Table Of ContentQuantum Friction of Micromechanical Resonators at Low Temperatures
Kang-Hun Ahn1 and Pritiraj Mohanty2
1 School of Physics, Seoul National University, Seoul 151-747, Korea
2 Department of Physics, Boston University, Boston, MA 02215
(Dated: February 2, 2008)
3
0 Dissipation ofmicro-andnano-scalemechanicalstructuresisdominatedbyquantum-mechanical
0
tunnelingoftwo-leveldefectsintrinsicallypresentinthesystem. Wefindthatathighfrequencies—
2
usually, for smaller, micron-scale structures—a novel mechanism of phonon pumping of two-level
n defectsgivesrisetoweaklytemperature-dependentinternalfriction,Q−1,concomitanttotheeffects
a observed in recent experiments. Due to their size, comparable to or shorter than the emitted
J phononwavelength,thesestructuressufferfromsuperradiance-enhanceddissipationbythecollective
5 relaxation of a large numberof two-level defects contained within thewavelength.
1
PACSnumbers: 62.65.+k,62.20.Dc,62.40.+i
]
l
l
a Physical properties of physically- and chemically- ample,emittedphononwavelengthbecomesmuchlonger
h
engineeredmicromechanicalsystemsareofimmensefun- (≫ 1µm) compared to the system size at low temper-
-
s damental and technical interest [1]. Some of the recent atures, bringing novel correlation effects[16], extremely
e
spectacular examples of micromechanical structures in- relavant to mesoscopic structures.
m
clude the measurement of vortex motion in high-T su- In this work, we evaluate internal friction Q−1 when
c
.
t perconductors[2],biomolecularrecognition[3],actuation the two-level defects operate non-linearly and non-
a
of sensors via Casimir force [4], and shuttling of electron adiabatically. We show that the nonlinear response of
m
charge in a quantum dot [5]. Important to all these ex- Q−1 results in a large contribution at low tempera-
-
d periments is the oscillation of a particular or a set of tures at high driving frequencies due to the process of
n micron-sized resonators at a resonance frequency deter- phonon pumping. We propose and demonstrate a mech-
o minedbythegeometryandmaterialproperties. Changes anism of how Q−1 can be significantly enhanced by the
c
in the ”resonant” oscillation frequency or the oscillation phonon pumping process through cooperative emission
[
amplitudemostlydeterminethemagnitudeoftheforceof of phonons. Cooperative emission becomes possible be-
1 interest, to which the micromechanical structure is cou- cause of the long wavelength of the emitted phonons,
v
pled. Detrimentaltothedetectionofforceisthedamping which allows the correlated decay of two-level systems
5
of the resonant structure, quantified by quality factor Q contained within the wavelength. Our theory provides a
5
2 or dissipation Q−1. mechanism for enhanced dissipation in micromechanical
1 The essential problem inherent to the types of force resonators within the context of two-level systems.
0 measurement mentioned above is the low quality fac- TheHamiltonianofaTLSisfullycharacterizedbythe
3
tor Q, observed in small resonators. In many experi- asymmetry energy ∆ and the tunneling matrix element
0
/ ments, intrinsic two-level defects[6, 7] are found to be ∆0. The asymmetryenergy of a TLS is the energy split-
at thedominantcauseofinternalfrictionQ−1 incrystalline ting due to the static strain±21∆s andthe appliedtime-
m resonators [8, 9, 10]. The observed linear temperature dependent strainfieldǫ0cosωt: ∆(t)=∆s+2γǫ0cosωt,
- dependence of Q−1 ∝ T [8] could be explained assum- where γ is the constantof coupling between the acoustic
d ingreasonabledensityofdefectswithin alinearresponse waveandtheasymmetryenergy. Frictionduetothermal
n
theory of two-level systems (TLS)[11, 12]. Despite the phonons is described by the Hamiltonian for the single
o
c success of the TLS mechanism, dissipation at low tem- TLS coupled to phonon fields[7];
iv: pobersaertvuaretisocnonotfinauweseatkohteomldpmeraantuyrcehdalelpenengedse,nncoet[a1b0l]yatnhde H = ∆s+2γ2ǫ0cosωtσz + ∆20σx+ ¯hωα(q)a†q,αaq,α
X the non-monotonic dependence of Q−1(T) in silicon res- Xq,α
r onators[13]. Mostrecent experiments merely addto the
a listPorefvpioroubsltehmesoryeettictaolbweoruknsdoenrstthooeda.coustic response of + Xq,αγαs2ρqV¯hvασz(aq,α+a†−q,α), (1)
TLS use the linear response approximation or adiabatic where V and ρ are the volume and the mass density of
approximation for nonlinear response[14, 15]. The ques- theresonator,v isthesoundvelocityforpolarizationα,
α
tion we would like to explore is whether the anomaly a (a† )isthephononannihilation(creation)operator.
