Table Of ContentQuantum inductance and negative electrochemical capacitance at finite frequency
Jian Wang ∗,1 Baigeng Wang,2 and Hong Guo3
1Center of theoretical and computational physics and Department of Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong, China
2Department of Physics, Nanjing University, Nanjing, China
7 3Department of Physics, McGill University, Montreal, Quebec, Canada
0
0 Wereporton theoreticalinvestigations offrequencydependentquantumcapacitance. Itisfound
2 that at finite frequency a quantum capacitor can be characterized by a classical RLC circuit with
n three parameters: a static electrochemical capacitance, a charge relaxation resistance, and a quan-
a tuminductance. Thequantuminductanceisproportionaltothecharacteristictimescaleofelectron
J dynamics and due to its existence, the time dependent current can accumulate a phase delay and
6 becomes lagging behind theapplied ac voltage, leading to a negative effective capacitance.
1
PACSnumbers: 72.10.-d,73.23.-b,73.21.La
]
l
l
a Understanding dynamic conductance of quantum co- high frequency. Due to L , electrons dwell in the neigh-
q
h
herent conductors is a very important problem in nano- borhood of the capacitor plates causing a phase delay.
-
s electronicstheory. Whentwoquantumcoherentconduc- At low frequencies, the dynamic response is capacitive-
e tors form a double plate “quantum capacitor”, its dy- like andvoltagelags current. At largerfrequencies when
m
namic conductance G(ω) is given by the frequency de- ω > 1/ C L , inductive behavior dominates and volt-
µ q
. pendent electrochemical capacitance1,2,3 C (ω), G(ω) = age leadps current: in this case the quantum capacitor
t µ
a −iωC (ω), here ω is the frequency. At low frequency, gives a negative capacitance value. It is, indeed, sur-
µ
m
C (ω) can be expanded in frequency and at the linear prising that a quantum capacitor can give an inductive
µ
- order, it is described2 by an equivalent classical circuit dynamic response. For the experimental setup of Ref.5,
d
consistingofastaticcapacitorC inserieswitha“charge we estimate that when ω ∼ 3GHz, the predicted high
n µ
relaxation resistor” R . For a conductor having a single frequency effects should be observable.
o q
c spin-resolved transmission channel, Rq was predicted2 Let’s first work out a simple expression for the fre-
[ to be half the resistance quantum, Rq = 1/2Go where quencydependentelectrochemicalcapacitanceCµ(ω)fol-
1 Go ≡h/e2. Thefactor1/2inRq isofquantumorigin2,4, lowing the work of Buttiker2. We consider a two-plate
v and has recently been confirmedexperimentally5. In the capacitor similar to the experiment of Gabelli et.al.5: a
0 experiment of Gabelli et.al.5, a submicron 2DEG quan- QD—labelled I, and a large metallic electrode—labelled
6 tum dot (QD) is capacitively coupled to a gold plate II. Each plate is connected to the outside world through
3 forming a double plate capacitor, where the QD con- its lead and a time dependent bias v is applied across
1,2
1 nectstotheoutsidereservoirbyasinglechannelquantum thetwoleads. Weconsidersmallamplitudesofv soas
0 1,2
point contact(QPC). The dynamic conductanceG(ω) is to focus on the linear bias regime. Under the action of
7
then measured at 1.2GHz, and the data is well fit to such a bias, the two capacitor plates develop their own
0
/ the equivalent circuit characterized by two parameters frequency dependent electric potential UI,II(ω). The
at (Cµ,Rq). charge on plate-I is equal to the sum of the injected
m The experiment of Gabelli et.al.5 opened the door for charge and induced charge: Q = Qinj +Qind. In the
I I I
elucidating important and interesting physics of high linear regime, the injected charge is proportionalto bias
-
d frequency quantum transport in meso- and nano-scale v (ω): Qinj =e2D (ω)v (ω)whereD (ω)isthegeneral-
1 I I 1 I
