Modeling the negative capacitance effect in dispersive organic materials using modified Drude theory

Citation for this paper:

Wu, Y., Lin, J. & Kwok, H.L. (2019). Modeling the negative capacitance effect in

dispersive organic materials using modified Drude theory. Solid State Electronics

Letters, 1(2), 105-109. https://doi.org/10.1016/j.ssel.2020.01.005

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Modeling the negative capacitance effect in dispersive organic materials using

modified Drude theory

You-Lin Wu, Jing-Jenn Lin, H.L. Kwok

July 2019

©2020 Published by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This

is an open access article under the CC BY-NC-ND license.

(

https://creativecommons.org/licenses/by-nc-nd/4.0/

).

This article was originally published at:

You-Lin

Wu

,

Jing-Jenn

Lin

,

H.L.

Kwok

a Department of Electrical Engineering, National Chi Nan University, 301 University Rd., Puli, Nantou, Taiwan b Department of Applied Materials and Optoelectronic Engineering, National Chi Nan University, Puli, Nantou, Taiwan c Department of Electrical and Computer Engineering, University of Victoria, Canada

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 7 November 2019 Revised 20 January 2020 Accepted 21 January 2020 Available online 13 February 2020 Keywords:

Complex carrier mobility Dispersive polymers Drude theory Negative capacitance

Organic polymer light-emitting diodes

a

b

s

t

r

a

c

t

Frequency- andmobility-dependent admittance have beenobserved inorganicpolymerlight-emitting diodes.Inthispaper,wedevelopedamodeltodescribethisdispersivebehaviorusingamodifiedDrude theory.Inthismodel,aphaseangledifferencebetweentheappliedelectric fieldandtheaverage dis-placementofthechargecarriersisintroducedratherthanusingacomplexmobility.Thisnewlyproposed modelsuccessfullydescribesthedispersivenature,aswellasthenegativecapacitanceeffect,atlow fre-quenciesinorganicpolymers.Thesimulationresultsofthismodelalsofitthenegativecapacitancedata reportedintheliterature,providedthatasuitablephaseangledifferenceisgiven.

© 2020PublishedbyElsevierB.V.onbehalfofKeAiCommunicationsCo.,Ltd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

It hasbeenreportedthat theadmittance oforganicpolymer light-emittingdiodes isfrequency-andcarriermobility-dependent,and becomesnegativeundercertainbiasedconditions[1,2].This dispersiveproperty hasbeenattributedto carriertrapping andhoppingin thepolymerlayer[1,3–5].BasedonDrudetheory[6,7],oneofourauthors,H.L.Kwok,proposedamodeltodescribethenegative capac-itance effectfoundindispersiveorganicpolymers[8,9].Inhis model,complexcarriermobilityisassumedtoaccount forthe dispersive behavior ofthe medium.The modelhas successfullydescribedthefrequency-dependentcapacitance andnegativecapacitance effectof organicpolymers.Inaddition,thesimulationresultsofKwok’smodelareingoodagreementwiththeexperimentaldatareportedinthe extant literature [1].However, the physicalconcept of complexcarriermobility isnot simple to understand.In thiswork, we propose analternativemodel,alsobasedonDrudetheorybutwithrealcarriermobility.Insteadofusingcomplexcarriermobility,aphaseangle difference betweentheapplied electricfield andtheaveragedisplacementofchargecarriersisintroduced toaccount forthedispersive natureofthemediuminournewlyproposedmodel.

2. Themodel

Here,wetake theorganiclayerasaplasmamedium,inwhichacollectionofpositive andnegativechargesexists.Thesechargesare assumedtooscillatewithaHook’slawforceconstantK0,andanacelectricfieldwithanangularfrequency

ω

,E=E0ejωt,isapplied.From

theDrudetheory,wecanobtainthefollowingequation(byconsideringpositivechargeonlyforconvenience)[8,9]:

md 2_x dt2 + q

μ

dx dt +K0x=qE (1)

where m isthe massof thepositive charges; xis the averageposition ofthe charges;q is the electronic charge;and

μ

isthe carrier mobility.Notethat,inKwok’smodel,acomplexparameter

γ

isusedtorepresentthefrequencyofoccurrenceofinelasticcollisionsper

∗ _{Corresponding author.}

E-mail address: ylwu@ncnu.edu.tw (Y.-L. Wu).

https://doi.org/10.1016/j.ssel.2020.01.005

2589-2088/© 2020 Published by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

