Materials aspects of quantum metrology

Alexander Tzalenchuk

445th WE-Heraeus SeminarQuantum Measurement and Metrology with Solid State Devices1-5 November 2009 Physikzentrum Bad Honnef, Germany

Scope

Real materials and precise measurements: friends or foes?

• Superconducting resonators vs. dielectrics• Quantum Hall effect vs. graphene

Superconducting resonatorsat low-temperatures

Tobias Lindström(NPL)

Joanne E. Healey, Chris M. Muirhead(University of Birmingham)

Yuichi Harada, Yoshiaki Sekine(NTT BRL)

Motivation

• To understand and quantify the loss mechanisms in superconducting resonators at low temperatures

• To build a high-throughput microwave loss measurement system based on the resonators

• To compare and quantify losses in practical materials coupled to this system

Astronomy instrumentation communityQubit community

In-line λ/2 co-planar resonator

f0

QIn Out

d

C

L… …

CcCc

f0 ∝ 1/d

d

In-line λ/2 co-planar resonator

superconductor

Where it all started: KIDs

•Photons break Cooper pair•Kinetic contribution to total inductance increases•Resonance frequency decreases

CLLf

kg )(1

0 +∝

In-line λ/2 co-planar resonator

W=10 μmS=5 μm

G=4 μm

In Out

-0.50 -0.25 0.00 0.25 0.500

300000

600000

900000

1200000

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-200

-150

-100

-50

0

50

100

150

200

-0.50 -0.25 0.00 0.25 0.50-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

QMagnetic flux density, mT

Freq

uenc

y, k

Hz

Magnetic flux density, mT

25 mK 50 mK 100 mK 100 mK reverse 280 mK300 mK, -21 dBm300mK, -27 dBm 500 mK

df0/d

B (G

Hz/

T)

Magnetic flux density, mT

Magnetic tuning of superconducting resonators

•Frequency tuned by magnetic field•No deterioration of Q – no additional losses

Magnetic tuning of superconducting resonators

More details in: J. Healey et al., APL 93, 043513 (2008)

Non-linear London equations

( )

( )( )

λ

βλλ

ββ

β

∝

⎥⎦

⎤⎢⎣

⎡+=

Δ−+=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=+−=

k

c

B

c

sssqpsss

LTH

HTTTH

TkT

vvTvTenJvenJ

2

2''

)()(1

),0(),(

exp1)0()(

1)(

( ) ( ) ( )( ) ( )( )224/0,10,, CTK HHTLTLTfHTf β−=

First conclusions

• Losses in superconducting resonators at low temperature are not determined by the superconductor

• Small magnetic field (current in a control line) can tune the resonance frequency of high quality resonators by hundreds of linewidths

Temperature dependence of resonance frequency

0 500 1000 1500 20005712.24

5712.26

5712.28

5712.30

5712.32

5712.34

5712.36

5712.38

Centre Freq Fr

eque

ncy,

MH

z

Temperature, mK

Superconductor Mattis-Bardeen

Shunting λ/4 co-planar resonator

Shunting λ/4 co-planar resonator

In Out

Frequency multiplexing

5.9850 5.9855 5.9860 5.9865 5.9870

-8

-6

-4

-2

0

Mag

nitu

de(d

B)

Frequency (GHz)

4000 4500 5000 5500 6000 6500 7000 7500 8000-10

-8

-6

-4

-2

0

Fre que ncy(MHz)

S21

(dB

)

• Can multiple on-chip resonators be made more compact while maintaining high Q?

