Dielectric Spectroscopy Centre - Manual 11. From complex conductivity to complex dielectric functionComplex dielectric function is experimentally obtained through the frequency, =2fdependent measurement (our range 10- - 10! "#$ of the complex admittance% &($='($(i )($*q+1-1,he real component of the admittance is conductance,'($, -hile the imaginarycomponent is usuallyexpressedthroughtheeffecti.e capacitanceC($=)($/(see0ec+1$+ 2n principle, measurement setup and the sample also posseses parasitic inductances,but their contribution to )($ is negligible and effecti.e capacitance C($ is .ery close tothereal one+ ,headmittance&($andthecomplexconducti.ity3($areproportionalthrough a factor, called the geometrical constant of the sample, l/S, -here % l - sample length/ distance bet-een the electrodes0 - sample cross section/ surface of the electrodes+ We have: ( ) $ ( YSl *q+1-22n the follo-ing -e -ill establish a relationship bet-een complex dielectric function,($and complex conducti.ity $ ( $ ( $ (0 0 i + Eq.1-3 ,his relationship may be deri.ed from a 4ax-ell equation%dtt dt t$ ($ ( $ (0DE H + Eq.1-4-here H(t$ is the magnetic field, 30 is the free carriers conducti.ity (dc-conducti.ity$,E(t$ is the ac-electric field, and D(t$ is the displacement .ector+ 5fter a 6ourier transform ofthe abo.e expression, -e may regard each frequency component separately (-here$ ( $ ( $ (0 E D $ and -e get%$ ( $7 ( $ ( 8 $ (0 0 E H i + Eq.1-50ince the total current in the circuit equals jtot($ = H($= ($E($ -e confirmthat the conducti.ity of the system under study, 3 ($ is gi.en byEq.1-3.30($represents the free-carrier component of the conducti.ity, -hich, in general,may be complex and frequency dependent - and thus not equal to the dc-conducti.ity+2. Contributions to dielectric function2n the pre.ious section ($ represented the dielectric function of the system understudy+ "o-e.er, -hen studying real materials, usually only one subsystem is of interest and-e ha.e to analy#e only the contribution of this subsystem to the dielectric function of the-hole sample+ ,hus, further throughout this text, ($ -ill represent only this contribution+9o-, for real materials -e -rite$ ( $ ( $ ( $ (exp 0 0 exp ip+ + Eq.2-1-here3exp($istheexperimentallymeasuredcomplexconducti.ityofthe-holesample+ 9ote that3exp($-ill al-ays be influenced by parasitic impedances of theexperimental setup, -hichmaybeexpressedthroughtheparasitic complexconducti.ity3p($+ ,heparasiticconducti.ityissmallerthantheothercomponents, andtherefore-eassumed that it combines additi.ely+ 5s before,30($ represents the free-carrier componentof the conducti.ity+,he dielectric function exp($may be understood (see 6igure 1$ as the polari#ationcurrent contribution of the subsystem under study plus the contributions of other subsystems,relaxing at higher frequencies (abo.e the experimental -indo-$% exp( $ ( $HF + *q+2-2

2n order to obtain the dielectric function of the subsystemunder study, theconducti.ities3p($and 30($ha.e to be subtracted from the measured conducti.ity+ ,heother contributions to the dielectric function of the -hole sample ha.e also to be subtracted+2norder toachie.ethis,backgroundconducti.ity,3bg($ismeasuredinabsenceof thesubsystem under study%bg p bgibg 0 0$ ( $ ( $ ( + + Eq.2-3:arasitic conducti.ity is assumed to be equal as in Eq.2-1+ ;epending on the physics ofthe material and experimental features of the measurement,30bg($and bgmay represent.arious contributions to the complex conducti.ity of the sample, -hich may be either muchlarger or much smaller than that of the subsystem under