Broadband dielectric microwave microscopy on micron length scales

Transcription

1 REVIEW OF SCIENTIFIC INSTRUMENTS 78, Broadband delectrc mcrowave mcroscopy on mcron length scales Alexander Tselev a and Steven M. Anlage b Center for Superconductvty Research, Department of Physcs, Unversty of Maryland, College Park, Maryland Zhengkun Ma and John Melngals Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Park, Maryland Receved 19 January 2007; accepted 1 March 2007; publshed onlne 3 Aprl 2007 We demonstrate that a near-feld mcrowave mcroscope based on a transmsson lne resonator allows magng n a substantally wde range of frequences, so that the mcroscope propertes approach those of a spatally resolved mpedance analyzer. In the case of an electrc probe, the broadband magng can be used n a drect fashon to separate contrbutons from capactve and resstve propertes of a sample at length scales on the order of one mcron. Usng a mcrowave near-feld mcroscope based on a transmsson lne resonator we maged the local delectrc propertes of a focused on beam mlled structure on a hgh-delectrc-constant Ba 0.6 Sr 0.4 TO 3 thn flm n the frequency range from 1.3 to 17.4 GHz. The electrostatc approxmaton breaks down already at frequences above 10 GHz for the probe geometry used, and a full-wave analyss s necessary to obtan qualtatve nformaton from the mages Amercan Insttute of Physcs. DOI: / I. INTRODUCTION The rapd development of nanotechnology and the shrnkng sze of the functonal elements of devces demand dverse tools for comprehensve materals characterzaton at the mcron and sub-mcron length scales. Near-feld mcrowave mcroscopy has become a useful tool for magng and measurement of electromagnetc propertes of materals at the mcron length scale. 1 6 Generally, the measurement prncple of a mcrowave near-feld mcroscope s based on alteraton of the probe electrcal mpedance due to the presence of a sample n close proxmty to the probe. To ncrease the mcroscope senstvty, the probe s often made as an ntegral part of a resonator ether dstrbuted or based on lumped elements. Usually, due to fundamental lmtatons of a specfc resonator desgn, the measurement frequency s lmted to just one frequency or to a small frequency range. Lttle work has been done on broadband mcrowave mcroscopy. Chang et al. 7 performed measurements of permttvty on thn flm Ba 1 x Sr x TO 3 BSTO phase spreads n a frequency range from 0.95 to 4.95 GHz wth the goal to nvestgate the compostonal dependence of the BSTO permttvty dsperson. However, n addton to the frequency-dependent propertes of materals, broadband measurements can also be mplemented to separate and study the contrbutons of dfferent sample propertes n the measured mcroscope response. For nstance, a broadband measurement can be used to dsentangle delectrc permttvty and conductvty for a leaky delectrc, as well as to separate the nfluence of a Present address: Department of Chemstry, Duke Unversty, Durham, NC, b Author to whom correspondence should be addressed; electronc mal: anlage@squd.umd.edu parastc factors and systematc uncertantes accompanyng, e.g., varyng probe-sample couplng. Such an approach, called mpedance spectroscopy, utlzng frequency sweeps s routnely used to study materal propertes at low frequences wth the use of LCR meters and mpedance analyzers see, e.g., Refs. 8 and 9. One needs to cover at least one decade of frequency to obtan a relable result of fttng to accomplsh ths task. Ideally, a mcroscope should acqure the full capablty of an mpedance analyzer whle mantanng the ablty to make spatally resolved measurements. In ths respect, the mcrowave mcroscopes based on transmsson lne resonators have an advantage over mcroscopes based on cavty or lumped element resonators snce the resonance mode structure n a transmsson lne resonator s mantaned over a sgnfcantly broader frequency range, and the frequency sweep can be realzed smply by jumpng from one resonance mode to another. An electrcally long transmsson lne resonator allows for a dense set of modes, creatng a quascontnuous frequency coverage. Fgure 1 a shows the most general electrcal schematc dagram