DTIC ADA530609: Intrinsic Capacitances and Inductances of Quantum Hall Effect Devices

13 Pages·0.18 MB·English

by  Defense Technical Information Center

Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
[J.Res.Natl.Inst.Stand.Technol.101,733(1996)]
Intrinsic Capacitances and Inductances of
Quantum Hall Effect Devices
Volume 101 Number 6 November–December 1996
M. E. Cage and A. Jeffery Analyticsolutionsareobtainedforthein- producesmallout-of-phaseimpedance
ternalcapacitances,kineticinductances,and correctionstothein-phasequantizedHall
magneticinductancesofquantumHall resistanceandtothein-phaselongitudinal
National Institute of Standards and
effectdevicestoinvestigatewhetherornot resistance.
Technology, thequantizedHallresistanceistheonly
Gaithersburg, MD 20899-0001 intrinsicimpedanceofimportanceinmea-
surementsoftheacquantumHalleffect. Keywords: impedancestandard;internal
Theinternalcapacitancesandinductances capacitance;intrinsicimpedance;kinetic
areobtainedbyusingtheresultsofCage inductance;magneticinductance;quantum
andLavine,whodeterminedthecurrent Halleffect;two-dimensionalelectrongas.
andpotentialdistributionsacrossthewidths
ofquantumHalleffectdevices.These
intrinsiccapacitancesandinductances Accepted: October22,1996
1. Introduction
TheintegerquantumHalleffect[1–3]requiresafully Impedancesarecomplexquantities,andcantherefore
quantizedtwo-dimensionalelectrongas(2DEG).Atlow have both real and imaginary components. If it is to be
currentsthereisnegligibledissipationwithintheinterior ausefulabsoluteorintrinsicacresistancestandard,the
of the 2DEG in the quantum Hall plateau regions of impedanceacrossthedevicemustbedominatedbythe
high-quality devices, and the longitudinal resistance real component, which is the quantized Hall resistance
R along the device is very small. Within these plateau R (i), and the impedance along the device must be
x H
regions the quantized Hall resistance R across the small and again dominated by the real component,
H
devicehasthevalueR (i)=h/(e2i)fortheithplateau, which is the longitudinal resistance R . The imaginary
H x
where h is the Planck constant, e is the elementary components(theinternalorintrinsicimpedancesdueto
charge, and i is an integer. We assume here that the capacitancesandinductancesofthequantumHalleffect
quantity iR (i) has the value of the von Klitzing deviceitself)mustprovidesmallcontributionsinorder
H
constant R =25812.807 (cid:86). toavoidasignificantout-of-phaseorquadraturesignal.
K
The quantum Hall effect has been used to realize a Thereare,ofcourse,externalcapacitancesandinduc-
device-independentresistancestandardofhighaccuracy tances in the sample probe arising from the sample
fordccurrents[4,5]andforverylowfrequencycurrents holder, bonding wires, and coaxial cables. We do not
below 4 Hz [6]. An ac quantum Hall effect impedance consider the external impedances here, but they must
standardisnowbeingdevelopedforalternatingcurrents also be accounted for. Signals that are in-phase with
havingfrequenciesoforder103Hzandangularfrequen- R (i) and R , due to products of external and internal
H x
cies of order 104 rad/s [7–10]. capacitances and inductances, can also be present.
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1. REPORT DATE 3. DATES COVERED
OCT 1996 2. REPORT TYPE 00-00-1996 to 00-00-1996
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Intrinsic Capacitances and Inductances of Quantum Hall Effect Devices
5b. GRANT NUMBER
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National Institute of Standards and Technology,Gaithersburg,MD,20899 REPORT NUMBER
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
These second-order effects must also be small if the
device is to be a useful intrinsic standard.
Seppa,Satrapinski,Varpula,andSaari[11]haveesti-
mated that the kinetic inductance contribution to the
imaginary (j) component of the impedance can be of
order 1(cid:86), which is very large compared with R . We
x
therefore calculate the internal kinetic inductance, as
wellasthemagneticinductance,alongthedevicetosee
if the imaginary components are indeed significant
compared with the longitudinal resistance R . We also
x
calculate the internal capacitance across quantum Hall
effect devices to see if it provides a significant out-of-
phase contribution to the quantized Hall resistance
R (i).(Wewillfindthattherearenocontributionsfrom
H
the internal capacitance along the device or from the
kinetic and magnetic inductances across the device.)