study+0ubtracting Eq.2-3 from Eq.2-1 , and using Eq.2-2 -e getexp 0 0 0( $ ( $ ( $ ( $ ( $bg HF bgbgi 1 11 ] ] 1 ] + *q+2-12n the follo-ing, -e -ill analy#e this equation for the t-o cases -e encounter in the;ielectric 0pectroscopy Centre+2.1. Small background- solid state samples,his limit occurs -hen the subsystem under study is a collecti.e electronic groundstate (density -a.e$ in a solid state sample+ "ere, experimentallymeasuredsampleconducti.ity,3exp($is dominatedbythecontributionsof free carriers(through 30($$and;< condensate (throughexp($$+6reecarrier conducti.ity is independent of frequency (-ithin the measurement error$ and equals todc-conducti.ity% 0($=30+ ,herefore,exp 0 0 exp( $ ( $ ( $pi + +Eq.2-5 ,he backgroundconducti.ity,3bg($isthe result of a separate measurement -hichhastobedoneintheabsenceofthe-holesample--ecannot remo.e=ust thedensity-a.e+++ )esides theparasitic conducti.ity,3p($, onlythedisplacement currentof the.acuum i0 (that is, bg=1$ contributes to 3bg($+ ,here are no free carriers in the absenceof the sample and 30bg($ is 0+ 0( $ ( $bg pi + Eq.2-65ccording to the abo.e considerations Eq.2-4 no- reads%exp 0 0( $ ( $ 8 ( $ 17bg HFi + Eq.2-7,a>ing the con.ention = ? - i @?? and using the notation 3($= 3exp($- 3bg($-e can separate the real and imaginary components% 02m8 ( $7?( $ 1HF + *q+2-!00Ae8 ( $ 7??( $ *q+2-B9ote that 3($ is calculated from the admittance &($, using Eq.1-1 and Eq.1-2, -here$ ( $ ( $ (exp bgC C C and exp( $ ( $ ( $bgG G G + :arameters 30='0Cl/0 and 0("6-1$= C"6Cl/0aredirectly obtained from the experiment+ ,hat is, '0 is the lo--frequency .alue ofexp( $ ( $ ( $bgG G G and C"6 is the high frequency .alue of $ ( $ ( $ (exp bgC C C + 9o-, -e can -rite explicit relationship bet-een the contributionof the subsystem under study to the dielectric function and the measured admittances -ith and-ithout the sample%0( $?( $HFC C lS *q+2-1000??( $( $ G G lS *q+2-112.2. Large background-aqueous samples,hislimit occurs-henthesubsystemunderstudy(colloid, polyelectrolyte$isinasolution (in an aqueous electrolyte$+"ere, experimentally measured sample conducti.ity,3exp($isthe result of ameasurement on asample solution-hich consists of the subsystem under study, differentionic species and -ater+exp 0 0 exp( $ ( $ ( $ ( $pi + + *q+2-12Dnli>eforthesolidstatesamples, freecarrierconducti.ity30($iscomplexandfrequency dependent+,hebackgroundconducti.ity,3bg($isgi.en bya separate measurement on abackground solution -a simple electrolyte (usually 9aCl in -ater$+,hebackgroundconducti.ityhas t-o large components% one is the ionic (free-carrier$ conducti.ity,30bg($,and the other comes from all the subsystems in the material -hich respond at frequenciesabo.e the experimental -indo- - and abo.e the response of the subsystem under study+ ,hisother component is described by the bac>ground dielectric constant,bg, -hich is, therefore,equal to "6%bg p bgibg 0 0$ ( $ ( $ ( + +

*q+2-1ground solution,30bg($ and3bg($can be matched, so subtracting the bac>ground from the sample conducti.ity (as inEq.2-4$ -e get%exp 0( $ ( $ 8 ( $ 7bg HF bgi + Eq.2-14Dnli>e for the solid state samples, here "6 is equal to @bg+ 0ince both the sample andbac>ground solutions are aqueous electrolytes, then"6 and @bg are approximately equal to@"2E F !0, the dielectric constant of pure -ater+ 0o, the abo.e equation readsexp 0( $ ( $ ( $bgi *q+2-1G,a>ingthe con.ention= ? -i @??-e canseparate the real andimaginarycomponents%02m8 ( $7?( $ *q+2-1H0Ae8 ( $7??( $ *q+2-1I9ote that 3($= 3exp($- 3bg($ is calculated from the admittance &($, using Eq.1-1 and Eq.1-2, -here $ ( $ ( $ (exp bgC C C and exp( $ ( $ ( $bgG G G +0( $?