of a transmsson lne resonator: the resonator a segment of a transmsson lne of length L s coupled to a feed lne wth a couplng element of mpedance Z c and termnated by a probe wth effectve mpedance Z t. Further, wthout sgnfcant loss of generalty, we consder n more detal an example of a mcroscope wth an electrc probe based on a coaxal lne resonator. Fgure 1 b dsplays, for ths partcular case, a schematc of the resonator open end wth a small-radus apex tp actng as a probe n contact wth a sample. Fgure 1 c shows a lumped element crcut representaton of the probe tp and a lossy delectrc sample. Such a sample can be descrbed by an RC par. 10,11 The capactor, C s, descrbes the frnge electrc feld energy stored n the /2007/78 4 /044701/7/$ , Amercan Insttute of Physcs

2 Tselev et al. Rev. Sc. Instrum. 78, broadband magng of a focused on beam FIB mlled structure on a Ba 0.6 Sr 0.4 TO 3 BSTO thn flm, coverng a frequency range from 1.3 to 17.4 GHz wth 1 m spatal resoluton. Usng a mcroscope system based on a coaxal transmsson lne resonator, we explore the possblty of dstngushng contrbutons of the flm permttvty and the flm conductance, partcularly through the resonant frequency shft, by mplementng broadband magng. FIG. 1. a Electrcal dagram of a transmsson lne resonator of length L termnated by a tp wth mpedance Z t ; both the feed lne and the resonator have a characterstc mpedance Z 0, and the resonator s connected to the feed lne through a couplng element wth an mpedance Z c ; b schematc of the open end of a coaxal resonator wth a protrudng probe tp and a sample; c smplfed equvalent crcut descrbng the probe tp wth a lossy delectrc sample nearby. C s and R s are the capactance and resstance presented by the sample, whch s coupled to the transmsson lne resonator through the resstance of the probe tp and the couplng capactance C sc. C str s the stray capactance between the probe tp and the outer conductor. sample, and the resstor, R s, stands for all channels of energy dsspaton n the sample. The sample s coupled to the resonator through the tp-sample couplng capactance C sc and the resstance of the tp R tp. The feld between the tp and the outer conductor, whch bypasses the sample, s represented by the parastc stray capactance C str. The effectve probe tp mpedance Z t can be calculated as the mpedance of the whole RC network. The values of the equvalent crcut elements depend on the local propertes of the sample under the probe tp and the set of parameters descrbng the tp-sample system. Generally, the dependence of the probe tp mpedance on sample propertes and the probe tp geometry s complcated and ts elucdaton consttutes the major problem of mcrowave mcroscopy. 2,6,12 16 A more detaled descrpton of a sample wll requre a more complcated equvalent crcut. When the tp s beng scanned over the sample, the varaton of the effectve tp mpedance causes shfts f of the resonant frequences and changes of the Q factors of the resonator modes, whch are montored and maged by a computer. Ideally, the broadband measurements could be accomplshed by swtchng from one resonant frequency to another whle the probe-sample geometry s fxed. By subsequent fttng of the mcroscope frequency-dependent response to a specfc electrcal model of the probe-sample system, the set of model parameters could be deduced, and further, the contrbutons of local propertes of the sample could be separated and determned. In ths artcle, we present an expermental effort of II. EXPERIMENT A 370-nm-thck Ba 0.6 Sr 0.4 TO 3 flm was deposted by pulsed laser deposton PLD at a substrate temperature of 800 C at an oxygen pressure of 120 mtorr on a 500- m-thck 001 LaAlO 3 LAO substrate. Mllng of the BSTO flm was performed wth a focused 50 kev Ga + on beam. The flm was removed from a 5 m wdth trench around a 20 m 20 m square sland. As a result of the mllng the materal nsde the trench as well as n some vcnty of the trench was doped wth Ga. The structure was annealed at 600 C n oxygen for 1 h after mllng to partally restore the damage caused by the mllng. Such an annealng treatment has been shown to sgnfcantly mprove the delectrc propertes of the BSTO flm n the damaged area. 17 Basc detals of the mcroscope system used n the present work were descrbed elsewhere. 