Thecalculationshaveanalyticsolutionswhichutilize
resultsfromtheworkofCageandLavine[12,13],who
calculated potential, electric field, current, and current Fig.1. A2DEGconductingsheet,withtheoriginofthecoordinate
densitydistributionsacrossthe400(cid:109)mwidthofaquan- systemlocatedatthesourceS,halfwayacrossthedevicewidthw.
The device length is L , and D is the drain. Conducting electrons
x
tum Hall effect device for applied currents between
extendacrossthedevicefromy toy .Bistheexternalmagnetic
0(cid:109)A and 225 (cid:109)A. The potential distributions of Cage fluxdensity,VHthequantumHmainllvolmtaaxge,ISDtheappliedcurrent,
and Lavine [12] are in excellent agreement with the and Jt(y') the total current density at point x', y', 0 in the 2DEG.
experimental measurements of Fontein et al. [14], who
usedalaserbeamandtheelectro-opticPockelseffectas
a contactless probe of the 2DEG.
The spatial extent of the conducting electrons lies
2. Potential and Current Distributions withinthedevicewidthw,andbetweenthecoordinates
y and y . We are interested in effects within the
min max
AcurrentI ,externallyappliedbetweenthesourceS device interior, so we will neglect the fact that the
SD
and the drain D of a mesa-etched quantum Hall effect applied current ISD enters and exits opposite corners of
device,isconfinedtoflowwithacurrentsheetdensity the device. The electrons are therefore flowing only in
J within the 2DEG layer of the device. This applied the positive x direction at this instant of time. This
t
currentinducesapotentialdistributionacrossthedevice corresponds to a current of positive charges moving in
in the presence of a perpendicular external magnetic the negative x direction. The potential on the left hand
fluxdensityB.Thetotalpotentialdifferenceacrossthe sideofthedeviceispositiverelativetothepotentialon
device width w is the quantum Hall voltage therighthandsidefortheseparticularcurrentandmag-
V (i)=R (i)I . netic flux density directions.
H H SD
Cage and Lavine [12] have calculated the potential
distributions for applied currents between 0 (cid:109)A and 2.1 Results of Cage and Lavine
225 (cid:109)A. Their potential distributions are composed of
parabolicallyshapedconfiningpotentials(duetohomo- Forconvenience,werepeatthoseresultsofCageand
geneouscharge-depletionregions)locatedoneitherside Lavine [12] that are used as the starting points of our
of the device, and a logarithmically shaped charge- calculations. The equations for the confining potential
redistributionpotential(duetotheLorentzforceexerted Vc at any point y' across the 2DEG are
ontheconductingelectronsinthe2DEGcausingdevia-
tionsfromtheaveragesurfacechargedensity)extending w
V (y')=–a(y'–(cid:108))2 for (cid:108)(cid:35)y' (cid:35) (1a)
across the device interior. c 2
Figure 1 is a schematic drawing of the 2DEG, with
theoriginofthecoordinatesystemlocatedatthesource V (y') = 0 for –(cid:108)<y' < (cid:108) (1b)
c
S and halfway across the 2DEG. On a quantum Hall
plateau, the conducting electrons within the 2DEG
w
occupy all the allowed states of filled Landau levels. V (y') = –a(y' + (cid:108))2 for – (cid:35) –y' (cid:35) –(cid:108), (1c)
c 2
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
where Figure 2 shows schematic diagrams of V (y') and
c
w V(y') for I = 0 and I >0. For clarity, the confining
(cid:108)= – (cid:68), (2) r SD SD
2 potentials V (y') have been stretched out over wide
c
regionsofthedevice.Thethickcurvesarethosepartsof
a=3.0(cid:51)1012V/m2,andthecharge-depletionwidth (cid:68) the potentials where electrons of the 2DEG occupy
is 0.5 (cid:109)m [15]. The confining potential and magnetic Landau states and contribute to the current. Refer to
fluxdensityproduceanelectroncurrentthatcirculates Figs. 7 and 9 of Ref. [12] for actual plots of V(y') and
t
aroundtheperipheryofthedeviceinacounterclockwise J(y') versus y' at I = 25 (cid:109)A.