q,α q,α
can be explained in terms of TLS if both assumptions γ is the coupling constant which relates the change in
α
are abandoned. For dissipation in micromechanical res- the asymmetry energy to the local elastic strain.
onators,suchissues becometerribly important,asanew In crystalline materials with many TLSs, it is known
parameter space emerges with decreasing size. For ex- thatthestaticasymmetryenergy∆ hasawidedistribu-
s
2
tion due to the randomness of local strain, while tunnel- right localized states respectively. The leading term for
ing splitting energy ∆ has a well-defined value[11]. For the transition from one state to the other is the second
0
simplicity, we consider a rectangular distribution[12] of order process involving the tunnelling and emission (ab-
∆ with n number of TLS per unit volume in 0<∆ < sorption)ofaphonon;atlowtemperaturessinglephonon
s s
∆max. The underlying assumption, h¯ω ≪ γǫ ≪ ∆max, processdominates. Thethermalaveragedrelaxationrate
s 0 s
holds in most experiments. is written as
The internal friction Q−1 is related to the energy loss
1 ∆
per unit volume δE in a cycle of the acoustic wave ; =A∆2∆coth , (5)
τ(∆) 0 2k T
Q−1 = δE , where E = 2ρǫ2v2 is the acoustic wave B
2πE0 0 0 (cid:0) (cid:1)
epnheorngoynsetmoriesdsiopnerpouwneirt Pv¯o(∆lusm,Te[)7]o.f aThTeLStimiser-ealvaetreadgetdo where A = α 2πh¯γ4α2vα5ρ. Here we assume the Debye
Q−1: Q−1(T) = n 1 ∆msaxP¯(∆ ,T)d∆ . Further- density of stPates of phonon with a linear dispersion:
2ρv2ǫ20ω 0 s s ωα =vαq.
more, we divide the two-level systems into two kinds
R Time evolution of the occupation number nr(= 1 −
basedontheirasymmetryenergy∆ : (i)∆ <2γǫ ,and
s s 0 n ) follows the rate equation[14] for the time-dependent
l
(ii) ∆ >2γǫ ; which contribute to the total dissipation
s 0 asymmetry energy ∆(t)=2γǫ cosωt+∆ :
according to Q−1(T)=Q−1+Q−1: 0 s
1 2
Q−11(T) = 2ρnv2ǫ21ω 2γǫ0P¯(∆s,T)d∆s (2) ddntr =−nτr(∆−(nt)(r0)); n(r0)(t)=1/(eβ∆(t)+1), (6)
0 Z0
n 1 ∆msax where β = 1/k T. The typical process is depicted in
Q−1(T) = P¯(∆ ,T)d∆ . (3) B
2 2ρv2ǫ20ω Z2γǫ0 s s Fig. 1. In the regime ∆s < 2γǫ0, the two energy levels
cross,andatthispointpopulationinversionarises. Note
Inlinearresponsetheorywhereǫ isinfinitesimallysmall,
0 that the population inversion process ( A → B → C )
all the TLS belong to category (ii), and Q−1 = 0. How-
1 is followedby the spontaneousemissionof phonon(C →
ever, even a small experimental value of ǫ = 10−6 will
0 D). The high-frequency assumption of Eq.(4) is crucial
giverisetononlinearbehavior[8],whichimpliesthatTLS
forthe pumping processnearthe level-crossingpoint. In
with ∆ <2γǫ may also play an important role[12].
s 0 the conventional adiabatic limit (γǫ ¯hω/∆ ≪∆ ) with
0 0 0
Usually a time-dependent unitary transformation U TLSenergiesof±1E(t),levelcrossingisavoidedandthe
is used for solving time-dependent Hamiltonian in the 2
TLSinthelow(high)energystateremainsmostlyinthe
nonlinear regime, so that the TLS has time-dependent
low(high) energy state.
energy levels given as ±1E(t),(E(t) = ∆(t)2+∆2).
