Electronic magnetization of a quantum point contact measured by nuclear magnetic resonance
Minoru Kawamura, Keiji Ono, Peter Stano, Kimitoshi Kono, Tomosuke Aono
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Electronic magnetization of a quantum point contactmeasured by nuclear magnetic resonance
Minoru Kawamura, ∗ Keiji Ono, Peter Stano, Kimitoshi Kono, and Tomosuke Aono RIKEN Center for Emergent Matter Science, Wako 351-0198, Japan Department of Electrical and Electronic Engineering, Ibaraki University, Hitachi 316-8511, Japan (Dated: November 8, 2018)We report an electronic magnetization measurement of a quantum point contact (QPC) basedon nuclear magnetic resonance (NMR) spectroscopy. We find that NMR signals can be detected bymeasuring the QPC conductance under in-plane magnetic fields. This makes it possible to measure,from Knight shifts of the NMR spectra, the electronic magnetization of a QPC containing only afew electron spins. The magnetization changes smoothly with the QPC potential barrier height andpeaks at the conductance plateau of 0.5 × e /h . The observed features are well captured by amodel calculation assuming a smooth potential barrier, supporting a no bound state origin of the0.7 structure. Quantum point contact (QPC) is a short one-dimensional (1D) channel connecting two electron reser-voirs. Its conductance is quantized to integer multiplesof 2 e /h , where e is electron charge and h is Planck’sconstant[1, 2]. The conductance quantization is well un-derstood within a model of non-interacting electrons[3].However, experiments have shown an additional conduc-tance feature, a shoulder-like structure at around 0.7 × e /h termed as 0.7 structure[4, 5]. Despite the simplic-ity of a QPC, a comprehensive understanding of the 0.7structure is still lacking [6–19].Theories proposed to explain the 0.7 structure can bediscriminated according to their predictions on the elec-tron spin arrangement, which include spontaneous spinpolarization[6, 7], antiferromagnetic Wigner crystal[8],Kondo screening[9–11], and local spin fluctuations ac-companied by van Hove singularity[12, 13]. Especiallyin the Kondo scenario, the existence of a localized mag-netic moment in the QPC is an inevitable ingredient. Onone hand, early experiments observing Fano resonancessuggested such presence of a local single spin trappedin a bound state regardless of magnetic fields[14]. Onthe other hand, an experiment measuring compressibil-ity contradicts such bound state formation[18]. Thus, thedegree of spin polarization of a QPC is one of the centralissues to understand the origin of the 0.7 structure.However, most experiments[4, 5, 14–17] to date havefocused on transmission properties, without the QPCspin polarization being addressed directly. Despite therecent progress in magnetic sensors[20], the magnetiza-tion measurement of a QPC containing only a few elec-trons is still very challenging. Recently, small magne-tizations of two-dimensional electron systems (2DESs)embedded in GaAs have been measured[21–23] by com-bining techniques of current-induced nuclear spin polar-ization [24–28] and resistance (conductance) detection ofnuclear magnetic resonance (NMR) signals of Ga and Asnuclei[27–29]. Because of the hyperfine interaction be-tween electronic and nuclear spins, an electronic magne-tization produces an effective magnetic field for nuclei, resulting in the shift of the NMR frequency, the Knightshift. From the Knight shift, the electronic magnetiza-tion can be determined[30].A recent transport experiment by Ren et al. [31] sug-gests such influence of nuclear spins on the QPC con-ductance. They observed hysteresis in the source-drainvoltage dependence of the differential conductance un-der magnetic fields, and attributed its origin to the dy-namical nuclear spin polarization (DNSP) induced in theQPC. However, NMR or other direct evidence show-ing involvement of nuclear spins has not been presentedso far. NMR signal detection in the QPC conductancewould constitute a novel experimental technique to probespin properties of QPCs or nanowires[32, 33].In this Letter, we report an electronic magnetizationmeasurement of a QPC defined in a GaAs/AlGaAs het-erostructure based on NMR spectroscopy. We find thatthe QPC differential conductance changes when the fre-quency of an applied oscillating magnetic field matchesthe NMR frequencies of Ga, Ga, and As. The re-sistive detection of the NMR signals allows us to measurethe electronic magnetization of the QPC from the Knightshifts of the NMR spectra. The Knight shift measure-ments are conducted at the QPC conductance between 0and 2 e /h by tuning gate and source-drain voltages. Themagnetization changes smoothly with the QPC potentialbarrier height and peaks at the conductance plateau of0.5 × e /h . The observed features are well captured bya model calculation assuming a smooth potential barrierwithout a bound state formed. Apart from the demon-stration of a new technique to measure a magnetizationof only a few electrons, the absence of a bound state inthe QPC is our main conclusion, directly relevant for theunderstanding of the 0.7 structure.QPCs studied in this work are fabricated from awafer of GaAs/Al . Ga . As single heterostructure witha 2DES at the interface. The mobility and sheet car-rier density of the 2DES at 4.2 K are 110 m /Vs and2.2 × m − , respectively. A QPC is defined electro-statically applying negative voltages ( V g1 , V g2 ) to a pair xy z V g1 (V)
210 -0.70 -0.60 G ( / h ) (a) G ( / h ) V g1 = -0.685 V (b) scan directions G ( / h ) t (s)0.630.62 2000 FIG. 1. (color online) (a) Linear conductance G as a func-tion of V g1 ( V g2 = − B = 0, 1.5, 3, 4.5, and 6 T,applied along the x direction. Inset shows a scanning electronmicroscope image of the device. (b) Differential conductance G as a function of source-drain bias voltage V sd at V g1 = − V sd in the positive (negative) direction at arate of 5.6 µ V/s. Inset shows the time dependence of G at B = 4.5 T and V g1 = − .
685 V after an instantaneous changeof V sd from 0 to − µ V. A slightly different value comparedto the one in (b) for the same parameters, B = 4.5 T and V sd = − µ V, arises due to a remaining DNSP created at large | V sd | during the V sd scan in (b). of Au/Ti gate electrodes patterned on the surface of thewafer. All data presented here are measured on a QPCwith lithographic dimensions of 300 nm length and 250nm width [inset of Fig. 1(a)], in a dilution refrigerator atthe mixing chamber temperature of 20 mK. The externalmagnetic field B is applied parallel to the 2DES planealong the current flowing direction [ x direction in the in-set of Fig. 1(a)] to avoid orbital effects and quantum Halledge channels. The differential conductance G = dI/dV sd (where I is the current and V sd is the source-drain biasvoltage) is measured using a standard lock-in techniquewith a typical excitation voltage of 20 µ V at 118 Hz.A single-turn coil is wound around the device to applyradio-frequency oscillating magnetic field B rf .The QPC shows a typical conductance quantizationbehavior. Figure 1(a) shows linear conductance G = G ( V sd = 0) as a function of gate voltage V g1 . In addi- f (MHz) As32.832.7 . x G / h FIG. 2. Differential conductance G as a function of frequency f of B rf at B = 4.5 T, V g1 = − .
685 V, and V sd = − µ V. B rf is applied perpendicular to B [ y direction in the inset ofFig. 1(a)]. f is scanned at a rate of 0.128 kHz/s. Data of 10subsequent measurements are averaged to improve the signalto noise ratio. tion to quantized conductance plateaus, the 0.7 struc-ture is observed at zero magnetic field, developing intoa plateau of 0.5 × e /h at high magnetic fields. Azero-bias conductance peak accompanying the 0.7 struc-ture is observed clearly in the G - V sd curve at B = 0 T[Fig. 1(b)]. With increasing B , the zero-bias conductancepeak is suppressed and turns into a dip above B = 3 T.Hysteresis is observed in the G - V sd curves when V sd isscanned slowly (5.6 µ V/s) in the positive and negativedirections [Fig. 1(b)]. The hysteresis is seen only at fi-nite magnetic fields. Typical time scale to develop thehysteresis is measured at B = 4.5 T by recording G afteran instantaneous change of V sd from 0 to − µ V [Insetof Fig. 1(b)]. The value of G continues to change over aperiod of 200 s. This time scale is consistent with nuclearspin relaxation or polarization times reported in GaAs-based devices[27, 28, 31, 34, 35]. Similarly as concludedin the earlier work[31], we interpret the slow change in G as the first indication for the DNSP in the QPC.To confirm the nuclear spin origin of the observed slowchange in G , we perform the NMR spectroscopy exper-iment. Scanning the frequency f of B rf , we observedecreases in G when f matches the NMR frequency of As (gyromagnetic ratio γ = 45.82 rad MHz/T) [Fig. 2].The obtained G - f curve represents the NMR spectrum of As, split into three dips due to the electric quadrupoleinteraction[36]. We observe signals at resonances of Gaand Ga, as well as analogous behavior in four otherQPC devices (not shown). These observations clearlyshow that the DNSP is induced in the QPC and that itschanges are measured by monitoring the QPC conduc-tance.Having established the method to probe the NMRspectra in transport, we now use it to determine the elec-tronic magnetization of the QPC. To this end, we per-form the following pump-probe experiment [Fig. 3(a)]. (b) V g1NMR = -0.667 V-0.673 V-0.679 V-0.685 V-0.691 V-0.697 V-0.706 V . x / h G -15 -10 -5 0 5 f - f (kHz) V g1 initialize NMR readout
20 s (a) timeoff-res.fV sd G NMR V g1 NMR -0.70 -0.69 -0.68 -0.67V g1 (V)1050 G ( / h ) K ( k H z ) (c) G ( / h ) m ( m - ) (d) G -10 0 f - f (kHz)
679 V assuming 3D (dashed) and 2D (solid) hard-wall confinement potentials. (d)Calculated magnetization density at the QPC center m plotted as a function of potential barrier height V . The red and bluecurves depict spin densities h n , ↑ i and h n , ↓ i , respectively. Calculated conductance G is plotted by a dotted curve referring tothe right axis. First, nuclear spins are initialized by inducing DNSP un-der a relatively large bias voltage V sd = − µ V at V g1 = − .
685 V. Then, V sd is set to 0 µ V and the QPC istuned to a state of interest by setting the gate voltageto V NMRg1 for a period of time (22 s), during which thefrequency of B rf is set to f for 20 s[37]. Finally, changesin the DNSP are read out by recording G with a smallac voltage excitation (20 µ V, 118 Hz) at V g1 = − .