t SD
direction. This is equivalent to an edge current of posi- I is an alternating current for impedance measure-
SD
tive charges in a clockwise direction. ments.ThefilledstatesandthicklinesofFig.2shiftto
The charge-redistribution potential V(y') is the right with increasing current. When I < 0, V (y')
r SD r
has the opposite sign, V (y') does not change sign, and
c
(cid:85) (cid:85)
the occupied states shift to the left. We will choose an
IR y'+w/2
V(y') = – r H Gln , (3) instant in time for the calculations when I has the
r 2 y'–w/2 SD
root-mean-square (rms) value and is in the negative x
direction, as shown in Fig. 1.
where I is that part of the total current I due to the
r SD
charge-redistributionpotential.Thevalueofthegeome- 2.2 Parameters Used in the Calculations
try factor G is 0.147 [12].
The total potential V(y') is A typical ac quantized Hall resistance device is
t
400 (cid:109)m wide, has a rms current of about 25 (cid:109)A, and
V(y') = V (y') + V (y') . (4) operatesonthei=2plateau.Thefollowingsetofvalues
t c r
found by Cage and Lavine [12] can therefore be used
The electric field equations are inourcalculations: I =25(cid:109)A,R =12906.4035(cid:86),
SD H
V = R I = 0.3227 V, B = 12.3 T, w = 400 (cid:109)m,
H H SD
w a = 3.0(cid:51)1012 V/m2, (cid:68)= 0.5 (cid:109)m, (cid:108)= 199.500 (cid:109)m,
E (y') = 2a(y' – (cid:108)) for (cid:108)(cid:35) y' (cid:35) (5a)
c 2 I = 24.74 (cid:109)A, G = 0.147, y = 199.564 (cid:109)m, y =
r max min
–199.554 (cid:109)m, V (y ) = –0.0122 V, V (y )
c max c min
=–0.0088V,V(y )=–0.1599V,V(y )=0.1594V,
E (y') = 0 for – (cid:108)< y' < (cid:108) (5b) r max r min
c E (y ) = 3.821 (cid:51) 105 V/m, E (y ) = –3.255 (cid:51) 105
c max c min
V/m,E(y )=5.380(cid:51)104V/m, E(0)=2.345(cid:51)102
r max r
w V/m, and E(y ) = 5.266 (cid:51) 104 V/m. Their device
E (y') = 2a(y' + (cid:108)) for – (cid:35) –y' (cid:35) –(cid:108), (5c) r min
c 2 length L was 4.6 mm. The reduced mass m* of the
x
electroninGaAsis0.068timesthefreeelectronmass,
and or 6.194 (cid:51) 10–32 kg.
We will also use the values y = –y =199.559
max0 min0
Er(y') = Ir2RH G[(w/2)2w– (y')2] , (6) (cid:109)Cmag,eaannddELc(eyvmianxe0)[=12–]Ewch(yemninI0)==3.054(cid:109)A(cid:51).105V/mfrom
SD
and
3. Internal Capacitance
E(y') = E (y') + E(y') . (7)
t c r
The quantum Hall voltage V (i) = R (i)I arises
H H SD
because the conducting electrons are shifted slightly
The total current density J(y') is
t towards one side of the device such that the Lorentz
forceev(cid:51)BequalstheCoulombrepulsiveforce–eE
1 t
J(y') = J (y') + J(y') = E(y'), (8) everywherewithinthe2DEG[16],wherevistheveloc-
t c r R t
H ityofaconductingelectronlocatedatcoordinatesx',y'.
This shift in position with applied current I causes a
and the applied current I is SD
SD deviation, –e(cid:100)(cid:115)(y'), or charge-redistribution of the
y(cid:69)max electronsinthe2DEGfromtheaverageelectronsurface
ISD = Jt(y')dy' . (9) charge density enS, where –(cid:100)(cid:115)is the surface density
deviation at coordinates x', y' and n = i(eB/h) is the
ymin average surface number density, e.g.S, 5.94 (cid:51) 1011/cm2
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
Fig. 2. Schematic drawings of the confining potentials V(y') and the charge-redistribution potential V(y')
c r
acrossthedevicewidthforI =0andI >0,where(cid:108)and–(cid:108)arethelocationswheretheconfiningpotentials
SD SD
begintodeviatefromzero.Thethicklinesbetweeny andy forI =0,andbetweeny andy for
min0 max0 SD min max
I >0aretheregionswherethe2DEGelectronsareconducting.Forclarity,theconfiningpotentialsextendfar
SD
intothedeviceinterior,asdothevaluesofy ,y ,y andy .
