: 6 by 3 piezoelectric tensor In component form: 1 3 and 1 6 Combine with Newton s law for stress:

13 Langevin sandwich transducer The Langevin sandwich transducer provides a good workable example (using many approximations): metal piezoelectric insulator metal For simplicity assume: Massless & zero thickness metal (only serves to define voltage) Isotropic mechanical response for piezoelectric: Elastic constant Piezoelectric response along : Piezoelectric constant

14 Langevin sandwich transducer (remainder of this example worked out on blackboard) dilatational resonator.pdf

15 Dilatational Resonator Andrew N Cleland IME - University of Chicago 1

16 Consider first a non-piezoelectric material. For dilatational vibration along the z-axis, the relevant strain component in 6-vector notation is S 3 (this is S 33 or S zz in tensor notation). We will use u(z, t) to represent the z-axis relative displacement, so S 3 = u/ z. We take this as the only non-zero strain, and also assume the only non-zero stress is T 3 = c 33 S 3, where c 33 is the relevant elastic constant (we are thus implicitly assuming that the Poisson ratio ν is zero). Note in terms of the Lamé constants, c 33 = 2µ + λ, and setting the Poisson ratio ν to zero is equivalent to setting λ = 0, so c 33 = 2µ. For a non-piezoelectric, isotropic solid, the relative displacement satisfies the dynamic equation ρ 2 u t = T 3 2 z = c S 3 33 z = c 2 u 33 z, (1) 2 where ρ is the material density. Note this is a wave equation, and the sound velocity is given by v s = c 33 /ρ. For the simplest case of a slab of thickness b with free-free boundaries at z = ±b/2, we need to have zero stress at each boundary, so T 3 (z = ±b/2) = 0 so z u z=±b/2 = 0. The solutions to the wave equation with these boundary conditions are then u n (z, t) = U sin(k n z)e iωnt for n odd, u n (z, t) = U cos(k n z)e iωnt for n even, with ω n = v s k n = c 33 /ρ k n and k n = nπ/b. The displacement amplitude is U. We now consider instead a piezoelectric material. We then must use the piezoelectric stress equations for the relevant z components: D z = ϵ 33 E z + e 33 S 3, T 3 = e 33 E z + c 33 S 3. Here D z and E z are the z components of the electric displacement and electric field, ϵ 33 is the relevant element of the dielectric tensor (in SI units, this would be ϵ 33 = ϵ r ϵ 0 ), e 33 is the relevant element in the piezoelectric modulus tensor, and T 3 is the relevant component of the stress 6-vector (T 3 = T zz ). Using Gauss law in the interior of the slab z D z = 0, and the dynamic relation ρ 2 u/ t 2 = z T 3, we obtain 2 u ϵ 33 z E z = e 33 z, 2 ρ 2 u 2 (3) u = e t 2 33 z E z + c 33 z, 2 yielding the self-consistent dynamical equation ( ) ρ 2 u t = c e u ϵ 33 z, (4) 2 2 (2)

17 + +σ T + z z=b/2 V D z - -σ T - z=-b/2 FIG. 1: Piezoelectric slab. Direction of D z corresponds to its algebraic positive value, but if the surface charge σ is positive, D z will be negative. yielding the same results as for (1) but with a (very slightly) modified effective stiffness c 33 = c 33 + e 2 33/ϵ 33. For a typical piezoelectric material the stiffness will change by at most a few percent. For again the slab of thickness b with free-free boundaries at z = ±b/2, zero stress boundaries gives z u z=±b/2 = 0, and solutions u n (z, t) = U sin(k n z)e iωnt for n odd, u n (z, t) = U cos(k n z)e iωnt for n even, with ω n = c 33 /ρ k n and k n = nπ/b. More generally, allowing for a normal external stress T ± on the top and bottom surfaces z = ±b/2, a surface charge ±σ at z = ±b/2, a voltage V across the slab, and displacements u ± at the top and bottom surface, with all components oscillating at frequency ω, there is a matrix relation similar to (2), T + u + T = Z u, (5) V σ where Z is the constitutive matrix relating the two vectors. These quantities are related to one another by (2), which can be written with S and D as the independent variables, E z = ϵ 1 33 D z h 33 S 3, (6) T 3 = h 33 D z + c 33 S 3, where h 33 = e 33 /ϵ 33. The external stress is T ± = T 3 (z = ±b/2), the surface charge is given by 3