106 Y.-L. Wu, J.-J. Lin and H.L. Kwok / Solid State Electronics Letters 1 (2019) 105–109

Fig. 1. Simulated capacitance versus frequency curves under various phase angle difference ( θ= 0–180 °) for mobility μequals to (a) 10 −11 , (b) 10 −10 , and (c) 10 −9 m 2 V −1 s −1 . unitcharge fordispersivemedium,andequals tothe reciprocalofthe carriermobility

μ

,i.e.,

γ

=1/

μ

=

γ

1 +j

γ

2 [8,9].However, the

physicalconceptofcomplexmobilityisnotsimpletocomprehend.Inourmodel,weintroducedaphasedifference

θ

betweentheapplied electricfieldandtheaveragepositionofthechargesx,ratherthanutilizingcomplexmobilitytoaccountforthedispersivenatureofthe medium.Therefore,thesolutiontoEq.(1)canbe expressedasx=xo ej(ωt+θ).Thephase angledifference indicateshowfastthecharge

carrierscanfollowthevariationoftheappliedfield.BysubstitutingE=E0ejωtandx=xoej(ωt+θ)intoEq.(1),onecaneasilyfindthat:

x0=

(

qE0/m

)

e− jθ



(

ω

2 0−

ω

2

)

+j

ω

q/

μ

m



(2)

where

ω

0=K0/misthecharacteristicfrequencyoftheplasma.ThepolarizationvectorPcanthenbeexpressedasfollows:

P=pqx0=

(

pq2_E 0/m

)

e− jθ



(

ω

2 0−

ω

2

)

+j

ω

q/

μ

m



(3)

Assumeaparallelplatecapacitorofarea AandplateseparationLhavingtheorganicdispersivemediumasthedielectric.Capacitanceis thengivenby:

C=Re



(

_EP 0

)(

A L

)



=Re



(

pq2_A/Lm

₎

_e− jθ



(

ω

2 0−

ω

2

)

+j

ω

q/

μ

m





=Re



C0e− jθ [

(

1−

ξ

2

)

_+j

ξ

_q/

μω

0m]

=Re



C0

(

cos

θ

− jsin

θ

)

[

(

1−

ξ

2

)

_+j

ξ

_q/

μω

0m]

(4)

Fig. 2. Calculated capacitance versus frequency curves with carrier mobility varying from 10 −8 to 10 −12 m 2 V −1 s −1 for phase angle difference (a) _{θ= 60 °, (b) θ= 90 °, and}

(c) θ= 120 °.

whereC0=pq2A/Lmand

ξ

=

ω

/

ω

0.Notethatthemobilityhereisreal.FromEq.(4),wecaneasilyfindthat:

C=C0



(

1−

ξ

2

₎

_cos

θ

₋

₍

ξ

_q/

μ

_m

ω

0

)

sin

θ

(

1−

ξ

2

)

2₊

(

ξ

_q/

μ

_m

ω

0

)

2

(5)

ItisobviousthatCbecomesnegativewhen:

(

1−

ξ

2

₎

_cos

_θ

_<

₍

_ξ

_q/

_μ

_m

_ω

0

)

sin

θ

(6) or

θ

>tan−1

(

1−

ξ

2

)

ξ

q/

μ

m

ω

0 (7)

Forthecaseofnon-dispersivemedium,

θ

=0,andthecapacitancegiveninEq.(5)reducesto:

C=C0



(

1−

ξ

2

₎

(

1−

ξ

2

)

2₊

(

ξ

_q/

μ

_m

ω

0

)

2

(8)

whichisexactlythesameastheonegiveninRefs.[8]and[9].

3. Resultsanddiscussion

ByusingEq.(5)andthesameparameter valuesgiveninreferences[8]and[9],i.e., A/L=100, p= 1× 1021 _(m−3_),

_ω

_{0 = 1× 10}12 (rads−1),m=0.91× 10−31_{(kg),andq=1.6× 10}−19_{(C),wecalculatedthecapacitanceoftheorganicmediumwithmobility}

_μ

₌₁₀−11_, 10−10,and10−9 (m2 _V−1 _s−1_{)forvarious phaseangle differencesandforthefrequencyrangingfrom10 to120,000(rad/s). Ingeneral,} thecarriermobilityfororganicpolymersisintherangeof10−8–10−12m2 _V−1 _s−1 _{[10,11].Thecalculated capacitanceversus frequency} curvesare showninFig. 1(a),(b)and(c). As observed,the capacitanceis afunction ofcarriermobility, frequencyoftheapplied field,

108 Y.-L. Wu, J.-J. Lin and H.L. Kwok / Solid State Electronics Letters 1 (2019) 105–109

Fig. 3. Comparison of the simulated capacitance data at low frequencies using the proposed model and the reported experimental data given in Ref. [1] . The parameter values for the simulation are presented in Table 1 .