min0 max0 min max
forthei=2plateauat12.3T.Thecharge-redistribution and L is the length of the device (neglecting corner
x
gives rise to separated charges and an internal capaci- effects). Appendix A of Ref. [12] showed that the
tance across the device width. surface charge-redistribution is
3.1 Calculations im* d2
(cid:100)(cid:115)(y') = V(y') , (11)
hB dy'2 t
Thereisanexcessofelectrons,withtotalcharge–Q,
on the right hand side of Fig. 1 and a depletion of where m* is the reduced mass of the electron in GaAs
electrons, with total charge +Q, on the left hand side, (0.068 times the free electron mass).
where NotethatonlythosepartsofthetotalpotentialV(y')
t
(cid:69)Lx w(cid:69)/2 whichchangewithappliedcurrentISDshouldbeusedin
Eqs.(10)and(11)tocalculatechargeseparationswithin
–Q = e(cid:100)(cid:115)(y')dy'dx' , (10a)
acapacitor.Thustheentirecharge-redistributionpoten-
0 0 tial V(y') contributes to Eq. (11), and the limits
r
of integration in Eqs. (10) are between y and 0 and
(cid:69)Lx (cid:69)0 between 0 and y . However, only thomisne parts of
max
Q = e(cid:100)(cid:115)(y')dy'dx' , (10b) the confining potential V (y') which differ from the
c
0 –w/2 ISD = 0(cid:109)A case contribute, i.e., the parts between
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
y and y and between y and y . The parts of devices.ThecapacitanceC isthusabout0.0029pFfor
min0 min max0 max H
V (y')betweeny andy donotcontributebecause the4.6mmlongGaAs/AlGaAsdeviceofRef.[12],and
c min max0
there is no difference from the I =0(cid:109)A case of about 0.0014 pF for the widely-used, 2.2 mm long
SD
Fig. 2. and 400 (cid:109)m wide, GaAs/AlGaAs BIPM/EUROMET
Thetotalcharges–Qand+Q,definedbyEqs.(10) devices [17].
and (11), are Two frequencies f often used in ac quantized Hall
(cid:83) (cid:68) resistance measurements are 1233 Hz and 1592 Hz.
im* These correspond to angular frequencies (cid:118)= 2(cid:112)f of
–Q = –e
hB 7747 rad/s and 10 000 rad/s, respectively, or about 104
rad/s. The impedance j/((cid:118)C ) due to the Hall capaci-
H
tanceC isabout7(cid:51)1010forBIPM/EUROMETdevices,
H
(cid:51) L {[E(y ) – E(y )] + [E(y ) –E(0)]} (12a) yieldingacorrectiontoR of2(cid:51)10–7R forthei=2
x c max c max0 r max r H H
plateau. This quadrature component correction to the
(cid:83) (cid:68) Hall impedance is small, but not insignificant.
im* WethuspredictaninternalcapacitanceC acrossthe
Q = e H
hB device width in parallel with the Hall resistance R .
H
There is no internal capacitance C along the device
x
length within the nearly dissipationless conduction
(cid:51) L {[E(y ) – E(y )] + [E(y ) – E(0)]}. (12b) region because the potential difference between V
x c min c min0 r min r x
probes is negligible and there is no charge separation
alongthesamplelength.Apotentialdifferenceslightly
UsingthevaluesfromSec.2.2,Eqs.(12)predictvalues greaterthanV doesoccurbetweenthesourceanddrain
H
of–Qand+Qof–9.36(cid:51)10–16Cand9.23(cid:51)10–16C, duetothecurrententeringandexitingatoppositedevice
respectively for the right and left hand sides of the corners, but this voltage arises from resistive heating
4.6mmlongdeviceatI =25(cid:109)AandB=12.3T.The ratherthanchargeseparation.Sotheimpedancedueto
SD
confining potential contributes 36 % to the charge Q. C along the device length is negligible compared with
x
Thereisa0.7%differenceinthevaluesofQbetween R = V /I .