n devices. An important question is what happens to izedglobaldensityofstates(DOS)ofplate-Iatfrequency
o
electrochemical capacitance at higher frequencies be- ω. The induced charge, on the other hand, is in general
c
yond the linear ω regime, and in particular whether the related to a nonlocal Lindhard function2 whose calcula-
:
v two-parameter(Cµ,Rq) equivalent circuit is adequate at tionis simplifiedby applying the Thomas-Fermiapprox-
i
X higher frequenciesto describe a quantumcapacitor. It is imation. Within this approximation, the induced charge
the purpose of this paper to address these issues. is proportionalto the induced potential U on the plate:
r I
a In the following, we report a microscopic theory for Qind =−e2D (ω)U wherethe minussignindicatesthat
I I I
high frequency quantum transport in a two-plate quan- the charge is induced. Putting things together, we ob-
tum capacitor. Our results show that to characterize its tain Q = e2D (ω)(v −U ). Clearly, the same charge
I I 1 I
high frequency dynamic response, one needs—in addi- Q canbecalculatedbytheusualelectrostaticgeometric
I
tion to C ,R , a new quantity L having the dimension capacitanceC : Q =C (U −U ). Wethereforeobtain
µ q q o I o I II
of inductance. L is found to have purely quantum ori- a relationship: C (U −U ) = e2D (ω)(v −U ). Ap-
q o I II I 1 I
ginandwillbenamed“quantuminductance”. Therefore, plyingthesameargumenttoplate-II,wesimilarlyobtain
thefrequencydependentelectrochemicalcapacitanceofa −C (U −U )=e2D (ω)(v −U ). Finally, the same
o I II II 2 II
quantumcapacitorC (ω)isequivalenttoaclassicalRLC charge Q can also be obtained from the definition of
µ I
circuitcharacterizedbythreeparameters(C ,R ,L ),at the electrochemical capacitance: Q = C (ω)(v −v ).
µ q q I µ 1 2
2
These three relations allow one to derive the following for different values of Γ , by setting ∆E and tempera-
L
expression for C (ω): ture to zero. We observe that C is positive at small
µ R
frequency andbecomes negative atlargerfrequency. For
e2 e2 1 1 instance, C becomes negative at a “critical” frequency
= + + (1) R
Cµ(ω) Co DI(ω) DII(ω) ωc ∼ 10GHz for ΓL = 10µeV. This critical frequency
can be smaller for smaller linewidth function Γ . We
L
This result resembles the one obtained by Bu¨ttiker for notethatitisnotdifficulttoachieveΓ =10µeV exper-
L
the static capacitance2. An important difference is that imentally: in Ref.8, Γ between 1µeV to 5µeV has been
L
the frequency dependent electrochemical capacitance in realized. As will be discussed below, the effective Γ in
L
Eq.(1) is a complex quantity: its real part is a measure the experiment of Ref.5 is tunable by a gate voltage so
totheelectrochemicalcapacitanceanditsimaginarypart that the critical frequency at which the negative capaci-
is proportionalto the frequency dependent charge relax- tance occurscanbe evensmaller. The inset ofFig.1also
ation resistance. showsthatasweincreaseω,theimaginarypartofC (ω)
µ
Toproceedfurther,weneedtocalculatethe frequency starts from zero, reaches a peak value around ω and
c
dependent DOS DI,II(ω). Following Ref.4, the gener- then decays to zero. The negative capacitance at large
alized local DOS of plate-α of the capacitor can be ex- frequency can be understood as follows. For a classical
pressed in terms of Green’s functions: capacitor, a charge is accumulated across the capacitor
induced by an external voltage. The current and volt-
dn (ω) dEf −f¯
α = [G¯rΓ Ga] (2) age has a fixed phase relationship: the voltage lags be-
dE Z 2π ~ω α xx hind currentwith a phaseπ/2. For a quantumcapacitor
at low frequency, there exists a charge relaxation resis-
where the subscript x labels space coordinates and
tance R = h/(2e2) for a single channel plate, therefore
D (ω)=Tr[dn (ω)/dE]. InEq.(2),f istheFermifunc- q
α α the charge build-up time is the RC-time τ = R C .