Shunting lumped-element resonator

Shunting lumped-element resonator

Transmission line

L C

Shunting lumped-element resonator

Fitting

yedcxxjQg

xjQR

QCZg

fffx

j

u

u

uc

−⎥⎦

⎤⎢⎣

⎡++

+++

=

=

−=

θ

πϖ

)21(2)21(2

:Minimize

)2( coupling

shift frequency normalised

200

0

0

Resonator3 parameters

Skewingparasitics

Magnitudeoffset

Phaserotation

Ideal Smith chart

Fitting

• Qu = 345514

• f0=6958658487 Hz

• g = 3.9074

• θ = −0.86441

• c = 17281

• d = 0.08

Low temperature frequency shift

Low-temperature frequency shift

0 1 2 3 4 5 6 7 8

-10

-5

0

5

x 10-7

hf/kBT

δ f

0 2 4 6 8 10 12 14

0

2

4

6

8x 10-5

hf/kBT

δ f

Frequency shift as a function of the normalized inverse temperature at several different powers

- λ/4- LE

SiO2 / Si

sapphire

- LE

Resonant process

TLF

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=Tk

PPc

nd

B

c

r 2tanh

13

4

0

22 ϖ

ε

ϖπα h

αr –resonant absorptionω -measurement frequencyn -density of TLFd -dipole momentc0 -speed of lightPc -critical powerT1, T2 -relaxation and dephasingΨ -digamma function significant when thermal energy is smaller than photon energy

212

02

23 where

and 1

TTdcP

PPconstPPP

c

ccr

ε

α

h=

<←>←∝

( )

⎭⎬⎫

⎩⎨⎧

+Ψ=

⎥⎦

⎤⎢⎣

⎡−−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−=

Δ

TikTg

TgTgTTnd

TTT

Bπϖϖ

ϖϖεε

εεεε

221Re),(

),(),(ln3

2)(

)()(0

0

2

0

0

hεεδ Δ

−=Δ

≡20

0 Ffff

F –filling factor

critical power

Permittivity from αr & Kramers-Kronig relations

0 1 2 3 4 5 6 7 8

-10

-5

0

5

x 10-7

hf/kBT

δ f

0 2 4 6 8 10 12 14

0

2

4

6

8x 10-5

hf/kBT

δ f

Low-temperature frequency shift

)(Tg⎟⎟⎠

⎞⎜⎜⎝

⎛

0

lnTT

SiO2 / Si

sapphire

Substrate losses

6

52

108.2106

tan3−

−

××

== πδε iFnFd

All data on a particular substrate is fitted using a single parameter:

SiO2 / Si

sapphire

More details in: T. Lindström et al., PRB 80, 132501 (2009)

Comparison of losses in different substrates

10-6 10-4 10-2 10010-6

10-5

10-4

10-3

tan

δ

W/W0

Sapphire50 mK

SiO2 / Si3 LE resonators

x4 temperatures

(50-340 mK)∝W-0.4

212

02

23 where

and 1

TTdcP

PPconstPPP

c

ccr

ε

α

h=

<←>←∝

W – energy in the resonator at f0W 0~10-16 J

02121)1(2 ϖinu PSSQW −=

A bit more subtle

• Can resonators probe thin films?

Current distribution in a lumped element resonator

μ-sensitive

ε -sensitive

Atomic Layer Deposition

•ALD involves the use of a pair of reagents.• each reacts with the surface completely• each will not react with itself

•Application of this AB Scheme•Reforms the surface•Adds precisely 1 monolayer

•Pulsed Valves allow atomic layer precision in growth•20 nm•Al(CH3)3 alternating H2O cycles•~100 C deposition temperature

ALD0 2 4 6 8

-2

0

2

4

6

8

10

12

14 x 10-6

hf/kB T

δ f

Larger nd2

Conclusions:

• Low-temperature losses in high quality superconducting microwave resonators are determined by the surrounding dielectrics

• The non-monotonic temperature dependence of the resonance frequency can in most cases be accurately described by resonant absorption of microwaves by two-level fluctuators (TLF), which couple to the resonator via their electric-dipole moments.

• Quantitative information about the distribution of two-levelfluctuators can be extracted from such measurements and different materials can be compared.

• This approach can be used for high-throughput (frequency-multiplexed) diagnostics of dielectric materials for quantum circuits at the temperatures, microwave frequencies and power levels relevant to their operation.