x x SD
the two sides of the device, but that does not violate
charge conservation. The solutions of Cage and Lavine 3.2 Line Charges Approximation
[12]areself-consistentbecausechargesonbothsidesof
the device are transferred between donor sites in the The total charges +Q and –Q are concentrated near
AlGaAslayerandthe2DEGintheGaAslayerinorder the positions y and y because that is where the
min max
tomaintainzeronetchargewithinthedevicevolume.It surface charge-redistributions e(cid:100)(cid:115)(y') are largest. (See
is this charge transfer between layers that gives rise to Fig. 11 of Ref. [12] for an example of the charge-
the charge separation +Q and –Q in the 2DEG. redistributions at I = 215 (cid:109)A.) One can closely ap-
SD
The separated total charges +Q and –Q within the proximatethecharge-redistributionsastwolinecharges
2DEG generate a potential difference V across the +Q/L and –Q/L , with radii (cid:114)that are about one-half
H x x
device width, producing an internal capacitance C the probability distribution thickness of the 2DEG [2],
H
across the device, where C is defined as and are separated by the device width w. It can
H
be shown using Gauss’s law, (cid:171)(cid:101)E(cid:63)dS = Q, and the
definitionsofpotential,V=–(cid:101)E(cid:63)dl,andcapacitance,
Q = C V , (13) C=Q/V,thatthecapacitancebetweentwolinecharges
H H
ofradii(cid:114)is(cid:112)(cid:171)L /ln[(w–(cid:114))/(cid:114)],where(cid:171)isthepermit-
or x
tivity of GaAs (which is 13.1 times larger than the
(cid:83) (cid:68) (cid:83) (cid:68) permittivity of a vacuum), (cid:114)is about 2.5 nm, dS is an
e im* elemental area of the integration surface, and dl is an
C =
H VH hB incremental length along the integration path. The ca-
pacitance between two line charges is 48 times larger
than that predicted by Eq. (14) for the 2DEG.
(cid:51)Lx{[Ec(ymax) – Ec(ymax0)] – [Er(ymax) – Er(0)]} . (14) Two line charges are not a good approximation of a
quantum Hall device, however, because it neglects the
large screening effects in the nearby AlGaAs layer of
Using the electric field values listed in Sec. 2.2, the theheterostructure.Chargesofoppositesigntotheline
capacitance per length is 0.63 pF/m for 400 (cid:109)m wide charges occur in the two regions of the AlGaAs layer
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
near the device sides because the 2DEG arises from PleasenotethatL isthekineticinductanceandL isthe
k x
electrons tunneling from the AlGaAs layer. There are device length.
really four line charges to consider, with charge densi- Eqs.(5)and(6)canbeusedinEq.(17),remembering
ties (cid:54)Q/L and (cid:55)Q/L located on either side of the thatonlythosepartsoftheelectricfieldswhichchange
x x
device.Thisgreatlyreducestheelectricfieldswithinthe withappliedcurrentI shouldbeincluded.Thismeans
SD
2DEG,andtherebydecreasesthecapacitanceacrossthe allofthecharge-redistributionelectricfieldE(y'),inte-
r
device.ThecapacitancepredictedbyEq.(14)accounts grated between the limits y and y , but only those
min max
forthesescreeningeffectsbecauseituseselectricfields partsoftheconfiningfieldE (y')whichdifferfromthe
c
derived from experimental results. I =0(cid:109)Acase,i.e.,thosepartsbetweeny andy
SD min0 min
and between y and y . The parts between y and
max0 max min
–(cid:108)and between (cid:108)and y do not contribute to L
4. Kinetic Inductance max0 k
because they result from an internal dc current which
circulates around the device periphery and is indepen-
The conducting electrons have an inertial mass m*,
dent of I .
which gives rise to a kinetic inductance [18] when the SD
The integrals of Eq. (17) are analytic, and have the
current is reversed. Seppa, Satrapinski, Varpula, and
solution
Saari [11] predict that this yields a large impedance
alongthedevicelength.Wewillexaminetheirresultsin L = A(B+C+D+E) (18a)
k
Sec. 4.2 after deriving our equations for the kinetic
where
inductance.