18 σ = D z, and the voltage is related to the electric field by V = E z dz. Note that as z D z = 0, D z = σ is constant in the slab. The boundary conditions allow displacements u(z) = (U s sin kz + U c cos kz)e iωt with ω = c 33 /ρ k c s k. The displacements u ± at z = ±b/2 are related to the amplitudes by U s = (u + u )/(2 sin(kb/2)) and U c = (u + + u )/(2 cos(kb/2)). The corresponding stress is S 3 = z u = (ku s cos kz ku c sin kz)e iωt, so the electric field is E z = ϵ 1 33 σ h 33 k(u s cos kz U c sin kz)e iωt. The voltage is then V = ϵ 1 33 σb + h 33 (u + u ). We can thus write T + kc 33 cot(kb) kc 33 csc(kb) h 33 T = kc 33 csc(kb) kc 33 cot(kb) h 33 V h 33 h 33 b/ϵ 33 σ u + u. (7) Assuming all this is correct, we can invert this matrix to get the displacements u ± and charge σ in terms of the stress and voltage. For a stress-free system, T ± = 0, and we find the relation between charge and voltage, kb cot(kb) + kb csc(kb) σ = (ϵ 33 /b) V, (8) g + kb cot(kb) + kb csc(kb) with dimensionless coupling constant g = 2h 2 33ϵ 33 /c 33. We also find that u + = u, so U c = 0 and only the sine part of the displacement is non-zero, U s = 1 h 33 g V. (9) g kb cot(kb) kb csc(kb) The corresponding electrical admittance can be obtained from I = Adσ/dt = Y (ω)v, with Y (ω) = iωϵ 33A b kb cot(kb) + kb csc(kb) kb cot(kb) + kb csc(kb) g. (10) If we set the piezoelectric coupling g 0, then we find Y (ω) = iωϵ 33 A/b Y C (ω), corresponding to a capacitance C = ϵ 33 A/b for a slab of area A, as expected. We thus separate the admittance Y (ω) = Y C (ω) + Y res (ω) with [ ] g Y res (ω) = Y C (ω) kb cot(kb) + kb csc(kb) g (11) = Y C (ω) h 33U s V. (12) The resonator admittance has sharp resonances whenever kb = bω m / c s (2m + 1)π 2g/b c s ω m (2m + 1)π. We focus only on the fundamental resonance m = 0, with ω 0 = π c s /b. Near this resonance, with x = bω/ c s π = π(ω/ω 0 1) = π(ω ω 0 )/ω 0, we can write 4

19 sin(kb) x and cos(kb) + 1 x 2 /2. The denominator of Y res is then kb cot(kb) + kb csc(kb) g π( x2 /2 1 x + 1 x ) g or approximately π( x/2) g πx/2 = (π2 /2)(ω ω 0 )/ω 0 and Y res (ω) 2g π Y ω 0 C(ω). (13) 2 ω ω 0 This is the first-order part of an undamped Lorentzian, 2ω 2 0/(ω 2 ω 2 0) ω 0 /(ω ω 0 ). Phenomenological damping can be added in to give ω 2 0 Y res (ω) (4g/π 2 )Y C (ω) ω 2 ω iβω, (14) with quality factor Q = 2β/ω 0. Defining the Lorentzian response function R(ω), R(ω) the total current through the resonator is ω 2 0 ω 2 ω iβω, (15) I res (ω) = Y (ω)v (ω) (16) = [ iωc (4g/π 2 )iωcr(ω) ] V (ω). (17) By a similar argument we can write U s = 1 4g R(ω)V (ω), (18) h 33 π2 and with V = iωv, U s = iωu s, we can write I res (t) = C V h 33 C U s. (19) 5

20 Les Houches School of Physics Lecture 2: Qubits coupled to resonators Institute for Molecular Engineering Andrew Cleland August 2015 Ecole de Physique Les Houches

21 Electrical harmonic oscillator C energy magnetic flux Φ Microwave frequency circuits: 2 ~ few GHz Electrical harmonic oscillator: SHO energy levels Operate at T = 10 mk: ~200 MHz 2 quantum ground state Need nonlinearity to enable quantum control!

22 The phase qubit: A nonlinear LC resonator Introduce Josephson junction in LC circuit ac & dc Josephson relations Ψ Ψ super conductor phase difference critical current capacitance ( ) highly nonlinear inductance thin insulator (~1 nm) ( ) will change by few % for each microwave photon in circuit

23 The phase qubit: A nonlinear LC resonator Magnetic flux energy Josephson nonlinear inductor energy magnetic flux /2 ~ 6 GHz ( )/2 ~ 200 MHz one flux quantum magnetic flux Φ quantum ground state < 300 mk qubit levels and splitting electrically tunable nonlinear at single photon level

24 The phase qubit: A nonlinear LC resonator Josephson nonlinear inductor Magnetic flux energy measure one flux quantum magnetic flux Φ Perform quantum operations, then: Separate from by tilting potential with external current, allowing to tunnel out then rapidly relaxes to lower energy well Flux measurement then distinguishes from

25 The phase qubit: A nonlinear LC resonator Flux bias Qubit Measurement SQUID 30 Tune qubit freq. Excite qubit Rotate qubit Destructive qubit readout Distinguish from (90-95% fidelity) Prepare qubit, evolve state Destructive projection distinguishing from Repeat ~1000 times for probability P(e) ~0.1 sec.