( $l CS *q+2-1!0??( $( $ l GS *q+2-1B,hese equations relate the contribution of the subsystem under study to the dielectric function of the sample -ith the experimentally measured sample and bac>ground admittances, for aqueous samples and aqueous electrolyte as bac>ground+6igure 1% ,he real part of exp($ is presented by the solid line+ ,he dotted line presents the .alue of 1 - the JcontributionJ of the .acuum+ ,he contribution ($ of the subsystem understudy and the contributions "6 of other subsystems, relaxing at higher frequenciesare sho-nby .ertical arro-s+ 5ngled arro-s denote ho- the bac>ground .alue of ? can be different, depending on the system under study+3. Te fittin! procedure,he relaxation processes of .arious subsystems under study are, naturally, .ery different+ "o-e.er, these processes are linear and may be described as harmonic oscillators, -ith respecti.e equations of motion%6requency?()"6bg"2E =bg=1aqueous samplessolid state0 mx x kx + + && &*q+2-1Ba,9o-, substituting the damping constant -ith the damping time constant m/, and relating the restoring force constant 20k m -ith the natural frequency 0 of the oscillator,-e can also -rite%200mmx x m x + + && & *q+2-1Bb,0tarting from the abo.e equation, e.entually -e come to a description of the contribution to the dielectric function from a gi.en process, described either by a Korent#ian (for underdamped processes, 0LL1$% 222 001 ( $( $/ i +*q+2-20,-here system responds at its natural frequency 0, or by a ;ebye function (for o.erdamped processes, 0MM1$%01( $i +*q+2-21,-here the system responds -ith the characteristic relaxation time 0=1/(02$, at frequencies 1/0 much lo-er than the natural one 0+2n the real systems that -e study a distribution of the relaxation times appears around the central relaxation time 0+ 0uch systems are better described by a generali#ed ;ebye function -a Cole-Cole function%101 ( $( $i+*q+2-22-here the parameter N0 describes the broadness of the distribution (the larger , the broader the distribution$+ 2n all cases, describes the strength of the dielectric relaxation of the subsystem under study+2f there is more than one relaxation process, the Cole-Cole functions are used additi.ely, as -e consider the subsystems under study to be linear - for example in case of t-o relaxation processes%( ) ( )1 21 21 11 21 ( $ 1 ( $( $i i ++ + *q+2-23.1. Choosing G0 & C! - solid state samples5s sho-n in 0ec 2+1 the bac>ground subtraction does not remo.e the free-particle conducti.ity '0 from the measured conducti.ity 'exp($+ ,he .alue of '0 has to be read off from the data sets manually, in order to con.ert ' and Cp into the dielectric function, using *qs 2-10 O 2+11+ Dsually '0 is read off at the lo-est frequencies, -here the ' data sho- no significant dependence on frequency+ "o-e.er, the noise le.els are high and this ma>es the extraction of the correct .alue of '0 rather difficult+ ,he best chec> is to see if ??($ data set obtained from 00( $ G G lS(see *q2+11$ sho-s a symmetric bell-shape, in -hich case the respecti.e choice of '0 may be accepted+6or practical reasons, high frequency .alue of capacity C"6 is not directly ta>en from the data,but it is allo-ed to enter the fit as a fit parameter b+ 9o- the data sets obtained from *q2+10 and 2-11 are fit to real and imaginary componenets of the Cole-Cole form% 10 0( $1 ( $bl Ci S + ' ;+ *q+2-21010 0( $1 ( $G G li S ' ;+ *q+2-2G-here b equals l/0CC"6+ 3.2. Choosing Gcorr- aqueous samples"ere, the fitting procedure is technically similar to the