tion and f¯≡f(E ) with E ≡E+~ω; Gr,a =Gr,a(E) RC q µ
+ + For C = 1fF and R = h/(2e2), this RC-time is about
is the retarded/advanced Green’s function at energy E µ q
τ = 13ps. If the external voltage reverses sign, the
and G¯r ≡ Gr(E + ~ω); Γ is the linewidth function RC
α charge accumulation will follow the voltage and also re-
describing the coupling strength between plate-α with
verse sign in due time. When frequency is low, namely
its lead. These quantities can be calculated in straight-
whenω <<1/τ =77GHz,thechargebuildupfollows
forward manner when the Hamiltonian of the capacitor RC
theacbiasalmostinstantaneouslyjustlikeaclassicalca-
model is specified6,7.
pacitor,thusthecapacitanceispositive. Whenfrequency
ForthequantumcapacitorofRef.5,plate-IisaQDand
is high, there is a phase difference between the ac bias
plate-II is a metal gate. Since the metal gate has much
and the charge build up. For frequencies comparable to
greater DOS, i.e. D >>D , we can safely neglect the
II I 1/τ , the charge build up can not follow the ac bias,
D term in Eq.(1). For a QD with one energy level E RC
II 0 thereby the capacitance may be negative.
and connected to one lead, its Green’s function Gr =
1/(E−E +iΓ /2) where Γ is the linewidth function
0 L L
of the lead. With this Green’s function, the frequency While the above argument explains why it is possible
to have negative capacitance, it would indicate a critical
dependent DOS canbe easily calculatedfromEq.(2), we
obtain: frequency to be near 77GHz. Our results (Fig.1) show
that the calculated ω is actually much smaller. This is
c
ΓL 1 ∆2 because there exists a second relevant time scale in the
D (ω)= [ ln
I 2π~ω(~ω+iΓL) 2 ∆+∆− QD,i.e. thedwelltimeτdwhichisthetimespentbyelec-
∆E−~ω ∆E+~ω tronsinsidethe QD.Thedwelltimeτd canbe calculated
−i(arctan −arctan )] (3) forspecific systems9. Importantly,atresonancethe elec-
Γ /2 Γ /2
L L
trons can dwell inside the quantum dot for a long time.
where ∆ = ∆E2 + Γ2/4, ∆ = (∆E ± ~ω)2 + For instance, for our QD with ΓL = 10µeV, we found
L ±
Γ2/4, and ∆E = E − E . At resonance ∆E = τd = 260ps while τRC = 12ps (since Cµ = 0.9fF). In
L F 0
0, we obtain Re(D (ω)) = [−xln(4x2 + 1) + other words,τd >>τRC. Such a τd correspondsto a fre-
I
2arctan(2x)]/(2πΓ x)/(x2+1) with x=~ω/Γ . Hence quency of 4GHz, much less than 1/τRC. In other words,
L L
whenacfrequencyisgreaterthan1/τ ,thechargesdwell
Re(D (ω)) is positive for small x and negative for large d
I
x, i.e. there is a sign change. A similar behavior is also inside the quantum dot and cannot respond to the ac
voltagechange. As a result,currentbecomes laggingbe-
found for the system away from the resonance. Due to
hindvoltage,leadingtoanegativecapacitance. Thispic-
this sign change of Re(D ), from Eq.(1) the frequency
I
dependent electrochemical capacitance C ≡ Re[C (ω)] ture agrees very well with the numerical results (Fig.1).
R µ
can become negative.
To be more specific, we fix the classical capacitance Having determined the general behavior of C (ω) for
µ
of QD C = 1fF which is a typical value for QD with the quantum capacitor, in the following we determine
0
area of ∼1µm2. Fig.1 plots C =Re[C (ω)] (real part) how to simulate this quantity using a classical circuit.
R µ
and C = Im[C (ω)] (imaginary part) versus frequency Expanding Eq.(1) into a Taylorseries to second order in
I µ
3
ω with the help of Eq.(3) at resonance, we obtain: The situation is somewhat different for quantum in-
ductance L when the system is off resonance (∆E 6= 0
q
C (ω)=C +iωC2 h −ω2C3 h2 +ω2C2 h2 (4) in Eq.(3)). In this case the dwell time τd becomes too
µ µ µ2e2 µ4e4 µ12πΓ e2 small to be relevant and another time scale becomes im-
L
portant,namely the tunneling time τ for electronsto go
t
where C = C (0) on the right hand side is the static
µ µ in/out of the QD. The further away E is from E , the
0
electrochemical capacitance. This result is equivalent to
longer is τ . Hence in Eq.(7) τ should be replaced by τ
t d t
that of a classical RLC circuit, as follows. For a classi-
foroffresonance. Ourresultshowsthatthefittingoffull
cal RLC circuit with capacitance C , resistance R and
µ q quantum capacitance C (ω) using classical parameters
µ
inductance L , the dynamic conductance is
q C , L and R (ω) is still quite good for off resonance.