(cid:83) (cid:68)
m*
4.1 Calculations A = L (18b)
eBR I2 x
H SD
The conducting electrons of Figs. 1 and 2 have a
velocity v (y') = E (y')/B = E(y')/B and a kinetic
x y z t z
energy 1m*v2(y'). Neglecting corner effects, the total IR G
2 x B = r H
kineticenergyKwithinthe2DEGofadeviceoflength w
L is
x
(cid:51)[y E(y )–y E(y )–V(y )+V(y )]
(cid:69)Lx w(cid:69)/2 max r max min r min r max r min
1
K = m*v2(y')ndy'dx' . (15)
2 x s (18c)
0 –w/2
C = 4a2
(cid:70) (cid:71)
Noting that J(y') = J (y') = n ev (y'), n = i(eB/h),
t x S x S 1
R = h/(e2i), and J(y') = E(y')/R , Eq. (15) can be (cid:51) (y3 +y3 )–(cid:108)(y2 –y2 )+(cid:108)2(y +y )
H t t H 3 max min max min max min
rewritten as
(cid:70)(cid:83) (cid:68) w(cid:69)/2 (cid:71) (18d)
1 m*R
K = H L J2(y')dy' I2
2 eBI2 x t SD
SD D = aIR Gw
–w/2 r H
(cid:72) (cid:74)
= 1 L I2 , (16) (cid:51) ln[(w/2)2–ym2in0] + ln[(w/2)2–ym2ax0]
2 k SD [(w/2)2–y2 ] [(w/2)2–y2 ]
min max
(18e)
where the kinetic inductance L is
k
E = 4a(cid:108)
(cid:83) (cid:68) w(cid:69)/2
Lk = meB*IR2H Lx [Jt(y')]2dy' (cid:51){[Vr(ymax) – Vr(ymax0)] –[Vr(ymin)–Vr(ymin0)]}.
SD
–w/2
(18f)
(cid:83) (cid:68) w(cid:69)/2
The kinetic inductance can be evaluated from Eqs.
m*
= eBR I2 Lx [Er(y') + Ec(y')]2dy' . (17) (18)usingthevalueslistedinSec.2.2,exceptforterm
H SD
–w/2
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
E where the values of V(y') need to be calculated to 1.6%ofn =5.94(cid:51)1011/cm2,andsatisfiesthefurther
r s
more significant figures using Eq. (3). The kinetic in- requirementinthemodelofCageandLavine[12]that
ductanceperlengthisabout15(cid:109)H/mfor400(cid:109)mwide the charge density varies slowly across the device
deviceswithi=2plateausat12.3T.Thetermsinvolv- width.
ing the confining potential (C, D, and E) contribute
33% of this value. The kinetic inductance L is thus
k 5. Magnetic Inductance
about 0.07(cid:109)H for the 4.6 mm long GaAs/AlGaAs
deviceofRef.[12],andabout0.04(cid:109)Hfor2.2mmlong
Determining the magnetic inductance L of a quan-
GaAs/AlGaAs BIPM/EUROMET devices [17], where m
tumHalleffectdeviceisnotquiteasstraightforwardas
thei=2plateauoccursatabout10T.Thesevaluesare
determining the Hall capacitance C or the kinetic in-
for I = 25 (cid:109)A. L is somewhat current dependent in H
SD k ductance L. The device can be treated as an isolated
this model because E(y') scales linearly with I but k
r SD objectwhencalculatingvaluesforC andL.Themag-
E (y') does not; so L decreases from 0.07 (cid:109)H to 0.05 H k
c k neticinductance,however,canonlybeevaluatedwhen
(cid:109)H between 25 (cid:109)A and 215 (cid:109)A. This difference over
thedeviceispartofacompletecurrent-carryingcircuit.
such a wide current range is small enough to ignore.
Therefore, L depends on the circuit geometry.
Theimpedancej(cid:118)L duetothekineticinductanceL m
k k We chose a geometry in which the device is repre-
is about 0.4 m(cid:86) for BIPM/EUROMET devices at
sented as a current sheet, with a return wire located
(cid:118)=104rad/sandI =25(cid:109)A,oronlyabout3partsin
SD below the middle of the sheet because this geometry
108 of R for the i = 2 plateau. This out-of-phase
H approximatesthesource-drainleadsofatypicalsample
impedancecomponentisalongthedevicelength,andis
probe.TheintegralequationsforL haveanalyticsolu-
comparable in magnitude to the longitudinal resistance m
tions for this geometry, and values of L can be com-
R = V /I . m
x x SD pared with the values for two parallel wires carrying
currents in opposite directions.