pre.ious case+ "o-e.er, the subtraction of aqueous electrolyte bac>ground has already remo.ed free-carrier conducti.ity '0 and high frequency capacitance C"6 contributions to the dielectric function+ 0till, as the bac>ground electrolyte may not be chosen -ith perfectly matching admittance there -ill al-ays be a small unremo.ed part of '0 and C"6+ 5s their origin is purely technical, -e label these .ariables as'corr and Ccorr+ 2t is no- 'corr -hich has to be read off at the lo-est frequencies, -here the ' data sho- no significant dependence on frequency+ 5gain, a good choice is ensured -ith the symmetry of the bell shape of??($ data set+Dnremo.ed capacitance is -ritten as b= l/0CCcorr and enters as a free parameter+ ,he fits for aqueous samples can be made -ith *qs 2-21 and 2+2G, ho-e.er since there are usually t-o relaxation processes, -e extend these according to *q2+23.3. !itting procedure summar"0tep 1% obtaining experimental 'exp($,Cexp($ and bac>ground 'bg($,Cbg($ datasets+0tep 2% subtracting these and obtaining ',C% exp( $ ( $ ( $bgG G G $ ( $ ( $ (exp bgC C C 0tep % con.erting to { } 0/0??( $( $corrG GlS

0( $?( $ bl CS + 0tep 1% fitting to Cole-Cole function{ } 0/10 0+++1 ( $( $corrG GlS i +' ;+ 10 0( $+++1 ( $bl CS i + +' ;+ loop, a so-called Jnull-loopJ+ ,he null amplifier commonly consists of an input amplifier, a narro--band, high-gain amplifier, and an output amplifier+ 2t is important that the input amplifier has a near-infinite input resistance in order to pre.ent current flo-ing from the Ko- terminal(s$ to ground+ ,he output of null amplifier is a .oltage signal -hich maintains the .irtual ground le.el at the Ko--:otential (Kp$ terminal and helps to pull the -hole ;D, current to a range resistor+ )y detecting the .oltage of the range resistors, the current -hich flo-s through the ;D, is measured+ 2mpedance analy#ers usually ha.e se.eral range resistors in order to achie.e high resolution for .arious current measurements+ 0ince the Ko--terminal .oltage is at the .irtual ground, the .oltage drop across ;D, is detected at the "igh-:otential ("p$ terminal of the instrument relati.e to ground+ ,ypically, the measurement circuit behind the "p terminal is isolated from the "c terminal, -hich enables more accurate detection of the .oltage applied to the ;D,+,he ;D, (either the aqueous sample inside the chamber or a solid sample crystal in the cryostat$ is connected to the analy#er using the 6our-,erminal :air, 1,: configuration, preferred2 for testing samples or de.ices by instruments -hich employ auto-balancing bridge measurement circuit+ 2n this configuration the ground terminal, -hich also ser.es as a return 2 "+ "aruta, $%e 'mpedance (easurement Handbook, 2nd edition (5gilent ,echnologies, D05/Uapan, 2000$connection at ambient temperaturecoax cabling% cryostat-10 cmchamber- G cm 1 test leads (1,:$, 2 probes ": 12!15, 5gilent12B152nstrument 6ront panel )9Cs;D ,:robes% electrodes, contacts "c "p Kp Kcpath for the current, is implemented as outer conductors of four coaxial cables, the inner conductor of each corresponding to one of the "c, "p, Kp, Kc terminals+ 1,: configuration reduces the effects of lead impedances because the signal current path and the .oltage sensingcables are independent+ 5lso, since the return current fows through the outer conductor of the coaxial cables, the magnetic fux generated by the inner conductor is canceled by the fux of the outer conductor, which minimizes the error due to mutual cable inductances. At the same time, outer conductors of coaxial cables also minimize stray capacitances between