18,19 Here we summarze the addtonal features of the broadband mcroscope. The mcroscope s based on a half-wavelength coaxal transmsson lne resonator of length 25 cm wth the nner conductor termnated by an scannng tunnelng mcroscope tungsten tp wth about 1 m radus apex Fg. 2. A mcrowave sgnal s generated by a syntheszed mcrowave source Aglent E8257C PSG analog sgnal generator. The mcrowave power reflected from the resonator back nto the feed lne s montored by a feedback crcut, whch locks the frequency of the mcrowave source to the chosen resonance mode. At a resonance frequency, the reflecton coeffcent Fg. 2 b passes a local mnmum. The couplng of the feed lne wth the resonator can be accomplshed by means of ether a capactor or an nductor col nserted n the nner conductor between the resonator and the feed lne. The best performance of the system can be acheved at modes close to the frequency of crtcal couplng of the resonator. At the crtcal couplng, the reflecton from the resonator at the resonant frequency s equal to zero. Ths condton dctated the use of several dfferent couplng elements for broadband magng, tuned for dfferent frequency subbands. For the broadband experments n ths work, we used a modfed SMA connector wth mnature gold wre cols n place of the central conductor to couple the resonator wth the feed lne. Three cols wth dfferent nductances were used to cover the frequency range from 1 to 17.5 GHz. Broadband magng was performed by swtchng from one resonance mode of the mcroscope to another upon completon of the scannng frame at a prevous frequency. The sample was mantaned n constant contact wth the probe tp by applcaton of a small force of N to the sample. For extracton of data on flm permttvty, we used the approach outlned by Stenhauer et al. 19 It s based on the

3 Broadband mcrowave mcroscopy Rev. Sc. Instrum. 78, FIG. 3. Mcrowave frequency shft mages of the FIB-mlled BSTO flm structure obtaned at four dfferent frequences. The frequences are shown n the correspondng mages. Scale bars n the mages of the lower panel correspond to 10 m. The explanaton of the features denoted wth letters A G s n the text. FIG. 2. a Schematc of the broadband near-feld mcrowave mcroscope. The mcrowave sgnal s generated by the source, enters the resonator through the couplng col, nteracts wth the sample, and s detected upon reflecton outsde the resonator. The source s frequency modulated by the feedback crcut, and sent a drect current correcton sgnal proportonal to the frequency shft of the mcroscope. Inset shows the tp-sample contact n detal. b Calculated absolute value of the reflecton coeffcent from the resonator,, vs frequency, f, for a purely nductve couplng between the feed lne and the resonator. The length of the resonator used n the calculatons was 25 cm, nductance of the couplng col was 3 nh, and the characterstc mpedance of both the feed lne and the resonator was equal to 50. Inset llustrates the scale of the resonance frequency shft due to a sample: sold lne s the reflecton coeffcent of the resonant mode at the mnmum of reflecton wthout a sample, and the dashed lne s the same wth a sample at the probe tp. perturbaton formulas for resonant cavtes and nvolves fnte element calculatons of felds around the probe tp n the ar and n the sample n the electrostatc approxmaton. The procedure requres probe calbraton aganst reference samples wth known propertes e.g., bulk delectrc substrates. Refnng that approach, n the present work, we more accurately descrbed the geometry of the probe and used the ANSOFT Maxwell 2D two-dmensonal fnte element analyss package to evaluate the perturbaton ntegrals. To reduce the sze of the fnte element model, the sample s placed on a conductng plane, whch s consdered to be at a floatng potental n the numercal smulatons. III. RESULTS AND DISCUSSION A. Frequency shft mages The upper panel of Fg. 3 shows a hgher contrast and lower resoluton frequency shft mage over an 80 m 45 m area of the FIB-mlled structure, obtaned by means of a blunter