4.2 Uniform Current Density Approximation
We consider only the magnetic inductance outside
the sheet and the wire. The self-inductance per length
Seppa, Satrapinski, Varpula, and Saari [11] consid-
insidealong,nonpermeable,cylindrically-shapedwire
ered the case of a uniform current density J = I /w
t SD is (cid:109)/(8(cid:112)) [19], where (cid:109) = 4(cid:112)(cid:51) 10–7 H/m is the
acrossthedevicewidthw.Auniformcurrentdensityin 0 0
permeability of free space.
Eq. (17) yields L = (m*R L )/(eBw) = (m*L )/
k H x x
(ne2w), where n = i(eB/h). For BIPM/EUROMET
s s 5.1 Calculations
devices with i = 2 plateaus at 10 T we find that L =
k
0.003(cid:109)Hforauniformcurrentdensityapproximation,
Figure 3 shows the circuit geometry. The current-
orabout13timessmallerthanthemorerealisticpredic-
carrying 2DEG sheet and parallel return wire each
tion in Sec. 4.1.
extend to (cid:54)(cid:96) along the x axis. The current density is
Seppa et al. [11] predicted a much larger value of
J(y') = E(y')/R within the conducting sheet, where
L =40(cid:109)Hforthisexamplethanwehavebecausethey t t H
k E(y')isgivenbyEqs.(5)and(6).Thereturnwirehas
assumed free electrons with mass m, rather than elec- t
e aradius(cid:114),andisseparatedfromthesheetbyadistance
trons with reduced mass m* in the 2DEG, and a con-
d from the origin. The wire carries a current
ducting electron number density that was 1000 times
smaller than n. This last assumption is inconsistent
s w(cid:69)/2
withtherequirementthattheaveragesurfacedensityis
I = J(y')dy' (20)
n = i(eB/h) on a quantum Hall plateau. SD t
s
Using our Eqs. (11), (1), and (3), the deviation –w/2
–(cid:100)(cid:115)(y') in the density of electrons from the average in the opposite direction to that in the sheet.
surface density ns is The magnetic flux (cid:102)m and magnetic inductance Lm
are defined by
(cid:83) (cid:68)(cid:72) (cid:74)
im* IR Gwy' (cid:69) (cid:82)
–(cid:100)(cid:115)(y') = 2a+ r H . (19)
hB [(w/2)2–(y')2]2 (cid:102) = B (cid:63)dS = A(cid:63)dl = L I , (21)
m m m SD
Thelargestdeviationinthe2DEGoccursaty'=y , whereB isthemagneticfluxdensitygeneratedbyboth
max m
and has the value –(cid:100)(cid:115)(y') = 9.30 (cid:51) 109/cm2 for the the conducting sheet and the return wire, dS is an
i = 2 plateau at 12.3 T and I = 25 (cid:109)A. This is only elementalareaoftheenclosedcurrent-carryingcircuit,
SD
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
Fig.3. Asketchofthe2DEGconductingsheetandaparallelcurrentreturn
wireofradius(cid:114),locatedadistancedfromthemiddleofthesheet.B(z)and
s
B (z)aremagneticfluxdensitiesgeneratedbytheconductingsheetandwire.
w
dS=dxdzisanelementalareainthex–zplanelocatedbetweentheconducting
sheetandthewire.
A is the vector potential, dl is an incremental length (cid:109) 1
B (z) = 0 I . (23)
alongthepatharoundthecircuitjustoutsidetheconduc- w 2(cid:112) SD(d –z)
tors, and I is the applied current [19].
SD
WechosedStobelocatedinFig.3inthex–zplane
B(z)isfoundbyconsideringtheconductingsheetasa
between the conducting sheet and the wire at y = 0; so s
series of wires carrying currents J(y')dy'
dS = dxdz. Therefore, only the y-components of B t
m
perpendicular to dS are needed to evaluate the surface
integral in Eq. (21). These y-components of B are (cid:109) dy' z
m dB(z)cos(cid:117)= 0 J(y')
s 2(cid:112) t (cid:207)(y')2+z2 (cid:207)(y')2+z2
B (z) = B (z) + B(z)cos(cid:117)= B (z) + B(z) + B(z),
m w s w r c
(24)
(22)
or
where Bw(z) is due to the return wire.Bs(z)cos(cid:117)is due (cid:109)z w(cid:69)/2 J(y')
to the conducting sheet, and is composed of charge- Bs(z)cos(cid:117)=Br(z)+Bc(z) = 2(cid:112)0 [(y')t2+z2] dy'.