685 Vand V sd = 0 µ V. The observed values of G at the begin-ning of the readout step reflect how much are the nuclearspins depolarized by B rf . Repeating this procedure withdifferent f , we obtain an NMR spectrum for a gate volt-age V NMRg1 as shown in Fig. 3(b).The bottom data of Fig. 3(b) is obtained by deplet-ing electrons from the QPC during the B rf application.Therefore, this spectrum is not affected by electrons, andhas a rather sharp dip at f = 32.755 MHz, the frequencycorresponding to the transition between the nuclear spinstates | I z = ± / i . As V NMRg1 is increased, the NMRinduced dips are shifted toward negative frequencies andbroadened. These shifts are the Knight shifts due to the electronic magnetizations in the QPC.We now evaluate the magnitude of the Knight shiftsby taking the spatial electron distribution into account.Extending earlier works[21, 22, 42], we adopt a modelof electrons confined in the y and z directions witha transverse wave function ψ ( y, z ). The Knight shiftfor an As nucleus at position ( y, z ) can be written as δf K ( y, z ) = α As m z | ψ ( y, z ) | , where α As = − . × − kHz m is the hyperfine coupling coefficient[36], and m z ≡ n ↑ − n ↓ is 1D electronic magnetization densitydefined as the difference in 1D spin densities. We make astandard assumption[42] that nuclear spins are depolar-ized by the rf-magnetic field according to the detuningfrom the resonance δf = f − ( f + δf K ) with a Gaussianprofile exp( − δf / γ ), where f and γ are the NMR fre-quency and the spectrum width without the influence ofthe Knight shift, respectively. Such depolarizations in-duce the change in the electron Zeeman energy which isgiven by an integral of local nuclear spin depolarizationmultiplied by electron distribution. Since these changesare small, we may expand the QPC conductance, whichis a function of the electron Zeeman energy, and get forits change δG ( f ) = A Z dydz exp( − δf / γ ) | ψ ( y, z ) | , (1)with A an unknown proportionality coefficient. To eval-uate Eq. (1), we approximate the transverse wave func-tion by the one of a two-dimensional (2D) hard-wallconfinement, ψ ( y, z ) ∝ cos( πy/w y ) cos( πz/w z ) with con-finement widths w y = (65 ±
5) nm and w z = (18 ±
3) nm[43]. The Knight shift becomes δf K ( y, z ) = − K cos ( πy/w y ) cos ( πz/w z ) with a parameter K pro-portional to m z via K = − α As m z | ψ (0 , | . The exper-imental data in Fig. 3(b) are fitted to Eq. (1) using K and A as fitting parameters with f = 32.755 MHz and γ = 1.36 kHz determined from the data measured at thedepletion configuration ( V NMRg1 = − .
706 V). As seen inthe figure, the agreement of the data and the model fittedfor each curve is excellent.We now consider an alternative fit, assumingthat the QPC transport occurs through a three-dimensionally (3D) confined electronic state ψ ( x, y, z ) ∝ cos( πx/w x ) cos( πy/w y ) cos( πz/w z ). A representative re-sult, using an analog of Eq. (1), is given in the insetof Fig. 3(c) and shows a much worse compatibility withthe data. We find that such discrepancy is not sensi-tive to the confinement details. As especially well vis-ible for large Knight shifts, the data show skewed lineshape, with steep (gentle) slopes on the low (high) fre-quency