and therefore more senstve tp. The mage was taken at a frequency of 1.4 GHz. The FIB-mlled trench looks lke a broad black strpe labeled A n Fg. 3 frnged wth whte lnes B. The bottom of the trench s heavly doped wth Ga ons and relatvely strongly conductng, producng a large frequency shft compared to the BSTO flm. The small-frequency-shft whte frngng s due to an edge effect when the tp apex s partally n contact wth the flm and partally hangs over the trench. The trench s surrounded on both sdes by darker regons C of Ga-doped BSTO flm. The FIB beam has substantal tals that dope regons outsde the man focus. The corners of the sland surrounded by the trench D are more heavly doped and are more conductng. Farther away s a broad brghter regon of damaged flm E wth a lowered delectrc permttvty producng a smaller frequency shft. A 1 m wde FIB-mlled lne F s also seen along the left edge of the mage. Brght spots G result from holes n the flm and partcles of the PLD target materal usually present on the surface of PLDgrown flms. The lower panel of Fg. 3 shows how the frequency shft contrast of the same regon evolves wth frequency. The mage for f =2.2 GHz looks smlar to that for f =1.4 GHz. At f =8.7 GHz, the edge effect s stll clearly seen, but the contrast between the trench bottom and the surroundng flm s almost absent. There s no vsble contrast between Ga-doped and undoped regons outsde the trench. A slght contrast between the damaged area and the ntact flm s vsble. At the hghest frequency used n the experments, f =17.4 GHz, the trench looks whte on a dark background. At ths frequency, the frequency shft at the trench bottom s notceably smaller than that for the surroundng and qute close to that correspondng to the LAO substrate wthout a flm. A contrast between the damaged area and the ntact flm s stll vsble. It s worth notng here that mages of flm delectrc

4 Tselev et al. Rev. Sc. Instrum. 78, tunablty see Ref. 19 for the measurement method detals obtaned smultaneously wth the frequency shft mages, all look smlar to each other, regardless of frequency. B. Theoretcal model To nterpret the transformaton of the contrast of the mages n Fg. 3, let us consder the relatonshp between the frequency shft and the sample propertes n terms of the equvalent crcuts n more detal. As mentoned earler, the resonant frequency of the resonator s determned by measurng the mcrowave power reflected from the resonator back nto the feed lne. The complex reflecton coeffcent for voltage can be expressed through the reflecton coeffcents of the resonator couplng element c and the termnatng tp t n the followng form: 20 = c + t e 2 L 2 c t e 2 L 1 c t e 2 L. 1 Here, = + j /c s the complex transmsson lne propagaton constant, L s the length of the resonator, s the angular frequency, =2 f, c s the speed of the lght n vacuum, s the attenuaton n the transmsson lne, and s the relatve permttvty of the transmsson lne delectrc. The expresson Eq. 1 can be obtaned by summng contrbutons from all partal waves multply reflected n the resonator and contrbutng to the wave gong back nto the feed lne. The condton for the resonant mnmum of the reflecton coeffcent d 2 d =0, d d defnes n mplct form the resonant frequency r as a functon of a set of the parameters c of the tp equvalent crcut. For example, the set of c s for the crcut n Fg. 1 c s: C sc, C s, C str, and R s. One s not nterested n the value of r, but rather n the change of the resonant frequency r caused by a change of the sample propertes. Under the condton of a weak frequency dependence of the reflecton coeffcents c and t, c, t f 0 1, f Arg c, t 0 1, 3 t can be found by a straghtforward calculaton wth use of Eqs. 2 that for a small perturbaton caused by the sample / dc r, 4 / where =Arg c t e 2 L and f 0 =c/ 2L s the lowest mode frequency of a free half-wave resonator wth length L. We note that s the change of the phase of a partal wave wthn the resonator for one round trp between the couplng element ant the tp. Equatons 3 demand a weak frequency dependence of the reflecton coeffcents c and t wthn the resonant frequency shft for each resonant mode, and thereby