redistribution and confining parts Br(z) and Bc(z), –w/2
respectively.B (z)iseasilyobtainedfromAmpere’slaw
w (25)
(cid:101)B (cid:63)dl = (cid:109)I
w 0 SD
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Volume101,Number6,November–December1996
Journal of Research of the National Institute of Standards and Technology
(cid:72) (cid:83) (cid:68) (cid:83) (cid:68)(cid:74)
This gives (cid:109)a y y
+ 0 (cid:108) arctan min –arctan min0
(cid:112)R z z
H
(cid:109)
B(z)= 0 IGwz
r 4(cid:112) r (cid:72) (cid:83) (cid:68) (cid:83) (cid:68)(cid:74)
(cid:109)a y y
– 0 (cid:108) arctan max –arctan max0 (29)
y(cid:69)max (cid:112)RH z z
1
(cid:51) (cid:70)(cid:83) (cid:68) (cid:83) (cid:68)(cid:71)(cid:70)(cid:83) (cid:68) (cid:71) dy', (26)
w 2 2 2
ymin 2 – y' y' +z2 fortheconfiningterm.TheintegralsinEq.(28)extend
only between y and y and between y and y
min0 min max0 max
because we are interested in the parts of B(z) that
and c
changewhenI changes.Thepartsbetweeny and–(cid:108)
SD min
and between (cid:108)and y provide a constant magnetic
(cid:109) w max0
B(z) = 0 IG (cid:70)(cid:83) (cid:68) (cid:71) flux density that does not contribute to the magnetic
r 4(cid:112) r w 2+z2 inductance.
2
Ifthequantummechanicalprobabilitydistributionof
the 2DEG extends over a distance 2(cid:114), if the device
(cid:72) (cid:83) (cid:68) (cid:83) (cid:68)(cid:74) length is Lx, and if we neglect the corner effects, then
y y Eqs. (21) and (22) yield
(cid:51) arctan max –arctan min
z z
L =L +L +L
m W r c
+ (cid:109)0IG(cid:70)(cid:83) (cid:68)z (cid:71) (cid:69)Lx d(cid:69)–(cid:114)
4(cid:112)r w2 2+z2 = I1SD [BW(z)+Br(z)+Bc(z)]dzdx . (30)
0 (cid:114)
(cid:53) (cid:51) (cid:52) (cid:51) (cid:52)(cid:54) UsingEqs.(23),(27),and(29)inEq.(30),wefindthat
w w
+y +y
2 max 2 min
(cid:51) ln –ln (27)
w w (cid:70) (cid:71)
2–ymax 2–ymin L = (cid:109)0 L ln d –(cid:114) , (31a)
W 2(cid:112) x (cid:114)
and
for the charge-redistribution term, and
(cid:72)(cid:70) (cid:83) (cid:68)(cid:71) (cid:70) (cid:83) (cid:68)(cid:71)(cid:74)
(cid:109) I w 2 w 2
(cid:109)a y(cid:69)min (y'+(cid:108)) Lr(cid:248)2(cid:112)0 ISrDGLx arctan 2(cid:114) – arctan 2(d–(cid:114))
B(z)= 0 z dy'
c (cid:112)R [(y')2+z2]
H
ymin0
(cid:53) (cid:51) (cid:52) (cid:51) (cid:52)(cid:54)
w w
+y +y
+ (cid:109)0a z y(cid:69)max (y'–(cid:108)) dy' , (28) +8(cid:109)(cid:112)0 IISrD GLx ln w2 –ymax –ln w2 –ymin
(cid:112)R [(y')2+z2] 2 max 2 min
H
ymax0
(cid:83) (cid:68)
and (cid:53) (cid:51) (cid:52)(cid:54)
w 2+(d –(cid:114))2
(cid:72) (cid:70) (cid:71) (cid:70) (cid:71)(cid:74) (cid:51) ln 2(cid:83) (cid:68) , (31b)
Bc(z)=(cid:112)(cid:109)R0aH z2 ln yym2m2aaxx0++zz22 +ln yym2m2inin0++zz22 w2 2+(cid:114)2
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