Quantum Resistance StandardBased on Epitaxial Graphene

Sergey Kubatkin, Samuel Lara-Avila, Alexei Kalaboukhov, Sara Paolillo (Chalmers)Mikael Syväjärvi, Rositza Yakimova (Linkoping)Olga Kazakova, T.J.B.M. Janssen (NPL)Vladimir Fal'ko (Lancaster)

Some unpublished figures have been deleted.More details in: A. Tzalenchuk et al., arXiv:0909.1220

On electrical metrology1

Dear Editor,

The formulation of the Ohm’s law has to be modified in the following way: “When carefully selected and perfectly prepared materials are used, then on a good day and with some skill one can construct an electric circuit for which the ratio of voltage to current measured over a finite time gives a constant value after introduction of relevant corrections.”

Signed

1 Physicists Make Jokes. Moscow, 1968, p. 218

Consistency of the Josephson effect

The voltage position of the microwave-induced Shapiro steps is independent of the composition and geometry of the junctions to a part in 1016

This is used in the SI realisation of the voltage standard.

Tsai et al., 1983

Junctions

1000

500

0

-500

-1000-10 0 10Array Voltage (mV)

Bia

s Cur

rent

(μA

)

The most precise law of physics (?)

Resistance metrology

Hall Effect

Edwin Hall, 1879

Uy = B Ix / neeRxy = UH / Ix = B /nee

Von Klitzing, 1980

Quantization of transverse (Hall) resistance in rational fractions of the resistance quantum

Ω== 557(18) 812.807 252K ehR

© D.R. Leadley, Warwick University 1997

Quantum Transport in 2DEG in Strong Magnetic Fields

L.Landau 1930© D.R. Leadley, Warwick University 1997

Materials

New materials are sought

MOSFET III-V

K. von Klitzing, Nobel lecture, December 9, 1985

Dirac Fermions in quantizing magnetic field

2nehRxy ±=

14/ 2

+±=

nehRxy

2/14/ 2

+±=

nehRxy

Landau level spacing

BBBcBLL

e

BBFBLL

F

kBkmBekkEmm

kBkBevkE

smv

/20)*(//067.0*

/420/2/

/106

≈==Δ=

≈=Δ

=

hh

h

ϖ

Graphene flakes

Pablo Jarillo-Herrero,

Experiment

Epitaxial graphene

LEEMphotoemission

Is SiC graphene sick?

C - terminated face: good mobility, stacked layers kill QHESi - terminated face: poor quality and mobility

Growth

AFM images of the sample reveal large flat terraces on the surface of Si-face of 4H-SiC(0001) substrate with graphene after high-temperature annealing in Ar atmosphere.

Patterning

Twenty Hall bar devices of different sizes, from 160 μm x 35 μm down to 11.6 μm x 2 μm were produced on 0.5 cm2 wafers. AFM shows that the Hall bars are patterned across many substrate terraces

Magnetotransport

300 K

μ≈ 2400 cm2/Vs at RT

4.2 K

μ≈ (4000-7500) cm2/Vs

ns ≈ (5-10)x1011 cm-2

Contact resistance

Rc=1.5 Ω

cf. kΩ in contacts to exfoliated graphene

Breakdown current

Non-dissipative transport

300 mKRxx<0.2 mΩ at Isd= 12 μA4.2 KRxx<2.4 mΩ at Isd= 2.5 μA

Cryogenic current comparator

B=0

Quantisation accuracy

Accuracy of resistance quantization:

3 ppb at 300 mK

Allan deviation from RK/2 vs. measurement time τ. The square root dependence indicates purely white noise.

0

12

1

2)]()([

)(ˆ0

PyyP

k kky

∑ = + τ−τ=τσ

Comparison

Quantisation 10000 times more accurate than in graphene flakes,comparable to semiconductor QH standards at 300 mK,still accurate at 4.2 K.

Conclusions

• Graphene Quantum Hall Resistance standard potentially superior to its seminconducting counteparts; optimization is routine!

• Scalable graphene technology!

• The role of substrate in doping and scattering processes of epi-graphene?

More details in: A. Tzalenchuk et al., arXiv:0909.1220

How about friends and foes?

Real materials and precise measurements: friends or foes?

The question is irrelevant.Important:

• Understanding real materials by precise measurements:– Superconducting resonators vs. dielectrics

• Making use of novel materials in quantum applications:– Quantum Hall effect vs. graphene