set certan requrements to the desgn of the resonator-feed lne couplng. Further, takng the dervatves of wth account of the second of Eqs. 3, we fnd / 1/f 0, / dc = Arg t / dc, and after that we obtan r f 0 Arg t dc. 5 The complex reflecton coeffcent t can be wrtten n the form t = 1 z / 1+z wth z=z 0 Y t, c =Z 0 G t, c + jb t, c. Here Y t =1/Z t, G t, and B t are the effectve tp admttance, conductance, and susceptance, respectvely. Z 0 s the characterstc mpedance of the resonator transmsson lne. Further, we neglect the actual tp resstance R t Fg. 1 c. Provded that z 1, one gets Arg t 4 2, where =Re z and =Im z and f, one can wrte for the frequency shft r 2Z 0 f 0 B t dc. And fnally, expressng the tp susceptance through the effectve tp capactance B t = C t, we obtan r r 2Z 0 f 0 C t dc. Note that as follows from the last expresson, the relatve frequency shft does not depend on frequency drectly n the approxmatons used here. Hence, the frequency dependence of the relatve frequency shft appears only through the frequency dependence of the propertes of the tp-sample system. Drect measurements and calculatons show that the condtons of Eqs. 3 and 7 settng the valdty range for the approxmatons are fulflled n our mcroscope. Beng equpped wth Eq. 9, we can analyze the frequency response of the mcroscope usng the equvalent crcut Fg. 1 c wth = C str, C sc, C s, R s. The change of the effectve tp capactance due to a sample s dc t = C t dc = C t C str dc str + C p f, where C p f s the frequency-dependent parallel capactance of the smpler crcut consstng of only three elements C sc, C s, and R s, as shown n the nset of the upper panel of Fg. 4. Ths crcut, wth well-known propertes n fact, a frst-order frequency flter, allows us to reproduce all essental features of the resonator frequency behavor n the regon of valdty of the electrostatc approxmaton. The change of the effectve tp conductance G p f due to a sample can be ntroduced smlar to C p f. For the followng analyss of the frequency response of the mcroscope, we neglect the stray capactance C str because t s a frequency-ndependent addtve to the parallel capactance C p f. The phase response of the crcut determned manly by the value of the parallel capactance C p f, governs the phase change of the wave n the transms-

5 Broadband mcrowave mcroscopy Rev. Sc. Instrum. 78, FIG. 4. Color onlne Calculated lumped element values C p and G p / 2 f vs frequency curves for the crcut shown n the nset wth frequencyndependent C sc, C s, and R s values, shown n a dmensonless coordnate system. The capactance and conductance are measured n unts of C sc, and the unt of frequency f s 2-1. Each panel contans three curves correspondng to three dfferent sample resstances R s =1/3 dashed, green, 1 sold, red, 3 dot-dashed, black. son lne upon reflecton from the tp, and hence, the resonant frequency shft. In the Appendx t s shown that the effectve tp conductance G p f determnes the change of the Q factor of the resonant modes. The frequency behavor of C p and G p 2 f wth frequency-ndependent values of C sc, C s, and R s s shown n Fg. 4 n a dmensonless coordnate system. The capactance and conductance are measured n values of C sc, and the unt of frequency f s 2 1. As can be seen from Fg. 4, at frequences much hgher than the pole frequency defned by the expresson C sc +C s R s =1, the value of C p s very weakly dependent on R s and the conductance contrast becomes nvsble for the mcrowave frequency shft. The transformaton of the contrast of the mages of Fg. 3 wth ncreasng frequency can be explaned by ths fact. At hgh enough frequences and under the condton C s C sc, the mcroscope response becomes senstve only to the delectrc propertes of the sample. Optmal frequences for magng of partcular propertes can be found wth use of Eqs. 9 and A5 as ponts of the maxmum absolute values of partal dervatves over the equvalent crcut parameters. C. Quanttatve analyss Fgure 5 shows the relatve frequency shft referenced to the bare LAO substrate versus frequency for four dfferent locatons around the mlled trench. The notatons for dfferent parts of the flm correspond to