cables.The keypoint in the 4T scheme is to connect the shields of coaxial cables as close as possible to the de!ice under test "t#electrodes with the sample droplet, or Au#leads of a solid sample$. %or the a&ueous samples chamber we achie!ed this by entering the coax#cables without outer insulation into the chamber body, which then ser!ed as a shield. The coresof cables were taken through the body and connected in pairs to the t#electrodes. Thus, the plates itself should be considered as the only source of the impedance, besides the sample. '(ecti!e cabling paths are only a couple of centimeters long, leading from the plates through the body and chassis to the )*+ connectors which attach the chamber directly to the ports of the instrument.,n the cryostats the coaxial cables lead down the sample holder, as close to the sample as possible - i.e. only Au#leads remain unshielded.hile applying the 4T connection scheme, the cable lengths ha!e to be kept as short as possible. This is to a!oid formation of standing wa!es at fre&uencies in and abo!e /0z range which can pre!ent the bridge circuit to balance. .e respected this re&uirement in the design of the a&ueous sample chamber. 0owe!er, for cryogenic measurements of solid samples total cable length reaches 1.23m, which reduces the upper fre&uency limit to about 4 /0z.%ormation of such standing wa!es is the main limiting factor which determines the upper fre&uency limit of 1 /0z for 0 4544A. ,n Agilent 4564A, this maximum fre&uency has been raised to 113 /0z by properly terminating each of the four coaxial terminals with a 73 V resistor at frequencies abo.e 1G 4"# (-hen no cable extension is used$+#.3.'gilent #2)#' (and$idth and '*eraging Settings4easurement functions are conductance '($ and the capacitance Cp($, -here =2, -hich combine into complex admittance &='(iCp (in.erse of complex impedance W$ + ,he respecti.e equi.alent circuit is the parallel connection of a resistor and a capacitor+ 5dmittance is sampled at 201 frequencies, i+e+ at 1 points per frequency decade for the frequency range 10"#-110 4"#+ 5 single frequency point is sampled -ith )ed by measurements of the capacitance ofthe cell containing a 100 K droplet of pure -ateri+ ,he measured, total capacitance (at 10 4"#$ -as subtracted for the empty cell capacitance and con.erted to r=I!+Bt0+H -hich is close to tabulated .alues for the dielectric constant of -ater "2E=I!+HG+ii"ere -e note that the relati.e error of the effecti.e cell constant deri.es from independent sources% pipetted .olume-0+2] and electrode separation-0+1]+ 0ince /=l210, the relati.e errors are additi.e% errY=2^0+1](0+2]=0+1]+ "o-e.er, this error is lo-er than -hat -e actually measure -hen the cell is calibrated -ith standard electrolytes+ ,he most plausible remaining source of error is the positioning of the droplet on the :t-plates, -hich are not flat and parallel to a certain degree+ 0o, a slightly .arying position of the droplet -ill lead to .ariations in the effecti.e cell constant+Kong term stability of the sample is achie.ed on the scale of up to 2 hours+ ,hat is, conducti.ity and capacitance measured for a standard electrolyte droplet stay -ithin the 2] error if the sample is >ept at 2GoC, or e.en cycled bet-een G and 10oC+Conducti.ity measurement range of this cell for aqueous solutions is 1+G - 2000 0/cm+ ,hat is, the lo-er limit is related to the conducti.ity of a droplet of pure -ater measured in the chamber+ ets/bioscience+nsf/homeiiC2C Handbook o3 C%emistr4 and 5%4sics, 87th edition, edited by ;a.id A+ Kide (,aylor O 6rancis )oo>s, 200H$+