26 The phase qubit: A nonlinear LC resonator (remainder of this discussion on blackboard from notes: Qubit QM & resonator QM)

27 Film bulk acoustic resonator (FBAR): A Langevin sandwich transducer metal piezoelectric insulator metal 0.25 mm Developed at HP then Agilent then Avago Compact GHz-frequency resonator Primary filter for cellphone 1.8 GHz Six in each iphone 6 Richard Ruby

28 Fabricating a piezo resonator 1. Silicon substrate Deposit & etch Al base layer 3 3. Deposit piezoelectric AlN 4 4. Etch via in AlN 5. Deposit & etch Al 6. Etch AlN and Al 7. Sacrificial XeF2 etch 5 7 6

29 Measuring a piezo resonator Al top & bottom electrodes Sputtered AlN piezoelectric XeF 2 substrate release 4-7 GHz fundamental resonance Integrable with qubit fabrication diameter 35 μm mass 20 ng resonator A.D. O Connell

30 Mechanical resonator coupled to qubit A.D. O Connell E. Lucero phase qubit

31 Dilution refrigerator

32 Qubit-mechanical resonator spectroscopy set qubit frequency apply weak microwave signal measure sweep qubit & microwave frequencies A.D. O Connell A.D. O Connell E. Lucero et al. Nature (2010) μwaves resonance: excite qubit state frequency (GHz) qubit excited state qubit tuning

33 Qubit-mechanical resonator spectroscopy frequency (GHz) level avoidance mechanical resonance f r = GHz coupling strength g =124 MHz 5.4 qubit tuning

34 Mechanical resonator ground state qubit as a quantum thermometer qubit in ground state tune into resonance, wait measure excited state probability tune qubit data

35 Exciting single phonons qubit resonator 2 e tunable 1 g Qubit in excited state Resonator ladder Qubit in ground state Resonator ladder Qubit splitting (tunable) state of system

36 Exciting single phonons Qubit off resonance (system in 0g state) Apply microwave π pulse to qubit (goes to 0e state)

37 Exciting single phonons Qubit off resonance (system in 0g state) Apply microwave π pulse to qubit (goes to 0e state) Tune qubit to resonator frequency Rabi oscillation: Transfer photon from qubit to phonon in resonator phonon in resonator 1 phonon rate: Ω

38 Exciting single phonons Qubit off resonance (system in 1g state) Apply microwave π pulse to qubit (goes to 1e state)

39 Exciting single phonons Qubit off resonance (system in 0g state) Apply microwave π pulse to qubit (goes to 0e state) Tune qubit to resonator frequency Rabi oscillation: Transfer photon from qubit to second phonon in resonator phonons in resonator 2 phonon rate: 2g 1 0 0

40 Single phonon excitations excite detuned qubit tune to resonance Rabi oscillation: Wait time, measure qubit excited state probability measure excite with one quantum tune qubit start expt theory

41 excite detuned qubit tune to resonance Rabi oscillation: Single phonon excitations excited state probability measure tune qubit start expt excited state probability stop at create one phonon theory

42 Single phonon excitations single phonon excitation: swap phonon into resonator detune & wait swap back to qubit measure qubit lifetime of excited state as expected: 6.1 ns phonon superposition: qubit in swap into resonator 0 1 wait, swap rotate to measure qubit superposition state 0 1 2

43 Half-wave coplanar resonator ( ) 2 s g n = 2 n = 1 n = 0 Photons in half-wave mode open terminations yield voltage antinodes Wavelength 2 8 mm Resonance frequency 2 7 GHz Quality factor ~10 10 Excitation lifetime ~ microseconds

44 Qubit coupled to EM resonator qubit capacitor microwave resonator Tunable Fixed qubit raising & lowering operators resonator lowering & raising operators Jaynes-Cummings Hamiltonian: coupling strength qubit-resonator photon swaps

45 Qubit coupled to EM resonator hofheinz et al. nature (2008) 25 mk

46 μwaves Qubit-resonator spectroscopy set qubit frequency, relax to weak microwave tone, measure qubit sweep qubit & microwave frequencies GHz 36 MHz hofheinz et al. nature (2008)

47 Qubit-resonator Rabi swaps excite detuned qubit tune to resonance Rabi oscillation: qubit resonator pulse qubit tune qubit & wait Excellent agreement with Jaynes-Cummings model: Rabi frequency: Δ /2 Swap probability: Δ

48 Pumping photons one-by-one Pumping photons: Excite qubit from to Transfer state to resonator Repeat N times for N photons Measurement: hofheinz et al. nature (2008) Qubit in State swap with resonator Rabi frequency scales with

49 Synthesizing arbitrary states prepare and measure 0 states in resonator theory exp. Hofheinz et al. Nature (2009) Law & Eberly PRL (1996) Banaszek &Wodkiewicz PRL (1996)

50 watching schrödinger s cat die Synthesizing arbitrary states

51 Synthesizing arbitrary states / /

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