those of Fg. 3. Pont D wth a strong dependence of the frequency shft on frequency below about 6 GHz s close to the sland corner and t receved the largest Ga dose outsde the trench. The dashed FIG. 5. Color onlne Relatve frequency shft f / f referenced to the bare LAO substrate vs frequency for four dfferent locatons around the mlled trench refer to Fg. 3 : black sold squares correspond to an ntact part of the flm; red hollow crcles correspond to a damaged flm wth small Ga dopng denoted E n Fg. 3 ; blue sold trangles correspond to the mddle of the sland; and green hollow trangles correspond to areas close to the sland corners denoted D n Fg. 3. The curved sold green lnesthe frequency shft relatve to the bare LAO substrate calculated for the tpsample system model of Fg. 1 c wth the followng parameters: Z 0 =50, C sc =50 ff, C str =11.7 ff, C s =5.7 ff, R s =10 k, C s LAO =3.9 ff, and R s LAO s equal to nfnty. The last two values are for a bare LAO substrate. The horzontal straght sold red lne s the result of the same calculaton wth R s =10 M. The dashed black and blue lnes are gudes to the eye. lnes n Fg. 5 are gudes to the eye. The factors wth the largest contrbuton to the measurement error are the calbraton error due to ncomplete representaton of the actual tp shape by numercal calculatons made n the probe calbraton process and the progressve damage of the tp durng scannng. The overall measurement error due to these factors was estmated by numercal calculatons and s reflected by the error bars n the plot. The curved sold green lne n Fg. 5 s the frequency shft relatve to the bare LAO substrate calculated usng Eq. 9 for the tp-sample system model of Fg. 1 c wth the followng parameters: Z 0 =50, C sc =50 ff, C str =11.7 ff, C s =5.7 ff, R s =10 k, C s LAO =3.9 ff, and R s LAO s equal to nfnty. The last two values are for a bare LAO substrate. The curve corresponds to the most Ga-doped and, therefore, most conductng part of the flm. The estmated pole frequency for ths case s 285 MHz. The ncrease of the frequency shft observed at lower frequences at the sland corners corresponds n fact to the end of the slope of the C p versus frequency curve of Fg. 4. The horzontal straght sold red lne n Fg. 5 s the result for the same model, but wth R s =10 M, representng the undoped area. The value of C str was determned by measurng the resonant frequency shft after nserton of the probe tp nto the open end of the resonator relatve to the resonance frequency of the resonator wthout the tp. The value of C s LAO was obtaned by numercal fnte element calculatons, and the values of C s and R s were chosen to ft the expermental data, and they are on the order of magntude expected. As we see, Eq. 9 and the equvalent crcut Fg. 1 c farly well descrbe the measurement results up to a frequency of about 11 GHz. The expermental ponts at frequences hgher than 11 GHz devate from the theoretcal

6 Tselev et al. Rev. Sc. Instrum. 78, curve and show smaller relatve frequency shfts. Two possble factors, whch mght cause the observed behavor of the frequency shft at hgher frequences are the dsperson of the delectrc permttvty of the flm, and the deformaton of the probe tp durng scannng. Based on the measured frequency shfts, we have calculated the relatve permttvty of the flm and found that the relatve permttvty of the ntact BSTO flm see Fg. 5 s 550± 70 at frequences below 11 GHz. The relatve permttvty of the damaged flm ponts D and E, Fg. 5 s about 100 lower. The relatve permttvty of the ntact flm at f =17.4 GHz would be close to 350 f t s calculated n the same way, whch means an unlkely drop of 200 upon frequency change from 10.9 to 17.4 GHz. Ths drop s too large and cannot be attrbuted to dsperson of the delectrc propertes of the BSTO of any orgn. Takng nto account the data of Refs. 7 and 21, one should expect a change n the relatve delectrc constant due to dsperson to be less than 100 n the whole frequency range from 1 to 18 GHz. In fact, the dsperson of the BSTO propertes could not be detected wth our setup, and therefore, t s less than the measurement error of about 140, whch s n agreement wth Refs.7 