µ q q
This further supports the conclusion that the frequency
G(ω)=−iωC /(1−ω2L C −iωC R ) . (5)
µ q µ µ q dependent quantum capacitance can be described by a
classical RLC circuit with static electrochemical capac-
Expandingthisexpressioninpowerseriesofω,weobtain
itance, charge relaxation resistance, and a quantum in-
G(ω)=−iωC +ω2C2R +iω3C3R2−iω3C2L . (6) ductance.
µ µ q µ q µ q
Finally, we perform a numerical calculation of the dy-
BecauseforacapacitorG(ω)=−iωCµ(ω),weobtainthe namic conductance for the device structure of Ref.5. In
resultthatourquantumcapacitanceEq.(4)is equivalent terms of scattering matrix, the DOS of Eq.(2) for a ca-
to the classicalRLC circuit model of Eq.(6). Comparing pacitor can be re-written as4,6
these two equations we readily identify R =h/(2e2)—a
result first obtained by Bu¨ttiker and co-qworkers2. Im- D (ω)=i dEf −f¯[1−s† (E )s (E)] (8)
portantly, a new quantity—the equivalent inductance, is I Z 2π ~2ω2 LL + LL
identified as L =h2/(12πe2Γ ). In terms of dwell time
q L with5 s† (E) = (r − eiφ)/(1 − reiφ), φ = 2πE/∆,
τd and charge relaxation resistance Rq, we obtain: r2 =1−LTL andT =1/[1+exp(−(V +V )/∆V )]
QPC QPC g 0 0
L =R τ /12 (7) which is the transmission coefficient of the QPC in the
q q d
experimental setup5. Fig.3 shows the dynamic conduc-
where τ =4~/Γ . tanceG(ω)vsgatevoltageV usingourtheorypresented
d L g
What is the reason that a quantum capacitor at fi- above. When ω = 1.2GHz (open circle), our results
nite frequency needs to be modeled by a classical RLC agrees very well10 with the experimental data of Ref.5.
circuit (instead of a RC circuit)? This is due to the Whenω =3GHz(opensquare),ourtheorypredictsthat
role played by the large dwell time τ of the QD. When the imaginary part of G(ω) which is the electrochemical
d
electrons dwells a long time τ inside the QD, the inter- capacitance, goes to negative. For even larger frequency
d
action between electrons become an important piece of ω =5GHz the effect is more significant. To understand
physics which, in our theory, is modeled by the induced why one canobservea negative capacitanceatsmall fre-
self-consistent potential U discussed above. Such an quency suchas 3GHz,we note that since electronenter-
I,II
interaction gives rise to the physics of induction, and re- ing the QD has to first pass the QPC5: this QPC serves
sulting to the quantity L of Eq.(7). Indeed, the explicit as a barrier (with an effective barrier height 1/Γ) that
q
dependence onτ byL alsoconfirmsthe importantrole is controlled by the gate voltage. At small gate voltage,
d q
playedbythedwelltime. BecauseL isdeterminedbyτ T is nearly zero and goes to one at large V . Hence
q d QPC g
as well as fundamental constants h and e, it is of purely theeffectiveΓforsmallgatevoltageisquitelargemaking
quantum origin and can be called quantum inductance. ω much smaller. Since the experiment of Ref.5 is per-
c
Fig.2 compares the fitting of classical RLC circuit formed at ω =1.2GHz, we assume that it is not too dif-
Eq.(5), with the full quantum result of Eq.(1). They ficulttopushthefrequencyto 3GHzsothattheeffectof
compareverywellfortheentirerangeofthefrequency— quantuminductancecanbeobservedexperimentally. In-
if we treat R as a function of ω. Indeed, while R has deed, a single-wall carbon nanotube transistor operated
q q
so far been a constant h/(2e2) as identified through the at2.6GHzhasbeendemonstrated11 andmeasurementof
Taylor expanded Eqs.(4,6), it is actually a function of ω current fluctuation at frequency from 5 to 90 GHz has
bythemoregeneralexpressionEq.(5). TheinsetofFig.3 been reported12.