2 0 The time-averaged phonon emission power[14] can be
In the standard adiabatic limit[14, 15] used to explain
p obtained by solving (Eq.6) for nr(t);
the low-frequency experiments, the acoustic field fre-
quency is assumed to be very low so that the additional 2π
ω ω dn (t)
term due to the unitary transformation −i¯hU†∂U = P¯(∆s,T)=− dt∆(t) r . (7)
∂t 2π dt
σ ∆0γǫ0¯hωsinωt can be safely neglected. In contrast, Z0
y E(t)2
here we consider the high frequency regime where the For the general case of asymmetric TLS, numerical in-
matrix element of this additional term can not be ne- tegration of Eq.(6) gives P¯ as a function of the static
agtletchteedleinveclo-cmropsasrinisgonpotointthEe(dti)a=gon∆a0l:term 12E(t)σz even sausybmsemqueetnrytlyengeivrgeys Q∆−1s1.anFdortethmepperaarttuicruelaTr,caasnedoEf sqy.(m2)-
metric TLS (∆ = 0), Eq.(7) can be rewritten using an
s
γǫ ¯hω
0 ≫∆0. (4) analytic solution of nr(t) at T =0;
∆
0
Under this condition, as the tunneling motion between P¯ = 2γ2ǫ20A∆20 2πω cos2ωt exp(siωnτωǫt) dt (8)
the two sites is not fast compared to the rapid time evo- πω Z−2πω cosh(1/ωτǫ)
lution of the acoustic wave, it is not useful to take the
TLS bonding states of the two sites as the basis states. where τǫ is a typical relaxation time scale of symmetric
Instead,it is morenaturalto considerthe localizedbasis TLS at zero temperature when ωτǫ >1;
states in each local potential minimum and consider the
1
tunneling term 21∆0σx as a perturbation. τ =2A∆20γǫ0. (9)
We treat the coupling to the acoustic wave adiabati- ǫ
cally and treat the tunneling term and the coupling to When ∆ >2γǫ , the TLS always stays in the lower en-
s 0
the thermal phonon pertubatively. The approximation ergy state at zero temperature, hence the TLS does not
isguaranteedbythe factTLS-phononinteractioncanbe emit a phonon in its steady state, P¯(∆ > 2γǫ ,T =
s 0
treatedpertubativelywheneverthetwo-levelapproxima- 0) = 0. Since P¯(∆ ,T = 0) is a smoothly-decaying
s
tion is valid[18]. Let |l > and |r > be the left and the function of ∆ , an analytic formula for Q−1(T = 0) can
s 1
3
3
A B C D
Q−1
Q−1
2 1
2υ] Q2−1
ρ
2γ/
n
C −1Q [
1
Energy
Phonon
Time
B
D 0
0 1 2
A
k T/2γε
B 0
FIG.1: Theschematicdiagramdescribestheenergy-pumping FIG.2: TheinternalfrictionQ−1 =Q−1+Q−1 asafunction
1 2
process of the two-level atom from the acoustic wave to of temperature for ωτǫ=1.
phonons. The crossing curves represent the time evolution
of the two energy levels with the energy gap of ∆(t) =
2γǫ0cosωt+∆s. The energy diagrams of thetwo-level atom
asafunction oftheconfiguration coordinates aredepictedin 12). Q−1showsaweaktemperaturedependencewithde-
theupperpanel. creasing temperature, as Q−1 becomes more important
1
than Q−1 at low temperatures. The low-temperature
2
saturation value of Q−1 for ωτ ≥ 1 is well approx-
be obtained using the approximation 02γǫ0P¯(∆s,T = imated by ∼ 0.3(1/ωτǫ)(nγ2/ρvǫ2). Now, let us esti-
0E)qd.∆(8s),≈thγeǫ0z(eP¯ro(0,te0m)+perP¯a(tγuǫr0e,0in))te=rnaγlRǫ0frP¯ic(t0i,o0n).Q−1Fr1omis mroautgehlτyǫthfreomhigehx-tpeemripmeernattaulredsaattau[r8a].tioSni[n6]cevanluγe2o/fρvQ2−1is,
found to be for the single-crystalline silicon data in Ref.[8], we esti-
mate nγ2/ρv2 ∼ 10−6 −10−5 from the experiments at
nγ2 1 1
Q−1(T =0)≈ sech( )I ( ), (10) ω ∼ 103 − 104Hz. Since the low temperature satura-
1 ρv2 ωτ 1 ωτ
ǫ ǫ tion value of Q−1 was found to be ∼ 10−7 − 10−6 for
these frequencies, we estimate ωτ ∼ 1, which means
where I (x) is the first kind modified Bessel function of ǫ
1
τ ∼ 10−3 − 10−4sec. However, these values of τ are
order 1. ǫ ǫ
Now let us turn to Q−1. For weak strain 2γǫ < ∆ , anomalously small compared to the direct estimate us-
2 0 s
ing Eq.(9)with reasonable values of parameters[6, 7, 11].
where the energy levels do not cross, phonon pumping
Byusingγ ∼1eV,ǫ ∼10−6,A=108s−1k−3intoEq.(9),
process does not take place. This is the regime where 0 B
we get τ ∼ 1 sec. In other words, the low-temperature
the usual linear response theory is a good approxima- ǫ
dissipation in experiments[8] is anomalously larger than
tion for describing internal friction. In the linear re-
sponse theory[6, 11], Q−1(T) in Eq.(3) for k T ≫¯hω > our direct theoretical estimation.