side. This is systematically reproduced by 2Dconfinement models, unlike 3D ones (see the Supplemen-tal Material[36]).In Fig. 3(c), K is plotted as a function of V g1 . A fi-nite K emerges near the conductance onset and increasessteeply as the conductance is increased to 0.5 × e /h .It keeps increasing gradually with increasing V g1 even inthe conductance plateau region of 0.5 × e /h . As V g1 isincreased further, K turns to decrease accompanied bya rise of conductance from 0.5 × e /h . As a result, apeak in K is formed at the high gate-voltage end of theconductance plateau. Using the relation between K and m z , the observed maximum value K = (11.7 ± m z = (16.5 ± × m − .We now show that the observed features are well re-produced by a model calculation. We model a QPC bya 1D tight-binding Hamiltonian, H = X j,σ ǫ j,σ c † j,σ c j,σ − t X j,σ c † j,σ c j +1 ,σ + X j U j n j, ↑ n j, ↓ . (2)Here c † j,σ creates an electron with spin σ ( σ = ↑ , ↓ ) at the j -th site of the tight-binding chain which has a hoppingamplitude t . We assume a short-range Coulomb interac-tion represented by the on-site Coulomb energy U j . Thepotential energy and the Zeeman energy are included inthe on-site energy, ǫ j, ↑ / ↓ = ǫ j ± gµ B B/
2, with the Bohr magneton µ B and the electron g -factor g . We assumea smooth parabolic potential barrier at the QPC centerwith a height V and a curvature Ω x . The interactionterm is treated by a mean-field approximation neglect-ing spin fluctuations. Then the mean-field spin density h n j,σ i is determined by a self-consistent Green’s functionmethod[45], where the on-site energy ǫ j,σ is shifted by U j h n j, ¯ σ i with ¯ σ , the opposite spin to σ . We calculatethe magnetization density profile m j = h n j, ↑ − n j, ↓ i andthe QPC conductance G . The values of U j and Ω x aredetermined from the conductance measurement data[36].The thick solid curve in Fig. 3(d) depicts the calculatedmagnetization density at the QPC center m = m j =0 as a function of V , resembling the observed V g1 depen-dence of K in Fig. 3(c). According to the calculation,the increase in m accompanied by the emergence of theconductance corresponds to the increase in the numberof up-spin electrons in the QPC. The value of m startsto decrease when down-spin electrons begin to populatethe QPC, lifting G from 0.5 × e /h . The gradual in-crease in m in the 0.5 × e /h plateau region is alsoreproduced. The maximum value of the calculated mag-netization density m = 9.3 × m − roughly agreeswith the value determined from the Knight shift. Spinpolarization P = h n , ↑ − n , ↓ i / h n , ↑ + n , ↓ i reaches 70.0% where m is maximal. Distribution of m j has a bell-shaped profile and extends over a length of about 100 nmaround the QPC center[36].The gradual change in m reflects the fact that thelocal density of states is continuous at the QPC centerunlike in a quantum dot. We therefore attribute the ob-served gradual change in K to be consistent with a QPCmodel without any bound states. This contradicts ear-lier observations claiming that a single electron spin istrapped in a bound state formed in the QPC[14]. Weestimate[36] that the observed magnitude of the magneti-zation density corresponds to the total magnetic moment(1 . ± .