and 21. The frequences durng the hgh-frequency experment were changed n the followng order: 17.4, 14.7, 8.7, 10.9, and 13.0 GHz. Hence, the downward trend seen n the curve at ncreasng frequency cannot be explaned by a progressve change of the tp shape n the flow of the experment, ether. We suggest that the devaton can be attrbuted to an enhanced contrbuton of radated and evanescent exctatons at the probe tp to the resonator response, whch s not captured by the electrostatc numercal calculaton. It should be noted that the relevant sample dmensons are not much smaller than the mcrowave wavelength at the hgher frequences. Indeed, at a frequency of 15 GHz, for nstance, LAO 4 mm n the LAO substrate wth a relatve permttvty 25, whereas the substrate thckness d /8 d =0.5 mm. A notceable devaton of the feld structure n the flm from the statc one can be expected n ths stuaton and a full-wave analyss s needed to clarfy the observed hghfrequency behavor of the frequency shft. Standard practcal restrctons on the applcaton of lumped-element models n the characterzaton of hgh frequency crcuts s that the dmensons of the crcut elements should be less than /20. If ths condton s appled to our mcroscope confguraton, the lumped element model and the near-feld electrostatc approxmaton can be used for measurements at frequences below 6 GHz only. Nevertheless, the broadband magng stll turns out to be useful n revealng the more conductng areas of the flms as s evdent from Fg. 3. Fnally, we would lke to note that n future versons of the broadband mcroscope, frequency should be swept at each pont pxel on the sample. The delectrc propertes of the materals can be extracted then by fttng the frequency dependences for each sample pont separately. One condton to realzaton of such an approach s desgn of a resonatorfeed lne couplng element capable of workng n a wde frequency range wth the couplng constant close to unty =1 corresponds to the crtcal couplng; at =1 the reflecton coeffcent from the resonator at the resonance s equal to zero. Further, desgn of the tp-sample system n such a way that ts pole frequency s close to the center of the frequency range covered by the mcroscope s another necessary condton for an effectve realzaton of the broadband approach. ACKNOWLEDGMENTS The work was supported by the Unversty of Maryland/ Rutgers NSF MRSEC under Grant No. DMR The authors thank V. V. Talanov Neocera, Inc. and R. Ramesh for frutful dscussons, as well as C. Canedy and A. Stanshevsky for sample fabrcaton. APPENDIX The parallel tp capactance C p determnes the resonant frequency shft of the mcroscope. Here we dscuss n more detal the role of the parallel tp conductance G p nset Fg. 4 n the mcroscope response, n partcular the qualty factor Q. At a resonance, the rato of the complex ampltudes of a partal wave wthn the resonator before, A 1, and after, A 2, one round trp between the couplng element and the tp can be expressed wth use of a complex frequency ˆ = + j as A 2 = exp j ˆ T 0 + t + c = t c exp 2 L, A1 A 1 where T 0 =1/f 0 s the tme of the round trp, and t, c are phase shfts of the wave upon reflecton from the tp and the couplng element, respectvely. It follows from the defnton of that and = f 0 2 L ln c ln t. A2 The Q factor of a resonator can be wrtten n the followng form: Q = 2. A3 Smlar to the steps leadng to Eqs. 5 and 8, we obtan d 2Z 0 f 0 G t dc. A4 For the valdty of the last approxmate expresson, t s requred that / /, and t s assumed that s fxed. G t G p f of the crcut n Fg. 4, and for changes of losses due to the sample we have Q 1 =4 1 f 0Z 0 G p. A5 The ampltude response of the crcut Fg. 1 c s determned manly by ts parallel conductance G p. The value G p / 2 f drectly determnes the losses and the change of the resonator qualty factor due to the sample. Ths value has a maxmum at the pole frequency defned by the expresson C sc +C s R s =1, correspondng to the maxmum of losses and to the mnmum of the Q factor. 1 B. T. Rosner and D. W. van der Wede, Rev. Sc. Instrum. 73, S. M. Anlage, V. V. Talanov, and A. R. Schwartz, n Scannng Probe

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