2 B
¯h/τ(∆ ) can be written as[6] Here we propose and discuss the origin of the anoma-
0
louslysmallrelaxationtimeτ . Webelievethatthecoop-
nγ2 1 ∆msax ∆2 E erativeemission[17]ofphonoǫnsprovidesanaturalmech-
Q−1(T)= sech2( )τ−1(E)d∆(,11)
2 ρv2k T ω E2 2k T anism for the short relaxation time. In Dicke superradi-
B Z2γǫ0 B
ance[17],N numberoftwo-levelatomswithinadistance
where E = ∆2+∆2. For ∆ ≪ γǫ ≪ ∆max, an shorter than the emitted photon wavelength emit pho-
0 0 0 s
expression for Q−1 is easily found: tons cooperatively, therefore the lifetime of the atom in
p 2
the excited state is effectively reduced by a factor of N.
nγ2 2 k T Inthe usualtreatmentofphononemissionofTLS,aswe
Q−1(T)= f B , (12)
2 ρv2 ωτ 2γǫ havedone sofar,the emissionofphononiscalculatedby
ǫ 0
(cid:0) (cid:1) assuming that the TLSs are totally independent. How-
where f(x) is an almost linear function, defined by ever,this assumptionisnotvalidwhenanumber ofTLS
f(x)=x ∞ dt t . are contained within a distance shorter than the wave-
1/x sinht
Fig. 2 shows the temperature dependence of Q−1 = length of the emitted phonon. This phonon wavelength,
R
Q1−1+Q−21 for ωτǫ =1, calculatedfromEqs.(2,3,6,7 and λph ∼ h¯γvǫ0s (vs isthe soundvelocity),canbe considerably
4
>
larger (∼µm), leading to the cooperative emission from
a large number of two-level systems.
Now let us consider N two-level systems within a vol- 2
ume λ3 . In the context of phonon emission, pumping
ph
of the two-level system is done by the acoustic wave
applied to the resonator. The many-TLS Hamiltonian
Cooperative
in Eq.(1) can be generalized to H = H + H + ω] Emission
HTLS−¯hωph(wqh)aer†e Ha0 =, HHTLS=+∆H0ph =N 21∆σ(t),0anNid=1σtuin,z + γεP[0N 1
q,α α q,α q,α tun 2 i=1 i,x P
P P Individual
N
q¯h Emission
H = γ σ (a +a† ).(13)
TLS−ph i=1 q,α αs2ρVvα i,z q,α −q,α
XX
0
The many-TLSstate l,l,l,...l ,whereallthe TLSare 0 5 10 15
inthelstate,isthemaximallyexcitedstateortheground
N
(cid:12) (cid:11)
state. Weconsiderthe s(cid:12)etofmany-TLSstatesgenerated
from H +H , which are
TLS−ph tun FIG.3: Thetime-averaged phononemission power P¯N of N
N/2+M symmetric TLSs (∆s = 0) at T = 0.1γǫ0/kB and ωτǫ = 20,
N
(N/2−M)! which was numerically calculated using Eq.(18).
M = σ+ l,l,l,...l (,14)
sN!(N/2+M)! i !
i=1
(cid:12) (cid:11) X (cid:12) (cid:11)
(cid:12) (cid:12) proportional to N2 for ωτ > 1. Our numerical results
where M = −N/2,−N/2+1,−N/2+2,...,N/2. These ǫ
indeed show that the P¯ has a quadratic dependence of
states satisfy H (t) M =M∆(t) M . As in the case N
TLS
N (see Fig. 3.)
of a single TLS, a perturbation analysis finds the transi-
(cid:12) (cid:11) (cid:12) (cid:11) When N number of TLS are within λ , one obtains
tion rate wM±1,M from(cid:12) M to M ±(cid:12) 1 : to be Q−1 by replacing P¯ with its effective valupehP¯∗ =P¯ /N.