plots the general R = R (ω) obtained numerically, and Insummary,wefoundthatatfinitefrequency,aquan-
q q
we observe it to be a slowly increasing function of ω. As tum capacitor consisting of a quantum dot and a large
expected, in the small frequency limit, R (ω) recovers metal conductor is equivalent to a classical RLC circuit
q
the result of half resistance quantum. For Γ = 50µeV, with three basic parameters: a static electrochemicalca-
L
R (ω)deviatesfromh/2e2atabout5GHz. Wehavealso pacitance C , a charge relaxation resistance R , and a
q µ q
attempted using three constant parameters C , L and quantuminductanceL . ItisfoundthatL ∼R τ where
µ q q q q
R =h/(2e2) into Eq.(5) to compare with the full quan- τ isthecharacteristictimescaleforthequantumdotsuch
q
tum result of Eq.(1), a reasonableagreementis obtained asthe dwelltime τ orthe tunneling time τ . Due to the
d t
(inset of Fig.2) although not as good as that shown in phase delay by the quantum inductance, the dynamic
Fig.2. current can lag behind of the applied ac voltage, giv-
4
ing rise to a negativecapacitance. Our numericalresults and 60390070 (B.G.W); NSERC of Canada, FQRNT of
showthatthiseffectshouldbedetectableexperimentally Qu´ebec and Canadian Institute of Advanced Research
using the present device technology. (H.G).
Acknowledgments. We thank Prof. K. Xia for
useful discussions. This work is supported by a RGC
grant(HKU7048/06P)fromtheHKSARandLuXinEn-
ergy Group (J.W); NSF-China grant number 10474034 ∗) Electronic address: [email protected]
6 Z.S. Ma et al, Phys.Rev.B 59, 7575 (1999).
FIG. 1: Frequency dependent capacitance Cµ(ω) versus fre- 7 B.G. Wang et al, Phys.Rev.Lett. 82, 398 (1999).
quency. Main figure is for CR and inset is for CI. Here 8 T. Fujisawa et.al., Science 282, 932 (1998).
ΓL = 10µeV (for the curve with the open circle), 50µeV 9 V. Gasparian et al, Phys.Rev. A 54, 4022 (1996).
(open square), 100µeV (open triangle), and 200µeV (open 10 In classical RC circuit theory, the convention is writing
diamond). Here Co =1fF. v(t)=v0exp(iωt),while in quantumtransport theory the
convention is v(t) = v0exp(−iωt). This is responsible for
the sign difference between our Fig.3 and the experimen-
FIG. 2: Comparison of the full quantum capacitance Cµ(ω) tally data of Ref.5.
(solidline)tothatobtainedbytheclassicalRLCcircuitmodel 11 S.D. Li et al, NanoLett. 4, 753(2004).
(dotted line) for ΓL = 50µeV. Here Re[Cµ(ω)] is indicated 12 R.Deblock et al, Science 301, 203(2003).
by open circle. Inset: similar fit but using three constant
parameters.
1 S. Luryi,Appl.Phys. Lett. 52, 501 (1998). FIG. 3: The dynamic conductance G(ω) (unit e2/h) versus
2 M.Bu¨ttiker,J.Phys.: Condens.Matter5,9361(1993);M. gate voltage at different frequencies ω = 1.2,3,5GHz. Solid
Bu¨ttiker et al, Phys.Lett. A 180, 364 (1993). line −Re[G]/ω, dotted line −Im[G]/ω. Other parameters:
3 T.P. Smith et al, Phys. Rev.B 32, 2696 (1985). ∆ = 500mK, α = 3000, V0 = −0.85, ∆V0 = 0.003, C0 =
4 S.E.Nigg, R.Lopezand M.Bu¨ttiker, Phys.Rev.Lett 97, 4fF, and temperature 50mK. For purpose of illustration,
206804 (2006). we divided G(ω) by ω. Inset: frequency dependent Rq (unit
5 J. Gabelli et.al.,Science 313, 499 (2006). h/e2) vs frequency.