45) in the QPC, exceeding the single-electron-spin magnetic moment which a bound state can support.Our measurement results of the NMR line shapes, thegradual change of K , and the magnetic moment valuesare consistent with a QPC model without bound states,such as Refs. [12, 13], which predicts a smooth increaseof the magnetization without saturation upon increasingthe magnetic field.In summary, we find that the NMR signals can bedetected by measuring the QPC conductance under in-plane magnetic fields. The resistive detection makes itpossible to measure the electronic magnetization of theQPC from the Knight shifts of the NMR spectra. Theelectronic magnetization changes smoothly with the gatevoltage and peaks at the conductance plateau of 0.5 × e /h . The gate voltage dependence of the Knight shift iswell explained by a model calculation assuming a smoothpotential barrier, supporting a no bound state origin ofthe 0.7 structure.This work was supported partially by Grant-in-Aid forScientific Research (No. 24684021) from JSPS, Japan.We thank T. Machida for valuable discussions. ∗ [email protected][1] D. A. Wharam et al., J. Phys. C , L209 (1988).[2] B. J. van Wees et al., Phys. Rev. Lett. , 848 (1988).[3] M. Buttiker, Phys. Rev. B , 7906 (1990).[4] K. J. Thomas et al., Phys. Rev. Lett. , 135 (1996).[5] K. J. Thomas et al., Phys. Rev. B , 4846 (1998).[6] C. -K. Wang and K. -F. Berggren, Phys. Rev. B , 4552(1998).[7] D. J. Reilly et al., Phys. Rev. B , 121311 (2001).[8] K. A. Matveev, Phys. Rev. Lett. , 106801 (2004).[9] S. M. Cronenwett et al., Phys. Rev. Lett. , 226805(2002).[10] Y. Meir, K. Hirose and N. S. Wingreen, Phys. Rev. Lett. , 196802 (2002).[11] T. Rejec and Y. Meir, Nature , 900 (2006).[12] C. Sloggett, A. I. Milstein, and O. P. Sushkov, Eur. Phys.J. B , 427 (2008).[13] F. Bauer et al., Nature , 73 (2013).[14] Y. Yoon et al.,
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3) nm is typical for the in-terface of GaAs/AlGaAs[44]. The parameter w y = (65 ±
5) nm, which is related to the inter-subband energy,is determined from the gate voltage width of the firstconductance plateau.[44] T. Ando, J. Phys. Soc. Jpn. , 3900 (1982).[45] S. Datta, Quantum Transport: Atom to Transistor (Cam-bridge University Press, Cambridge, England , 2005). ❙(cid:0)✁✁✂✄☎✄✆✝✞✟✠ ▼✞✝✄✟✡✞✂ ♦✆❊☛☞✌✍✎✏✑✒✌ ♠✓✔✑☞✍✒✕✓✍✒✏✑ ✏✖ ✓ q✗✓✑✍✗♠ ♣✏✒✑✍ ✌✏✑✍✓✌✍♠☞✓✘✗✎☞✙ ❜② ✑✗✌☛☞✓✎ ♠✓✔✑☞✍✒✌ ✎☞✘✏✑✓✑✌☞✚✛✜✢✣✤✥✢ ✦✧★✩✤✧✪✫✛✜ s✪✛✥✣✣✥✬✭ ✫✮ ✣t✜ ◆✯✰ s✪✜✢✣✤★❚✱✲ ✳✴✵ ✷✸✲❡✹✺✻✼ ✽✾ ✿❀❆✷ ✷✱✽✇❁ ✐❁ ❋✐❂❃ ❄ ✽✾ ✹✱✲ ✼❅✐❁ ✹✲❇✹ ✐✷ ✷✸❈✐✹ ✐❁✹✽ ✹✱✺✲✲ ❡✽❁❝✻❡✹❅❁❡✲ ❝✐✸✷❃ ❉✐✼✐❈❅✺❈● ✹✱✲✷✸✲❡✹✺❅ ✾✽✺ ❍■❏❅ ❅❁❝ ✿❑❏❅ ❡✽❁✷✹✐✹✻✹✲ ✽✾ ✹✱✺✲✲ ❡✽❁❝✻❡✹❅❁❡✲ ❝✐✸✷ ❅✷ ✺✲✷✸✲❡✹✐▲✲❈● ✷✱✽✇❁ ✐❁ ❋✐❂✷❃ ❉✶❖❅P ❅❁❝ ❖◗P❃ ❚✱✲✷✲✷✸❈✐✹✹✐❁❂✷ ❅✺✲ ❝✻✲ ✹✽ ✹✱✲ ✲❈✲❡✹✺✐❡ ❘✻❅❝✺✻✸✽❈✲ ✐❁✹✲✺❅❡✹✐✽❁❯ ✹✱✲ ✐❁✹✲✺❅❡✹✐✽❁ ◗✲✹✇✲✲❁ ✹✱✲ ✲❈✲❡✹✺✐❡ ❘✻❅❝✺✻✸✽❈✲ ✼✽✼✲❁✹✽✾ ✹✱✲ ❁✻❡❈✲✐ ❅❁❝ ✹✱✲ ✲❈✲❡✹✺✐❡ ❱✲❈❝ ❂✺❅❝✐✲❁✹ ❅✹ ✹✱✲ ✸✽✷✐✹✐✽❁ ✽✾ ✹✱✲ ❁✻❡❈✲✐❃ ❚✱✲ ❅✼✸❈✐✹✻❝✲ ✽✾ ✹✱✲ ✲❈✲❡✹✺✐❡ ❘✻❅❝✺✻✸✽❈✲✐❁✹✲✺❅❡✹✐✽❁❯ ✱✲❁❡✲ ✹✱✲ ❅✼✸❈✐✹✻❝✲ ✽✾ ✷✸❈✐✹✹✐❁❂✷❯ ✐✷ ✲❇✸✲❡✹✲❝ ✹✽ ◗✲ ✸✺✽✸✽✺✹✐✽❁❅❈ ✹✽ ✹✱✲ ✲❈✲❡✹✺✐❡ ❘✻❅❝✺✻✸✽❈✲ ✼✽✼✲❁✹ ✽✾❁✻❡❈✲✐❃ ❚✱✲ ✽◗✷✲✺▲✲❝ ❅✼✸❈✐✹✻❝✲✷ ✽✾ ✹✱✲ ✳✴✵ ✷✸✲❡✹✺✻✼ ✷✸❈✐✹✹✐❁❂✷ ❅✺✲ ❲❢ ❳ ❨❩ ❦❬❭❯ ❄❪ ❦❬❭❯ ❅❁❝ ✶❫ ❦❬❭ ✾✽✺ ✿❀❆✷❯❍■❏❅❯ ❅❁❝ ✿❑❏❅❯ ✺✲✷✸✲❡✹✐▲✲❈●❃ ❚✱✲ ✺❅✹✐✽ ✽✾ ✹✱✲ ❲❢ ✺✽✻❂✱❈● ❅❂✺✲✲✷ ✇✐✹✱ ✹✱✲ ✺❅✹✐✽ ✽✾ ✹✱✲ ✲❈✲❡✹✺✐❡ ❘✻❅❝✺✻✸✽❈✲ ✼✽✼✲❁✹❴ ✐❁ ❈✐✹✲✺❅✹✻✺✲❧✶❵❛ ❴❞❣❤❥ ❳ ♥❃❄r ✉✶♥✈①③ ✼①❯ ❴④⑤⑥⑦ ❳ ♥❃✶r ✉✶♥✈①③ ✼①❯ ❅❁❝ ❴❞⑧⑥⑦ ❳ ♥❃✶❄ ✉✶♥✈①③ ✼①❃ ⑨❁ ❅ ❡✻◗✐❡✷●✼✼✲✹✺✐❡ ◗✻❈❦ ❏❅❆✷ ❡✺●✷✹❅❈❯ ✹✱✲ ✲❈✲❡✹✺✐❡ ❘✻❅❝✺✻✸✽❈✲ ✐❁✹✲✺❅❡✹✐✽❁ ✐✷ ✻✷✻❅❈❈● ❭✲✺✽❃ ⑩✲ ✐❁✾✲✺ ✹✱❅✹ ✹✱✲ ✷✹✺❅✐❁ ✐❁❝✻❡✲❝◗● ✹✱✲ ✸❅✐✺ ✽✾ ❚✐❶❆✻ ❂❅✹✲ ✲❈✲❡✹✺✽❝✲✷ ◗✺✲❅❦✷ ✹✱✲ ❡✻◗✐❡ ✷●✼✼✲✹✺● ❅❁❝ ✐❁❝✻❡✲✷ ✹✱✲ ✲❈✲❡✹✺✐❡ ❘✻❅❝✺✻✸✽❈✲ ✷✸❈✐✹✹✐❁❂✷ ✐❁ ✹✱✲✽◗✷✲✺▲✲❝ ✳✴✵ ✷✸✲❡✹✺❅❃ ❷✣★❸✥✛✥✣❹ ✫✮ ✜❺✣✜✤✬★✛ ❻★✭✬✜✣✥✢ ❼✜✛✩❚✱✲ ✳✴✵ ✷✸✲❡✹✺❅ ✷✱✽✇❁ ✐❁ ❋✐❂❃ ❪❖◗P ✽✾ ✹✱✲ ✼❅✐❁ ✹✲❇✹ ❅✺✲ ✹❅❦✲❁ ❅✹ ❽ ❳ ❨❃❫ ❚❃ ❚✱✲ ✼❅❂❁✲✹✐❡ ❱✲❈❝ ✐✷ ✸✺✽❝✻❡✲❝◗● ❅ ✷✻✸✲✺❡✽❁❝✻❡✹✐❁❂ ✷✽❈✲❁✽✐❝ ✻✷✐❁❂ ❅ ✸✲✺✷✐✷✹✲❁✹ ✼✽❝✲ ❅❁❝ ✹✱✲ ❱✲❈❝ ✐✷ ❁✽✹ ❡✱❅❁❂✲❝ ✹✱✺✽✻❂✱✽✻✹ ✹✱✲ ✷✲✺✐✲✷ ✽✾ ✹✱✲✼✲❅✷✻✺✲✼✲❁✹✷❃ ⑨✹ ✹✽✽❦ ❅◗✽✻✹ ❾♥ ✱✽✻✺✷ ✹✽ ❡✽✼✸❈✲✹✲ ✹✱✲ ✷✲✹ ✽✾ ✹✱✲ ✳✴✵ ✷✸✲❡✹✺❅ ✻❁❝✲✺ ▲❅✺✐✽✻✷ ❂❅✹✲ ▲✽❈✹❅❂✲✷ ✷✱✽✇❁ ✐❁❋✐❂❃ ❪❖◗P❃ ⑩✲ ✹✽✽❦ ✹✱✲ ❝❅✹❅ ✾✽✺ ❿ ➀➁➂➃❑ ❳ ➄♥➅➆❾❫ ➇ ✹✇✐❡✲ ❅✾✹✲✺ ❅❁ ✐❁✹✲✺▲❅❈ ✽✾ ❅◗✽✻✹ ❫♥ ✱✽✻✺✷ ❅❁❝ ❡✽❁❱✺✼✲❝ ✹✱❅✹ ✹✱✲✾✺✲❘✻✲❁❡● ✷✱✐✾✹ ✇❅✷ ❈✲✷✷ ✹✱❅❁ ✶❃❫ ❦❬❭❃ ❚✱✐✷ ✐❁❝✐❡❅✹✲✷ ✹✱❅✹ ✹✱✲ ✼❅❂❁✲✹✐❡ ❱✲❈❝ ✇❅✷ ✷✹❅◗❈✲ ✲❁✽✻❂✱ ✹✽ ✺✲✷✽❈▲✲ ✹✱✲ ➈❁✐❂✱✹✷✱✐✾✹ ✽✾ ✹✱✲ ➉➊➋❃ ➌❹✪✜✤❼✬✜ ✢✫✧✪✛✥✬✭ ✢✫✬s✣★✬✣❬●✸✲✺❱❁✲ ✐❁✹✲✺❅❡✹✐✽❁ ◗✲✹✇✲✲❁ ❅ ❁✻❡❈✲❅✺ ✷✸✐❁ ➍➎➏ ❅❁❝ ✲❈✲❡✹✺✽❁ ✷✸✐❁✷ ➍➐➑ ✐✷ ❝✲✷❡✺✐◗✲❝ ◗● ❅ ❬❅✼✐❈✹✽❁✐❅❁➒❖➍➎➏P ❳ ➓➑ ❨➔→❪➎ ➔➣➔↔ ↕➙➑❖ ➍➛➏P↕① ➍➐➑ ➜ ➍➎➏➅ ❖✶P❬✲✺✲ ➔→ ✐✷ ❅ ✼❅❂❁✲✹✐❡ ✸✲✺✼✲❅◗✐❈✐✹●❯ ➔➣ ✐✷ ✹✱✲ ➝✽✱✺ ✼❅❂❁✲✹✽❁❯ ❅❁❝ ➔↔ ✐✷ ✹✱✲ ✼❅❂❁✲✹✐❡ ✼✽✼✲❁✹ ✽✾ ✹✱✲ ❁✻❡❈✲✻✷ ✇✱✐❡✱❝✲✸✲❁❝✷ ✽❁ ✹✱✲ ✐✷✽✹✽✸✲ ➞❃ ➙❖ ➍➛➏P ✐✷ ✹✱✲ ✲❈✲❡✹✺✽❁✐❡ ✇❅▲✲ ✾✻❁❡✹✐✽❁ ❅✼✸❈✐✹✻❝✲ ❅✹ ✹✱✲ ❈✽❡❅✹✐✽❁ ➍➛➏ ✽✾ ✹✱✲ ❁✻❡❈✲✻✷❃ ⑨❁❅ ✷✲✼✐❡✽❁❝✻❡✹✽✺ ❡✺●✷✹❅❈❯ ✹✱✐✷ ✇❅▲✲ ✾✻❁❡✹✐✽❁ ✐✷ ❅ ✸✺✽❝✻❡✹ ✽✾ ❅ ✸✲✺✐✽❝✐❡ ❅✹✽✼✐❡ ✇❅▲✲ ✾✻❁❡✹✐✽❁ ➟❖➍➠P❯ ❅❁❝ ❅❁ ✲❁▲✲❈✽✸✲✾✻❁❡✹✐✽❁ ➡➑❖➍➠P❯ ❁❅✼✲❈● ➙➑❖➍➠P ❳ ➡➑❖➍➠P➟❖➍➠P❃ ❚✱✲❁ ✇✲ ❡❅❁ ✺✲✇✺✐✹✲ ✹✱✲ ❬❅✼✐❈✹✽❁✐❅❁ ❅✷➒❖➍➎➏P ❳ ➔↔ ➢➤➓➑ ❨➔→❪➎ ➔➣➥↔ ↕➡➑❖ ➍➛➏P↕① ➍➐➑➦➧ ➜ ➍➎➏ ❳ ➔↔ ➍❽➨ ➜ ➍➎➏ ❖❄P✇✐✹✱ ➥↔ ❳ ↕➟❖ ➍➛➏P↕①❃ ➇❅❈✻✲✷ ✽✾ ➥↔ ❝✲✸✲❁❝ ✽❁ ❁✻❡❈✲❅✺ ✷✸✲❡✐✲✷ ❅✷ ✇✲❈❈ ❅✷ ✹✱✲ ❡✺●✷✹❅❈ ✷✹✺✻❡✹✻✺✲❃ ⑩✱✲❁ ✲❈✲❡✹✺✽❁✷ ❅✺✲✸✽❈❅✺✐❭✲❝❯ ✹✱✲ ✷✻✼ ✐✷ ❁✽✹ ❭✲✺✽ ❅❁❝ ✺✲✷✻❈✹✷ ✐❁ ❅❁ ✲➩✲❡✹✐▲✲ ✼❅❂❁✲✹✐❡ ❱✲❈❝ ❖➈❁✐❂✱✹ ❱✲❈❝P ➍❽➨ ❅❡✹✐❁❂ ✽❁ ✹✱✲ ❁✻❡❈✲❅✺ ✷✸✐❁➍➎➏❯ ❈✲❅❝✐❁❂ ✹✽ ✹✱✲ ➈❁✐❂✱✹ ✲❁✲✺❂● ✷✱✐✾✹ ✽✾ ➔↔ ↕ ➍❽➨↕❯ ✲❘✻✐▲❅❈✲❁✹ ✹✽ ❅ ✾✺✲❘✻✲❁❡● ✷✱✐✾✹ ➫❢➨ ❳ ➔↔ ↕ ➍❽➨↕➭➯❃❋✽✺ ✶➲ ✲❈✲❡✹✺✽❁✷ ❡✽❁❱❁✲❝ ✐❁ ✹✱✲ ➳ ❅❁❝ ➵ ❝✐✺✲❡✹✐✽❁✷❯ ✹✱✲ ➈❁✐❂✱✹ ✷✱✐✾✹ ✱❅✷ ❅ ✷✸❅✹✐❅❈ ▲❅✺✐❅✹✐✽❁ ✺✲➸✲❡✹✐❁❂ ✹✱✲ ✲❁▲✲❈✽✸✲✾✻❁❡✹✐✽❁ ✸✺✽❱❈✲❯ ➫❢➨❖➳➺ ➵P ❳ ➻↔ ❖➼❑➽➾ ➄ ➼❑➽➚ P↕➡❖➳➺ ➵P↕① ❳ ➻↔ ➪❑➽↕➡❖➳➺ ➵P↕①➺ ❖❪P✇✱✲✺✲ ➻↔ ➶ ➔↔ ➥↔ ✐✷ ❅ ❁✻❡❈✲❅✺➹✷✸✲❡✐✲✷ ❝✲✸✲❁❝✲❁✹ ❡✽✻✸❈✐❁❂ ❡✽✲➘❡✐✲❁✹❯ ➼❑➽➾ ❅❁❝ ➼❑➽➚ ❅✺✲ ✷✸✐❁➹✺✲✷✽❈▲✲❝ ✶➲ ✲❈✲❡➹✹✺✽❁ ❝✲❁✷✐✹✐✲✷❯ ❅❁❝ ➪❑➽➴ ➷ ➼❑➽➾ ➄ ➼❑➽➚ ✐✷ ✹✱✲ ✶➲ ✼❅❂❁✲✹✐❭❅✹✐✽❁ ❝✲❁✷✐✹●❃ ❆✷✷✻✼✐❁❂ ✾✽✺ ✷✐✼✸❈✐❡✐✹● ❅ ✱❅✺❝➹✇❅❈❈ G G Ga Ga . x / h0 . x / h (a) (b) ❋(cid:0)✁✂ ❙✄ ✂ ❘☎✆✝♦✞✆☎ ♦✟ ❞✠✡☎☛☎✞t✠☞✌ ❝♦✞❞✍❝t☞✞❝☎ ● t♦ ☛✟r✎☞✏✞☎t✠❝ ✑☎✌❞ ❇✒✓ ✝✌♦tt☎❞ ☞✆ ☞ ✟✍✞❝t✠♦✞ ♦✟ ✟☛☎❢✍☎✞❝✔ ✕✂ ❋☛☎❢✍☎✞❝✔☛☞✞✏☎✆ ❝♦☛☛☎✆✝♦✞❞✠✞✏ t♦ ◆✖❘ ✆✝☎❝t☛☞ ✟♦☛ ✻✗✁☞ ✭☞✘ ☞✞❞ ✼✙✁☞ ✭✚✘ ☞☛☎ ✆s♦✇✞✛ ☛☎✆✝☎❝t✠❡☎✌✔✂ ❚s☎ ❞☞t☞ ☞☛☎ ♦✚t☞✠✞☎❞ ☞t ts☎ ✆☞✎☎❝♦✞❞✠t✠♦✞ ☞✆ ❋✠✏✂ ✜ ♦✟ ts☎ ✎☞✠✞ t☎✢t ✭❇ ❂ ✹✂✣ ❚✛ ❱❣✙ ❂ ✤✵✿✥✦✣ ✧✛ ☞✞❞ ❱★✩ ❂ ✤✣✵ ✪✧✘✂✫✬✮✯✮✰✱✰✮✲ ✐✮ ✲✳✰ ◗✴✶ ✫✸✬✺✺✽✺✰✫✲✐✬✮✾ ✲✳✰ ✲✸❀✮✺❁✰✸✺❀❃ ❄✸✬❅✮❆ ✺✲❀✲✰ ❈❃✬❉✰✺✲ ✺❅❊❊❀✮❆❍ ❉❀❁✰ ■❅✮✫✲✐✬✮ ✐✺ ❏❈②❑ ③❍ ▲♣✷▼❖P♣✷▼❖❯ ✫✬✺❈❲②▼❖P❍ ✫✬✺❈❲③▼❖❯❍ ❉✐✲✳ ✫✬✮✯✮✰✱✰✮✲ ❉✐❆✲✳✺ ❖P ❀✮❆ ❖❯❳ ❨✳✰✮ ✲✳✰ ✱❀♠✐✱❅✱ ❁❀❃❅✰ ✬■ ✲✳✰ ❩✮✐❄✳✲✺✳✐■✲ ❬ ❊✰✫✬✱✰✺ ❬ ▲ ❭❪ ❫❴❵ ❛ ✷❖P ❜ ❛ ✷❖❯ ❜ ❤ ❈❥❍❦✐✱✐❃❀✸❃❧✾ ✐✮ ✲✳✰ ✫❀✺✰ ✬■ ✷♥ ✰❃✰✫✲✸✬✮ ✺❧✺✲✰✱ ✫✬✮✯✮✰❆ ✐✮ ✲✳✰ ③ ❆✐✸✰✫✲✐✬✮✾ ✲✳✰ ❩✮✐❄✳✲ ✺✳✐■✲ ❊✰✫✬✱✰✺❬ ▲ ❭❪❈q✉❵✈ ① q✉❵④ ❍ ❛ ✷❖❯ ❜ ▲ ❭❪❫✉❵ ❛ ✷❖❯ ❜ ❑ ❈⑤❍❅✺✐✮❄ ✲✳✰ ✺❀✱✰ ✫✬✰⑥✫✐✰✮✲ ❭❪ ❀✺ ✐✮ ✲✳✰ ⑦♥ ✫❀✺✰❳ ⑧✰✸✰ q✉❵✈ ❀✮❆ q✉❵④ ❀✸✰ ✺⑨✐✮✽✸✰✺✬❃❁✰❆ ✷♥ ✰❃✰✫✲✸✬✮ ❆✰✮✺✐✲✐✰✺✾ ❀✮❆❫✉❵❯ ⑩ q✉❵✈ ① q✉❵④ ✐✺ ✲✳✰ ✷♥ ✱❀❄✮✰✲✐❶❀✲✐✬✮ ❆✰✮✺✐✲❧❳ ❷❀✺✰❆ ✬✮ ✲✳✰ ❩✮✐❄✳✲ ✺✳✐■✲ ✱✰❀✺❅✸✰✱✰✮✲✺ ✐✮ ❸❀❹✺ ❺❅❀✮✲❅✱ ❉✰❃❃✺❉✐✲✳ ✷♥ ✰❃✰✫✲✸✬✮ ✺❧✺✲✰✱✺✾ ✲✳✰ ✫✬✰⑥✫✐✰✮✲ ■✬✸ ❻ ▲ ❼❴❸❀ ✐✺ ✸✰⑨✬✸✲✰❆ ❀✺ ❭❽❾❿➀ ▲ ①❈❥❤⑤ ➁ ➂❤✷❍▼✷ ➃ ⑦➂➄✉✉ ➅⑧❶ ✱➆ ➇✷➈❈❨✳✐✺ ✐✺ ✳❀❃■ ✲✳✰ ✬✸✐❄✐✮❀❃ ❁❀❃❅✰ ✐✮ ➉✰■❳ ➇✷➈ ❊✰✫❀❅✺✰ ✬■ ✲✳✰ ❆✐➊✰✸✰✮✫✰ ✐✮ ✲✳✰ ❆✰✯✮✐✲✐✬✮ ✬■ ❭❍❳ ❨✳✰ ✮✰❄❀✲✐❁✰ ✺✐❄✮ ✱✰❀✮✺✲✳❀✲ ✮✰❄❀✲✐❁✰ ■✸✰❺❅✰✮✫❧ ✺✳✐■✲ ✐✺ ✐✮❆❅✫✰❆ ❊❧ ❅⑨✽✺⑨✐✮ ✰❃✰✫✲✸✬✮✺❳ ✴❅✲✲✐✮❄ ✲✳✐✺ ❁❀❃❅✰ ❀✮❆ ❁❀❃❅✰✺ ■✬✸ ➋❪ ❀✮❆ ➌❪ ❈➋❽❾❿➀ ▲✷❳⑤➍✷ ➎➏▼✷❫➐✾ ➋❽➑➒➓ ▲ ⑦❳❥➔→ ➎➏▼✷❫➐✾ ➌❽❾❿➀ ▲ ✷❳➣➃⑦➂➆✾ ❀✮❆ ➌❽➑➒➓ ▲ ❥❳⑤➃⑦➂➆ ➇➔✾ ❥➈❍ ✐✮✲✬ ❀ ⑨✸✬⑨✬✸✲✐✬✮❀❃✐✲❧ ✸✰❃❀✲✐✬✮❭❽➑➒➓❭❽❾❿➀ ▲ ❛ ➋❽➑➒➓➋❽❾❿➀ ❜ ❛ ➌❽➑➒➓➌❽❾❿➀ ❜ ❑ ❈➍❍✲✳✰ ✫✬✰⑥✫✐✰✮✲ ■✬✸ ❼↔❹✺ ✐✺ ❆✰✲✰✸✱✐✮✰❆ ❀✺ ❭❽➑➒➓ ▲ ①✷❤⑦ ➃ ⑦➂➄✉✉ ➅⑧❶ ✱➆❳↕➙➛➛➙➜➝ ➞➟➠➡➢➛➠ ➤➙➛➥ ➦➧➞➙➨➡➠ ➩➨➜➫➜➟➭➟➜➛ ➯➨➛➟➜➛➙➧➢➠➲✮ ✲✳✰ ✱❀✐✮ ✲✰♠✲✾ ❉✰ ✺✲❀✲✰ ✲✳❀✲ ❀ ➔♥ ✫✬✮✯✮✰✱✰✮✲ ✱✬❆✰❃ ✫❀✮ ✮✬✲ ❊✰ ✸✰✫✬✮✫✐❃✰❆ ❉✐✲✳ ✲✳✰ ✱✰❀✺❅✸✰❆ ❆❀✲❀❳ ⑧✰✸✰❉✰ ❆✰✱✬✮✺✲✸❀✲✰ ✲✳❀✲ ✲✳✐✺ ✫✬✮✫❃❅✺✐✬✮ ✐✺ ✐✮✺✰✮✺✐✲✐❁✰ ✲✬ ✲✳✰ ❉❀❁✰ ■❅✮✫✲✐✬✮ ✺✳❀⑨✰✾ ❀✮❆ ✲✳✰✸✰■✬✸✰ ✲✳✰ ⑨✸✰✫✐✺✰ ✫✬✮✯✮✰✱✰✮✲⑨✸✬✯❃✰❳ ❨✬ ✲✳✐✺ ✰✮❆✾ ❉✰ ✯✲ ✲✳✰ ➳➵➉ ✺⑨✰✫✲✸❀ ➇➸✐❄❳ ➔❈❊❍ ✬■ ✲✳✰ ✱❀✐✮ ✲✰♠✲➈ ❅✺✐✮❄ ❁❀✸✐✬❅✺ ❉❀❁✰ ■❅✮✫✲✐✬✮ ⑨✸✬✯❃✰✺ ❀✲ ❀✸✬❅✮❆✲✳✰ ✫✰✮✲✰✸ ✬■ ✲✳✰ ◗✴✶❳ ➺✰ ❀✺✺❅✱✰ ✲✳❀✲ ✲✳✰ ✫✬✮❆❅✫✲❀✮✫✰ ✐✺ ❀ ■❅✮✫✲✐✬✮ ✬■ ✲✳✰ ➻✰✰✱❀✮ ✰✮✰✸❄❧❳ ❷✰✫❀❅✺✰ ✲✳✰ ✫✳❀✮❄✰ ✐✮✲✳✰ ➻✰✰✱❀✮ ✰✮✰✸❄❧ ➼➽➾ ✐✮❆❅✫✰❆ ❊❧ ✲✳✰ ✸■✽✱❀❄✮✰✲✐✫ ✯✰❃❆ ✐✺ ✺✱❀❃❃✾ ❉✰ ✫❀✮ ✰♠⑨❀✮❆ ✲✳✰ ✫✬✮❆❅✫✲❀✮✫✰ ✲✬ ✯✮❆ ✐✲✺ ✫✳❀✮❄✰➼➚❈➪ ❍ ❊✰✐✮❄ ⑨✸✬⑨✬✸✲✐✬✮❀❃ ✲✬ ➼➽➶❳ ❨✳✰ ✫✳❀✮❄✰ ➼➽➶ ✫❀✮ ❊✰ ❉✸✐✲✲✰✮ ❊❧ ❀✮ ✐✮✲✰❄✸❀❃ ✬■ ✲✳✰ ❃✬✫❀❃ ➹❁✰✸✳❀❅✺✰✸ ✯✰❃❆ ✫✳❀✮❄✰➼➘➴❈➷➬➮ ➪ ❍ ✱❅❃✲✐⑨❃✐✰❆ ❊❧ ✰❃✰✫✲✸✬✮ ❆✐✺✲✸✐❊❅✲✐✬✮ ➱❏❈➷➬❍➱✉ ❉✐✲✳ ❀✮ ❅✮➅✮✬❉✮ ⑨✸✬⑨✬✸✲✐✬✮❀❃✐✲❧ ✫✬✰⑥✫✐✰✮✲ ✃❳ ❨✳✰✮ ➼➚ ✫❀✮ ❊✰❉✸✐✲✲✰✮ ❀✺ ➼➚❈➪❍ ❐ ➼➽➶ ❈➣❍▲ ✃ ❒ ❮➷➬ ➼➘➴❈➷➬➮ ➪ ❍➱❏❈➷➬❍➱✉ ❈❰❍▲ ✃ ❒ ❮➷➬ Ï➇➪ ① Ð➪Ñ Ò ➼➪Ó❈➷➬❍Ô❑ Õ➈➱❏❈➷➬❍➱✉❑ ❈→❍ (a) G ( a r b . un i t s ) (b) -15 -10 -5 0 5f - f (kHz) (c) ❋(cid:0)✁✂ ❙✄ ✂ ❋☎✆✆☎✝✞ r✟✠✡☛✆✠ ✆t ✆☞✟ ◆✌✍ ❞✎✆✎✏ ❢rt✑ ❋☎✞✂ ✒✓✔✕ ☎✝ ✆☞✟ ✑✎☎✝ ✆✟✖✆✏ ❢tr ❱ ✗✘✙❣✚ ❂ ✛✵✿✻✜✢ ✣ ✎✠✠✡✑☎✝✞ ✻ ❞☎✤✟r✟✝✆ ✇✎✈✟❢✡✝✥✆☎t✝ ♣rt✦☛✟✠✂ ❚☞✟ ✇✎✈✟ ❢✡✝✥✆☎t✝ ♣rt✦☛✟✠ ✎r✟ ☛☎✠✆✟❞ ☎✝ ❚✎✔☛✟ (cid:0)✂ ✓✎✕ ❚☞✟ r✟✠✡☛✆✠ ❢tr ★✧ ✓✔☛✎✥❦✕ ✎✝❞ ★✩ ✓r✟❞✕✂ ✓✔✕ ❚☞✟ r✟✠✡☛✆✠❢tr ★✄ ✓✔☛✎✥❦✕ ✎✝❞ ★✪ ✓r✟❞✕✂ ✓✥✕ ❚☞✟ r✟✠✡☛✆✠ ❢tr ★✒ ✓✔☛✎✥❦✕ ✎✝❞ ★✻ ✓r✟❞✕✂ ❚☞✟ ②☎✟☛❞✟❞ ✦✆✆☎✝✞ ♣✎r✎✑✟✆✟r✠ ✎r✟ ❑ ❂ ✽✂✢✏ ✢✂✄✏✢✂✄✏ ✧✧✂✒✏ ✧✄✂✧✏ ✎✝❞ ✧✒✂✵ ❦✫✬ ✎✝❞ ❆ ❂ ✛✩✿✻ ✭ ✧✵✮✺✏ ✛✩✿✵ ✭ ✧✵✮✺✏ ✛✒✿✩ ✭ ✧✵✮✺✏ ✛✧✿✻ ✭ ✧✵✮✯✏ ✛✧✿✄ ✭ ✧✵✮✯✏ ✎✝❞ ✛✧✿✩ ✭ ✧✵✮✯ ❢tr★✧ ④ ★✻✏ r✟✠♣✟✥✆☎✈✟☛② ✓❚☞✟ ✇✎✈✟ ❢✡✝✥✆☎t✝ ♣r✟✰❢✎✥✆tr ❈ ☎✠ ✥☞t✠✟✝ ✠t ✆☞✎✆ ✌✎✖✱✲✳✲✷✴ ❂ ✧✕✂(cid:0)✝❞✟✖ ❲✎✈✟ ❢✡✝✥✆☎t✝ ♣rt✦☛✟ Pt✆✟✝✆☎✎☛★✧ ❈ ✥t✠✓✶✹✼✾❀✕ ✥t✠✓✶③✼✾❁✕ ✄❃ ☞✎r❞✰✇✎☛☛★✄ ❈ ✟✖♣✓✛✹✷✼✄❧✷❀✕ ✥t✠✓✶③✼✾❁✕ ♣✎r✎✔t☛☎✥ ✓✹✕ ✎✝❞ ☞✎r❞✰✇✎☛☛ ✓③✕★✒ ❈ ✟✖♣✓✛✹✷✼✄❧✷❀✕❄☎✓③✼❧❁ ❅ ♥❇✕ ♣✎r✎✔t☛☎✥ ✓✹✕ ✎✝❞ ✆r☎✎✝✞✡☛✎r ✓③✕★✩ ❈ ✥t✠✓✶①✼✾❉✕ ✥t✠✓✶✹✼✾❀✕ ✥t✠✓✶③✼✾❁✕ ✒❃ ☞✎r❞✰✇✎☛☛★✪ ❈ ✟✖♣✓✛①✷✼✄❧✷❉✕ ✟✖♣✓✛✹✷✼✄❧✷❀✕ ✥t✠✓✶③✼✾❁✕ ♣✎r✎✔t☛☎✥ ✓①❊ ✹✕ ✎✝❞ ☞✎r❞✰✇✎☛☛ ✓③✕★✻ ❈ ✟✖♣✓✛①✷✼✄❧✷❉✕ ✟✖♣✓✛✹✷✼✄❧✷❀✕❄☎✓③✼❧❁ ❅ ♥❇✕ ♣✎r✎✔t☛☎✥ ✓①❊ ✹✕ ✎✝❞ ✆r☎✎✝✞✡☛✎r ✓③✕❚❄●❍■ (cid:0)✂ ❲✎✈✟ ❢✡✝✥✆☎t✝ ♣rt✦☛✟✠ ✡✠✟❞ ❢tr ✆☞✟ ✦✆✆☎✝✞ ✎✝✎☛②✠☎✠ ✎✝❞ ✆☞✟ ✥trr✟✠♣t✝❞☎✝✞ ✥t✝✦✝✟✑✟✝✆ ♣t✆✟✝✆☎✎☛✠✂ ❄☎✓③✕ ✎✝❞ ♥❇❞✟✝t✆✟ ✆☞✟ ❄☎r② ❢✡✝✥✆☎t✝ ✎✝❞ ☎✆✠ ☛✎r✞✟✠✆ ✬✟rt ♣t☎✝✆✏ r✟✠♣✟✥✆☎✈✟☛②✂ ❚☞✟ ♣✎r✎✑✟✆✟r✠ ✎r✟ ✾❉ ❂ ✾❀ ❂ ✻✵ ✝✑✏ ✾❁ ❂ ✧✪ ✝✑✏ ❧❉ ❂ ❧❀❂ ✧✢✂✽ ✝✑✏ ✎✝❞ ❧❁ ❂ ✪✂✪ ✝✑✂❏▲▼❖▼ ◗❘❯❳⑦❨❩ ❬ ❭❪❫♠❴❥❵❳⑦❨❩❥❛ ✐❜ ❝▲▼ ❡❤✐♦▲❝ ❜▲✐s❝ q❝ ❝▲▼ ✉⑤❜✐❝✐⑤❤ ⑦❨⑥ ⑧ ⑨q⑩❜❜✐q❤ s⑩❤❶❝✐⑤❤ ❷❳◗❘❸ ❹❩ ❬ ▼❺✉❳❻◗❘ ❛❼❽❹❛❩❾▼❜❶❖✐❿▼❜ ❝▲▼ ❾▼✉⑤➀q❖✐➁q❝✐⑤❤ ⑤s ❤⑩❶➀▼q❖ ❜✉✐❤❜ ❿➂ ❖s➃➄q♦❤▼❝✐❶ ➅▼➀❾ q❝ q s❖▼➆⑩▼❤❶➂ ❾▼❝⑩❤✐❤♦ ◗❘ ➇ ❘ ❻ ❳❘➈ ➉ ◗❘❯❩ ❏✐❝▲❘➈ ❿▼✐❤♦ ❝▲▼ ➊➋➌ s❖▼➆⑩▼❤❶➂ ❏✐❝▲⑤⑩❝ ❝▲▼ ✐❤➍⑩▼❤❶▼ ⑤s ❝▲▼ ❡❤✐♦▲❝ ❜▲✐s❝⑥ ➌▼❏❖✐❝✐❤♦ ◗❘❯❳⑦❨❩ ❬ ❻➎❥❵❳⑦❨❩❥❛❼➋q❺➏❥❵❳⑦❨❩❥❛➐q❤❾ ⑩❜✐❤♦ ➎ q❤❾ ➑ q❜ ➅❝❝✐❤♦ ✉q❖q➄▼❝▼❖❜➒ ❝▲▼ ▼❺✉▼❖✐➄▼❤❝q➀ ❾q❝q ✐❤ ➓✐♦⑥ ✸❳❿❩ ✐❤ ❝▲▼ ➄q✐❤ ❝▼❺❝ q❖▼ ➅❝❝▼❾ ❝⑤ ➔➆⑥ ❳→❩❏✐❝▲ ❘➈ ❬ ✸❽⑥➣↔↔ ➋↕➁ q❤❾ ❹ ❬ ➙⑥✸➛ ➜↕➁➒ ❝▲▼ ❜q➄▼ q❜ ✐❤ ❝▲▼ ➄q✐❤ ❝▼❺❝⑥➓⑤❖ ❝▲▼ ➅❝❝✐❤♦ q❤q➀➂❜✐❜➒ ❏▼ q❜❜⑩➄▼ ➛ ❾✐➝▼❖▼❤❝ ❏q➞▼ s⑩❤❶❝✐⑤❤ ✉❖⑤➅➀▼❜ ❵❳⑦❨❩ ➀✐❜❝▼❾ ✐❤ ❝q❿➀▼ ➟⑥ ➠▲▼ ➅❝❝✐❤♦ ❖▼❜⑩➀❝❜ ❝⑤ ❝▲▼➡ ➢➤➥➦➧ ❬ ❻➨➩➛➣→ ➫ ❾q❝q q❖▼ ❜▲⑤❏❤ ✐❤ ➓✐♦⑥ ➭❽⑥ ➠▲▼ ❏q➞▼ s⑩❤❶❝✐⑤❤ ✉❖⑤➅➀▼❜ ➯➙ ➃ ➯✸ ❶⑤❖❖▼❜✉⑤❤❾ ❝⑤ q ❽➲ ❶⑤❤➅❤▼➄▼❤❝✉⑤❝▼❤❝✐q➀ ❜⑩✉✉⑤❜✐❤♦ ❝▲q❝ ❤⑤ ❿⑤⑩❤❾ ❜❝q❝▼ ✐❜ s⑤❖➄▼❾ q❝ q❖⑤⑩❤❾ ❝▲▼ ➳➵➸ ❶▼❤❝▼❖⑥ ➠▲▼ ❏q➞▼ s⑩❤❶❝✐⑤❤ ✉❖⑤➅➀▼❜ ➯➺ ➃ ➯➛❶⑤❖❖▼❜✉⑤❤❾ ❝⑤ q ✸➲ ❶⑤❤➅❤▼➄▼❤❝ ✉⑤❝▼❤❝✐q➀ ❜⑩✉✉⑤❜✐❤♦ ❝▲q❝ q ❿⑤⑩❤❾ ❜❝q❝▼ ✐❜ s⑤❖➄▼❾ q❖⑤⑩❤❾ ❝▲▼ ➳➵➸ ❶▼❤❝▼❖⑥ ⑧❜ ❜▼▼❤✐❤ ❝▲▼ ➅♦⑩❖▼➒ ❝▲▼ ❶⑤❖❖▼❜✉⑤❤❾▼❤❶▼ ✐❜ ❜➂❜❝▼➄q❝✐❶q➀➀➂ ➄⑩❶▲ ❏⑤❖❜▼ ✐❤ ❝▲▼ ➀q❝❝▼❖ ❶q❜▼❜ ❶⑤➄✉q❖▼❾ ❝⑤ ❝▲▼ s⑤❖➄▼❖ ⑤❤▼❜⑥ ⑧❜q ♦▼❤▼❖q➀ s▼q❝⑩❖▼➒ ✸➲ ❶⑤❤➅❤▼➄▼❤❝ ➄⑤❾▼➀❜ ❖▼❜⑩➀❝ ✐❤ ♦▼❤❝➀▼ ❜➀⑤✉▼❜ ⑤❤ ❝▲▼ ➀⑤❏➃s❖▼➆⑩▼❤❶➂ ❜✐❾▼ q❤❾ ❜❝▼▼✉ ❜➀⑤✉▼❜ ⑤❤ ❝▲▼▲✐♦▲➃s❖▼➆⑩▼❤❶➂ ❜✐❾▼ ❖▼♦q❖❾➀▼❜❜ ⑤s ❝▲▼ ❾▼❝q✐➀❜ ⑤s ❝▲▼ ❏q➞▼ s⑩❤❶❝✐⑤❤ ✉❖⑤➅➀▼❜⑥ ➠▲✐❜ ❝❖▼❤❾ ✐❜ ⑤✉✉⑤❜✐❝▼ ❝⑤ ❝▲▼ ▼❺✉▼❖✐➄▼❤❝q➀➀➂⑤❿❜▼❖➞▼❾ ❜✉▼❶❝❖⑩➄ ❜▲q✉▼❜➒ ▼❜✉▼❶✐q➀➀➂ s⑤❖ ❝▲⑤❜▼ ❏✐❝▲ ➀q❖♦▼ ❡❤✐♦▲❝ ❜▲✐s❝❜➒ q❤❾ ❽➲ ❶⑤❤➅❤▼➄▼❤❝ ➄⑤❾▼➀❜ ❶⑤❖❖▼❶❝➀➂ ♦q❜✉❝▲✐❜ ❿▼▲q➞✐⑤❖⑥➠⑤ ⑩❤❾▼❖❜❝q❤❾ ❝▲✐❜ ❖⑤❿⑩❜❝ ❾▼✉▼❤❾▼❤❶▼ ⑤s ❝▲▼ ❶⑤❤❾⑩❶❝q❤❶▼ ❾✐✉ ❜▲q✉▼ ⑤❤ ❝▲▼ ❏q➞▼ s⑩❤❶❝✐⑤❤ ❶⑤❤➅❤▼➄▼❤❝ ❾✐➄▼❤❜✐⑤❤➃ r [r M ] F ( r ) F F F f-f [-a As m r M ] F * e x p [ a . u ] n=1n=2n=3 (a)(b) ❋(cid:0)✁✂ ❙✄ ✂ ✭☎✆ ❚✝✞ ❢✟✠✡☛☞✌✠✍ ✎♥✭✚✆ ☞✠ ❊✏✂ ✭✑✒✆✂ ✭✓✆ ❚✝✞ ✡✌✠✈✌♦✟☛☞✌✠ ❬❊✏✂ ✭✑✒✆✔ ✞✈☎♦✟☎☛✞❛ ✠✟✉✞✕☞✡☎♦♦✖ ❢✌✕ ♣☎✕☎✉✞☛✞✕✍ ☎✍ ❣☞✈✞✠☞✠ ☛✝✞ ☛✞t☛ ❢✌✕ ♠✵✱ ♠✶✱ ☎✠❛ ♠✷ ❂ ♠✶✗✘✒ ✠✉✱ ✙❆✛ ❂ ✜✺ ✢ ✑✒✣✷✷ ❦✤✥ ✉✸✱ ✦ ❂ ✑✿✄✧ ❦✤✥✱ ❧✵ ❂✺✂✺✱ ✑✒✂✺✱ ☎✠❛ ✑✄ ✠✉ ❢✌✕ ★ ❂ ✑✱ ✘✱☎✠❛ ✄✱ ✕✞✍♣✞✡☛☞✈✞♦✖✂✩✪✫✬②✮ ✇❡ r❡✇r✫✬❡ ✬✯❡ s✰✩✬✫✩✪ ✫✐✬❡✲r✩✪ ✫✐ ✳✴✻ ✼✽✾ ✫✐✬❀ ✩✐ ✫✐✬❡✲r✩✪ ❀❁❡r ✬✯❡ ❡✪❡❃✬r❀✐ ❞❡✐s✫✬② ❄ ❅ ❥❇❥❈✮❉●✼❍ ✾ ■ ❩ ❏▼❑ ❞❄ ▲◆✼❄✾ ❡❖✰ P◗ ❘❍ ◗ ❍❑ ◗ ❯❱❲❳❨❭◆❄❪❈❫❴❈ ❵ ❜ ✼❝❤✾✇✯❡r❡ ✇❡ ✫✐✬r❀❞q❃❡ ✬✯❡ ①✩✲✐❡✬✫③✩✬✫❀✐ ❞❡✐s✫✬✫❡s ④❀r ✬✇❀ ✼⑤ ❅ ❫✾ ✩✐❞ ✬✯r❡❡ ✼⑤ ❅ ⑥✾ ❞✫①❡✐s✫❀✐✩✪ ❃❀✐⑦✐❡①❡✐✬ ✩s❳⑧ ⑨ ❝⑩❶❷ ❸①❭⑧ ✼❀qr ❡❖✰❡r✫①❡✐✬✾ ✩✐❞ ❳❑ ❅ ❝ ✼✩ ✰❀✪✩r✫③❡❞ ✪❀❃✩✪ ①❀①❡✐✬✾✮ r❡s✰❡❃✬✫❁❡✪②✻ ❹✯❡ ❞✫s✬r✫❺q✬✫❀✐ ❀④ ✬✯❡❞❡✐s✫✬② ❄ ❺❡✬✇❡❡✐ ✫✬s ①✫✐✫①✩✪ ❄ ❅ ❤ ✩✐❞ ①✩❖✫①✩✪ ❁✩✪q❡ ❄ ❅ ❄❻ ✫s ✲✫❁❡✐ ❺② ✬✯❡ ✇❡✫✲✯✬ ④q✐❃✬✫❀✐ ▲◆✼❄✾✻ ❼❀r✫✪✪qs✬r✩✬✫❀✐✮ ✇❡ ✬✩❽❡ ✩ s✰✯❡r✫❃✩✪✪② s②①①❡✬r✫❃ ❾✩qss✫✩✐ ❞❡✐s✫✬② ✰r❀⑦✪❡✮ ❥❇✼❿➀✾❥❈ ❅ ❄❻ ❡❖✰✼◗➀❈➁➂❈❑✾✮ ④❀r ✇✯✫❃✯ ▲◆✼❄✾❃✩✐ ❺❡ ❃✩✪❃q✪✩✬❡❞ ✩✐✩✪②✬✫❃✩✪✪② ✩s ▲◆✼❄✾ ■ ❘✪✐✼❄❻ ➁❄✾❪➃◆❭❈➄➅❈✻ ❹✯✫s ④q✐❃✬✫❀✐ ✫s ✰✪❀✬✬❡❞ ❀✐ ❼✫✲✻ ➆⑥ ✼✩✾✮ ④r❀① ✇✯❡r❡ ✬✯❡❞✫①❡✐s✫❀✐✩✪✫✬② ❡➇❡❃✬ ✫s ❃✪❡✩r✪② ❁✫s✫❺✪❡✻ ➈✐❞❡❡❞✮ ④❀r ⑤ ❅ ❫ ✼r❡❞ ❃qr❁❡s✾ ✬✯❡ ✇❡✫✲✯✬ ④q✐❃✬✫❀✐ ✫s ❃❀✐s✬✩✐✬✮ s❀ ✫✬s ❃❀✐❁❀✪q✬✫❀✐✇✫✬✯ ✬✯❡ ❾✩qss✫✩✐ ④r❡✴q❡✐❃② ❞✫s✬r✫❺q✬✫❀✐ ✇✫✪✪ r❡sq✪✬ ✫✐ ✩ s②①①❡✬r✫❃ ✰r❀⑦✪❡✮ ✩s s✯❀✇✐ ❀✐ ❼✫✲✻ ➆⑥ ✼❺✾✻ ➉❀①✰✩r❡❞ ✬❀✬✯✩✬✮ ✩ ✬✯r❡❡ ❞✫①❡✐s✫❀✐✩✪✪② ❃❀✐⑦✐❡❞ ✇✩❁❡ ④q✐❃✬✫❀✐ ✯✩s ①❀r❡ ✇❡✫✲✯✬ ✩✬ s①✩✪✪ ❄✮ ✇✯✫❃✯ ✫✐ ❀✬✯❡r ✇❀r❞s ①❡✩✐s ✬✯✩✬✫✬ ✯✩s ✩ ①❀r❡ ✇❡✫✲✯✬ ✫✐ ✬✯❡ ✬✩✫✪✻ ❹✯✫s r❡sq✪✬s ✫✐ ✩✐ ✩s②①①❡✬r✫❃ s✯✩✰❡ ❀④ ✬✯❡ ❃❀✐❁❀✪q✬✫❀✐✮ s❽❡✇❡❞ ✬❀ ✬✯❡ ✪❡④✬ ✼❺✪q❡❃qr❁❡s✾✻ ➊❡❃✩qs❡ ❯❱❲ ✫s ✐❡✲✩✬✫❁❡✮ ✬✯❡ s✯✩✰❡ ✫s ①✫rr❀r r❡➋❡❃✬❡❞ ✇✯❡✐ ❃❀✐❁❡r✬✫✐✲ ✬✯❡ ✯❀r✫③❀✐✬✩✪ ✩❖✫s ④r❀① ✬✯❡ ➌✐✫✲✯✬s✯✫④✬ ✫✐ ❼✫✲✻ ➆⑥ ✼✩✾ ✬❀ ✬✯❡ ➍➎➏ ④r❡✴q❡✐❃② ✫✐ ❼✫✲✻ ➆⑥ ✼❺✾ ✩✐❞ ❼✫✲✻ ➆❫✻ ❹✯✫s ✩✪✪❀✇s qs ✬❀ ❡❖❃✪q❞❡ ✬✯❡ ✰❀ss✫❺✫✪✫✬② ❀④ ✩ ⑥➐❃❀✐⑦✐❡①❡✐✬ ①❀❞❡✪✮ ✩s ❀qr ❞✩✬✩ s②s✬❡①✩✬✫❃✩✪✪② s✯❀✇ ❡✫✬✯❡r s②①①❡✬r✫❃✮ ❀r s❽❡✇❡❞ ✬❀✇✩r❞s ✪❀✇❡r ④r❡✴q❡✐❃✫❡s s✯✩✰❡s✮❃❀rr❡s✰❀✐❞✫✐✲ ✬❀ ❫➐✮ ✩✐❞ ❝➐ ❃❀✐⑦✐❡①❡✐✬ ①❀❞❡✪s✮ r❡s✰❡❃✬✫❁❡✪②✻➑❡ ✩✪s❀ ✐❀✬❡ ✬✯✩✬ ✬✯❡ ❺✪✩❃❽ ❃qr❁❡s ✼➒❝ ➓ ➒⑥✾ ✫✐ ❼✫✲✻ ➆❫ ✪❀❀❽ ❁❡r② s✫①✫✪✩r✮ sq✲✲❡s✬✫✐✲ ✬✯✩✬ ✬✯❡ ⑦✬✬✫✐✲ ❞❀❡s ✐❀✬❞❡✰❡✐❞ s✬r❀✐✲✪② ❀✐ ✬✯❡ ❞❡✬✩✫✪s ❀④ ✬✯❡ ✇✩❁❡ ④q✐❃✬✫❀✐ ✰r❀⑦✪❡s✮ ✩✐❞ ②✫❡✪❞ r❀❺qs✬ ❁✩✪q❡s ❀④ ➔✮ ❺❡✫✐✲ →✻✽ ❽➣③✮ ✽✻❫ ❽➣③✮✩✐❞ ✽✻⑥ ❽➣③ ④❀r ➒❝✮ ➒❫✮ ✩✐❞ ➒⑥✮ r❡s✰❡❃✬✫❁❡✪②✻ ❹✯❡r❡ ✫s ✩ r❡❃❀✲✐✫③✩❺✪❡ ✬r❡✐❞✮ ✐✩①❡✪② ✬✯❡ ❁✩✪q❡s ❀④ ➔ ✬❡✐❞ ✬❀ ✫✐❃r❡✩s❡✩s ✬✯❡ ✬✩✫✪s ❀④ ✬✯❡ ✇✩❁❡ ④q✐❃✬✫❀✐ ✰r❀⑦✪❡ ❡❖✬❡✐❞s✻ ❹✯❡r❡④❀r❡ ✬✯❡ ①❀❞❡✪ ❀④ ✬✯❡ ❫➐ ✯✩r❞➓✇✩✪✪ ✰❀✬❡✐✬✫✩✪ ✇❡ ✩❞❀✰✬ ✫✐ ✬✯❡①✩✫✐ ✬❡❖✬ ✲✫❁❡s ✩ ❺❀q✐❞ ④r❀① ❺❡✪❀✇ ④❀r ➔✮ ✯❡✐❃❡ ④❀r ❳↔ ✇✯✫❃✯ ✩✪r❡✩❞② ❡❖❃❡❡❞s ✬✯❡ ①✩✲✐❡✬✫❃ ①❀①❡✐✬ ④❀r ✩ s✫✐✲✪❡❡✪❡❃✬r❀✐ s✰✫✐✻ ❹✯✫s ④qr✬✯❡r ❃❀rr❀❺❀r✩✬❡s ✬✯✩✬ ❀qr ❃❀✐❃✪qs✫❀✐s ✩r❡ ✐❀✬ ✩r✬✫④✩❃✬s ❀④ ✩ s✰❡❃✫⑦❃ ❃❀✐⑦✐❡①❡✐✬ ①❀❞❡✪✻↕➙➛➜➝ ➞➟➝➞➠➝➟➡➢➙➤➈✐ ❀qr ✬✫✲✯✬➓❺✫✐❞✫✐✲ ①❀❞❡✪ ❃✩✪❃q✪✩✬✫❀✐✮ ✇❡ ✩ssq①❡ ✩ s①❀❀✬✯ ✰✩r✩❺❀✪✫❃ ✰❀✬❡✐✬✫✩✪ ❺✩rr✫❡r ✩✬ ✬✯❡ ❃❡✐✬❡r ❀④ ✬✯❡ ➥➦➉✇✫✬✯ ✩ ✯❡✫✲✯✬ ➧❑ ✩✐❞ ✩ ❃qr❁✩✬qr❡ ➨➩✻ ➈✐✬❡r①❡❞✫✩✬❡ r❡✲✫❀✐s ✼➫❑ ➭ ❥➫❥ ➯ ➲ ✾ ✩r❡ ✫✐s❡r✬❡❞ ❺❡✬✇❡❡✐ ✬✯❡ ❃❡✐✬r✩✪ ✰✩r✬ ❀④✬✯❡ ➥➦➉ ✩✐❞ ✬✯❡ ✪❡✩❞s✮ ✇✯❡r❡ ➳➵ ❅ ❤✮ ✬❀ ❃❀✐✐❡❃✬ ✬✯❡ ✰❀✬❡✐✬✫✩✪ s①❀❀✬✯✪② ❘❼✫✲✻ ➆✹✼✩✾❪❘❷❪➸➳➵ ❅ ➺➻➼➻➽ ➧❑ ◗ ➾➚➪➶➹ ➫❈❜ ❥➫❥ ➯ ➫❑➘✼➫ ◗ ➲ ✾❈ ➴ ➷✼➫ ◗ ➲ ✾➶❜ ➫❑ ➭ ❥➫❥ ➯ ➲❤❜ ➲ ➭ ❥➫❥ ❜ ✼❝❝✾ e j / t (a) U j / t (b) ❋(cid:0)✁✂ ❙✄ ✂ ❚☎✆✝t✞✟☎✠✡☎✠✆ ♠☛✡❞☞ ♣✌✍✌♠❞t❞✍✎ ❢☛✍ ◆ ❂ ✷✏✑ ❥✵ ❂ ✶✒✑ ✓①✔✕ ❂✖✂✖✗✘✑ ❱✵✔✕ ❂ ✖✂✏✘✙✑ ✌✠✡ ❯✔✕ ❂ ✖✂✚✒✂ ✭✌✛ P☛t❞✠t☎✌☞❞✠❞✍✆❡ ✜✢ ✌✎ ✌ ❢✣✠✤t☎☛✠ ☛❢ ✎☎t❞ ☎✠✡❞✐ ❥✂ ✭✟✛ ❙☎t❞ ✡❞♣❞✠✡❞✠t ❈☛✣☞☛♠✟ ☎✠t❞✍✌✤t☎☛✠ ❯✢ ✌✎ ✌ ❢✣✠✤t☎☛✠ ☛❢ ❥✂✲✥✳✻✾✲✥✳✻✻✲✥✳✻✸✲✥✳✻✥✲✥✳✦✼✲✧✳✥ ✲✥✳✦ ✥✳✥ ✥✳✦ ✧✳✥★✩✩✪●✴✪✫❣✬ ✮✯✰✱✴❤✹✽ ✿❀❀ ❁❀❃❄❀ ❁❀❃❅✿ ❁❀❃❅❀ ❁❀❃✿✿❆❇❉❆❊ ❍■❏❑▲▼ ❉❖◗❘❲❳❨ ❲❩❨ ❬ ❭ ❪❫❴ ❵❛s❜ ❭ ❝ ❦❧❬ ❭ ❝ ❵ ✫❣✬ ✮✹✽❊ ❍■❏◗❘ ✫♥♦ ✮q✹✽❋(cid:0)✁✂ ❙✏ ✂ ✭✤☛☞☛✍ ☛✠☞☎✠❞✛ ✭✌✛ ❚✍✌✠✎✤☛✠✡✣✤t✌✠✤❞ ✭r✉✔r❱✈✇✛ ♣☞☛tt❞✡ ☎✠ ✤☛☞☛✍ ✎✤✌☞❞ ✌✎ ✌ ❢✣✠✤t☎☛✠ ☛❢ ❱②③ ✌✠✡ ❱✈✇ ❢☛✍ ④ ❂ ✖✂✖ ❚✂❚✝❞ ✡☛tt❞✡ ☞☎✠❞✎ t✍✌✤❞ t✝❞ ✌☞☎✆✠♠❞✠t ☛❢ ✎✣✟✟✌✠✡ ❞✡✆❞✎ ⑤☎t✝ t✝❞ ✎☛✣✍✤❞ ✌✠✡ ✡✍✌☎✠ ❞☞❞✤t✍☛✤✝❞♠☎✤✌☞ ♣☛t❞✠t☎✌☞✎✂ ⑥ ☞❞⑦❞✍✞✌✍♠ ❢✌✤t☛✍✤☛✠⑦❞✍t☎✠✆ ❢✍☛♠ ✆✌t❞ ⑦☛☞t✌✆❞ t☛ ❞✠❞✍✆❡ ☎✎ ✡❞✍☎⑦❞✡ ❢✍☛♠ t✝❞☎✍ ✎☞☛♣❞✎ ✌✠✡ ☎✠t❞✍✎❞✤t☎☛✠✎✂ ✭✟✛ ❚✍✌✠✎✤☛✠✡✣✤t✌✠✤❞ ❢☛✍ ❱②③ ❂ ✖ ⑧⑨ ✌✠✡④ ❂ ✄✂✏ ❚ ♣☞☛tt❞✡ ✌✎ ✌ ❢✣✠✤t☎☛✠ ☛❢ ❱✈✇✂ ❚✝❞ ✎t✍❞✠✆t✝ ☛❢ ❈☛✣☞☛♠✟ ☎✠t❞✍✌✤t☎☛✠ ❯ ☎✎ ✎❞t ✟❡ ✍❞⑩✣☎✍☎✠✆ t✝✌t t✝❞ ✤✌☞✤✣☞✌t❞✡ ✍❞✎✣☞t✍❞♣✍☛✡✣✤❞✎ t✝❞ ♠❞✌✎✣✍❞✡ ❞✠❞✍✆❡ ✡☎❶❞✍❞✠✤❞ ✟❞t⑤❞❞✠ t✝❞ ✎♣☎✠✞✎♣☞☎t t✍✌✠✎✤☛✠✡✣✤t✌✠✤❞ ♣❞✌❷✎✂❸❹❺❻❺ ❼ ❽❾❿ ➀ ❽❻❺ ❿❺➁❺❻➂➃❾❺❿ ➄➅ ➁❹❽➁ ➆➇ ❽❾❿ ➈➆➇➉➈➊ ❽❻❺ ➋➅❾➁➃❾➌➅➌➄ ❽➁ ➊ ➍ ➊➎➏❼ ➍ ➐➑➇➒➓➔→➣ ↔↕➎ ➙ ➛➣➜➝➞ ➊➐➎ ➟ ➛➣➜➠➞ ➊➎➡➊➎ ➙ ➢➤➥ ➦ ➡➧➨➤➀ ➍ ➩➝➑➇➒➓➔→➣ ↔➙ ➛➣➜➐➞ ➊➐➎ ➙ ➨❼➡➊➎ ➙ ➢➤➥ ➫ ➡➧➭➤➯❹❺ ➅❾➲➄➃➁❺ ➳➅➌➵➅➂➸ ❺❾❺❻➺➻ ➼➇ ➃➄ ❾➅❾➲➽❺❻➅ ➅❾➵➻ ➃❾ ➁❹❺ ➋➅❾➄➁❻➃➋➁➃➅❾ ➾❽❻➁ ❽❾❿ ➋❹❽❾➺❺➄ ➄➂➅➅➁❹➵➻ ➁➅ ➼➇ ➍ ➚ ➃❾ ➁❹❺ ➵❺❽❿➪➶➃➺➹ ➘➴➡➸➤➷ ➪✺➬ ➮➷➹ ➼➇ ➍ ➱ ➼ ❺✃➾ ↔➙ ➑➇❐➔→❒➩➓➑➇❐➔→➣ ➥ ➦ ❮➊❮ ❰ ➢➚➦ ❮➊❮ Ï ➢➫ ➡➧➴➤➯❹❺ ➾❽❻❽➂❺➁❺❻➄ Ð➅❻ ➁❹❺ ➂➅❿❺➵ ➋❽➵➋➌➵❽➁➃➅❾ ❽❻❺ ❿❺➁❺❻➂➃❾❺❿ ❽➄ Ð➅➵➵➅❸➄➹ ➯❹❺ ❹➅➾➾➃❾➺ ❽➂➾➵➃➁➌❿❺ Ñ ➃➄ ❻❺➵❽➁❺❿ ➁➅ ➁❹❺➶❺❻➂➃ ❺❾❺❻➺➻ ➆Ò ➸➻ ➆Ò ➍ ➨Ñ➡➧ ➙ ➋➅➄ ➨Ó❼➉ÔÒ➤➬ ❸❹❺❻❺ ÔÒ ➃➄ ➁❹❺ ➶❺❻➂➃ ❸❽Õ❺ ➵❺❾➺➁❹ ❽❾❿ ❼ ➃➄ ➁❹❺ ➵❽➁➁➃➋❺ ➋➅❾➄➁❽❾➁ ➅Ð ➁❹❺➁➃➺❹➁➲➸➃❾❿➃❾➺ ➋❹❽➃❾➹ ➯❹❺ ❾➌➂❺❻➃➋❽➵ ➋❽➵➋➌➵❽➁➃➅❾➄ ❽❻❺ ❿➅❾❺ Ð➅❻ ➢ ➍ ➨✺ ❽❾❿ ➊➎ ➍ ➧➭➹ ➯❹❺ ➵❽➁➁➃➋❺ ➋➅❾➄➁❽❾➁ ❼ ➃➄ ➄❺➁ ➁➅❼ ➍ ➧➉ÖÔÒ ➪×➷➬ ❸❹➃➋❹ ➵❺❽❿➄ ➁➅ ➁❹❺ ØÙ➳ ➵❺❾➺➁❹ ➅Ð ➭➴➚ ❾➂ Ð➅❻ ➢ ➍ ➨✺➹ Ú ➵❺Õ❺❻➲❽❻➂ Ð❽➋➁➅❻ Û ➍ ➭➴ ➂❺ÜÝÜ ➋➅❾Õ❺❻➁➃❾➺➺❽➁❺ Õ➅➵➁❽➺❺ ↕Þ➩ ➁➅ ❺❾❺❻➺➻ ➃➄ ❺✃➁❻❽➋➁❺❿ Ð❻➅➂ ➁❹❺ ➁❻❽❾➄➋➅❾❿➌➋➁❽❾➋❺ ➡ßà➉ß↕Þ➩➤ ❿❽➁❽ ➃❾ ➶➃➺➹ ➘✺➡❽➤➹ ➯❹❺❾➬ ❽➄➄➌➂➃❾➺❽ ➁❻❽❾➄➂➃➄➄➃➅❾ ➾❻➅➸❽➸➃➵➃➁➻ Ñá➡➆➤ ➍ ➧➉➡➧ ➟ â➐ã➑äå➓ä→❐➛➜➤➪Ö➷➬ ➁❹❺ ➋➌❻Õ❽➁➌❻❺ ➅Ð ➁❹❺ ➾➅➁❺❾➁➃❽➵ ➸❽❻❻➃❺❻ æç ➃➄ ❺Õ❽➵➌❽➁❺❿➸➻ è➁➁➃❾➺ ➁❹❺ ➵➃❾❺❽❻ ➋➅❾❿➌➋➁❽❾➋❺ ➁❻❽❾➄➃➁➃➅❾ ➸❺➁❸❺❺❾ ➨â➐➉é ❽❾❿ ➴â➐➉é➹ ➯❹❺ ➾❽❻❽➂❺➁❺❻ ➼ ➃➄ ❿❺➁❺❻➂➃❾❺❿ ➄➅ ➁❹❽➁ ➁❹❺➋❽➵➋➌➵❽➁❺❿ ❻❺➄➌➵➁ ❻❺➾❻➅❿➌➋❺➄ ➁❹❺ ❺❾❺❻➺➻ ❿➃ê❺❻❺❾➋❺ ➸❺➁❸❺❺❾ ➁❹❺ ➁❻❽❾➄➋➅❾❿➌➋➁❽❾➋❺ ➾❺❽ë➄ ➋➅❻❻❺➄➾➅❾❿➃❾➺ ➁➅ ➁❹❺ ➄➾➃❾➲➄➾➵➃➁ m j ( m - ))