1 N
(cid:12)(cid:12) N(cid:11) (cid:12)(cid:12) N(cid:11) The enhanced phonon emission power due to superradi-
wM±1,M(t)=w(∓∆(t)) ∓M ±M +1 ,(15) anceP¯∗ ∼NP¯thenprovidesanaturalexplanationofthe
2 2
(cid:18) (cid:19)(cid:18) (cid:19) anomalously-large internal friction at low temperatures.
where Including the phonon superradiance effect, Eq.(10) for
ωτ >1 can be rewritten as
∆ ǫ
w(∆)=A∆2 (16)
01−exp(−∆/kBT) Q−1(T =0)=Q−1(T =0)∼ 1 nγ2, (19)
1 ωτ∗ ρv2
Here, we have used the relation M ±1 N σ± M = ǫ
i=1 i
where τ∗ ≈ τ /N is the renormalized relaxation time
N/2∓M) N/2±M +1). N(cid:10)ow th(cid:12)ePfollowin(cid:12)g r(cid:11)ate ǫ ǫ
(cid:12) (cid:12) and N is the number of cooperating TLSs. Thus, the
qequation for the many-TLS remains to be solvednumer-
(cid:0) (cid:0) enhancement of Q−1 at low temperature by a factor of
ically:
103−104comparedtoourinitialestimateintheindepen-
dn dent TLS model (Eq.(10)) indicates that a largenumber
M
dt = wM,M+1(t)nM+1−wM−1,M(t)nM of∼103−104 TLS emit phononscooperatively. By tak-
+ wM,M−1(t)nM−1−wM+1,M(t)nM, (17) ing λph ∼ 10µm from λph = h¯γvǫ0s, the total number of
TLSs in the volume λ3 is ∼ 105 according to Ref.[11]
The time-averaged power P¯N emitted by N number of or ∼103 according to Rphef.[12], which is reasonably large
TLS can be obtained as: enoughnumberforthecooperativeemissionby103TLSs.
Inconclusion,weformulateatheoryofinternalfriction
ω 2π/ω N/2 dn (t)
P¯ =− dt M∆(t) M . (18) ofmicro-andnano-mechanicalresonators,whichinvokes
N
2π Z0 M=X−N/2 dt intrinsictwo-leveldefects inthenonlinearregimeathigh
frequencies. As temperature decreases, the mechanical
Note that for phonons independently emitted by the friction crosses over from the linear regime to the non-
TLS, P¯ is simply proportionalto the number N; P¯ = linear regime, resulting in the saturation behavior as T
N N
NP¯. However,incooperativeemission,thelifetimeofthe goes to zero. Because of the phonon superradiance, the
excitedstateofasingleTLSisN timesshorter,hencethe low-temperature saturation value of Q−1 is strongly en-
phononemissionpowerofasingleTLSshouldbeN times hanced by a factor givenby the number of two-levelsys-
larger when ωτ >1[19]. This means that P¯ should be tems contained within the emitted phonon wavelength.
ǫ N
5
This work was supported by Brain Korea 21 project. [9] R. E. Mihailovich and J. M. Parpia, Phys. Rev. Lett.,
KHA is thankful to the Boston University Physics De- 68, 3052 (1992).
partment for the kind hospitality during his visit, where [10] P.Mohanty et al., Phys. Rev.B 66, 085416 (2002).
[11] W. A. Phillips, Phys.Rev. Lett. 61, 2632 (1988).
apartofthisworkwasperformed. PMthankstheDean’s
[12] R. W. Keyes,Phys. Rev.Lett. 62, 1324 (1989).
Office of Boston University for the support of this work.
[13] D.S. Greywall et al.,Europhys. Lett., 34(1), 37 (1996).
[14] Yu. M. Galperin, Zh. Eksp. Teor. Fiz. 85, 1386 [Sov.
Phys. 58, 804 (1983)], M. Kirkengen, Y.M. Galperin,
Phys. Rev.B 62, 16624 (2000).
[15] J.T.Stockburger et al., Phys.Rev.B 51, 2835 (1995).
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[4] H.B. Chan et al.,Science 291, 1941 (2001). [19] Since the energy gap of our TLS has time variation, if
[5] A.Erbe et al., Phys.Rev.Lett. 87, 096106 (2001). ωτǫ<1,reductionoftherelaxationtimegivesrisetored
[6] W. A.Phillips, Rep.Prog. Phys. 50, 1657 (1987). shift of emitted phonon. So the phonon emission power
[7] Tunneling systems in amorphous and crystaline solids, of N numberof TLS is not simply proportial to N2.
edited by Pablo Esquinazi (Springer, 1998).
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