Table 1

Parameter values used for the calculation of Fig. 1.

ω (rad s −1 ) _{P (m} −3 ) _{ω0 (rad s} −1 ) _{θ( °) μ(m} 2 V −1 s −1 ) 30 4.50 × 10 21 _{2.30 × 10} 21 _{91.67 3.00 × 10} −13 100 4.50 × 10 21 _{2.30 × 10} 21 _{95.68 3.35 × 10} −13 300 4.50 × 10 21 _{2.30 × 10} 21 _{106.00 3.00 × 10} −13 1000 4.00 × 10 21 _{2.30 × 10} 21 _{111.00 2.30 × 10} −13 Table 2

Comparison of the negative capacitance of the reported experimental data shown in Ref. [1] , the previous data from Kwok’s model given in Ref. [9] , and the calculation results using the model proposed in this work. The numbers shown in the parentheses are the percentage error compared with the reported data.

ω (rad s −1 ) _{C (nF)}

Reported [1] Kwok’s Model [9] This work

0 −66.6 −66.9 (0.45%) −66.6 (0%)

100 −28.8 −23.0 (20.13%) −24.0 (16.67%)

300 −69.8 −65.0 (6.88%) −69.7 (0.14%)

1000 −14.8 −14.0 (5.41%) −14.1 (4.73%)

andphaseangledifference.Forthecaseofnon-dispersivemedium,i.e.,

θ

=0,weseethatthecapacitancesaturatesatlowfrequencies, andthesaturationvaluesbecomehigherasthecarriermobilityincreases.Itisreadilyunderstoodthathighercarriermobilitycausesless carriercollisionortrapping,andthusthecapacitanceishigher.Itisalsonoticedthat,exceptfor

θ

₌0,thecapacitancebecomesnegative afteracriticalfrequencyisreached.Thiscriticalfrequencydecreases withincreasingphase angledifference whenthemobilityisfixed, whileitincreaseswithincreasingcarriermobilitywhenthephaseangledifferenceisfixed.Foradispersivemediumwithfixedmobility, ahigher

θ

valuemeanshigherpolarizationcharge,andalowerfrequencyisneededtoyieldthesamecapacitance. Whenthe

θ

valueis fixed,inamediumwithhighercarriermobility,the polarizationcharge moreeasilyfollowsthechangesoftheapplied field,andhence negativecapacitance occursathigherfrequency.Moreimportantly,itisfound thatthecapacitance becomesall negativewhen

θ

> 90°. Thesebehaviorsarebasicallyinagreementwiththenatureofthedispersivemedium.Alloftheargumentsdiscussedabovecanbemore clearlyseen inFig.2,whichshowsthecapacitanceversus frequencycurvesundervariouscarriermobilitiesfor

θ

= 60°,90°,and120°. Wecomparethecalculationresultsofourmodelandtheexperimentaldataofthenegativecapacitancereportedinreference[1]toverify theproposedmodel,asshowninFig.3.ThevaluesoftheparametersusedforthecalculationaregiveninTable1.Itisobviousthatthe calculationresultsagreewell withtheexperimental data.As observedinTable 1, the

θ

valuesincrease withincreasingfrequency. This isbecausehigherpolarization chargesare neededwhenthefrequencyis higher.Inaddition,the valuesoftheparameters usedforthe calculationareallreasonable.Table2comparesthenegativecapacitancevaluesobtainedfromthereportedexperimentaldatagiveninRef.

[1],thecalculateddatafromKwok’spreviousmodel[9],andthesimulationresultsofthepresentwork.Wealsospecifiedinparentheses thepercentageerrorofKwok’smodelandourmodelcomparedwiththereportedexperimentaldata.Itcanbeseenthatourmodelgives smallerdeviationerrorthandoesthepreviousKwok’smodel.Forconjugatepolymers,ithasbeenshownthatcarriermobilityvariesina field-dependentmannerandcanbewrittenas

μ

₌

μ0

(₋

β

E1/2_),where

μ

0isthezerofieldmobility,

β

isthefieldactivationfactor,andE istheappliedfield[2,12].Weutilizethiscorrelationbetweentheappliedfield(voltage)andthecarriermobilityinEq.(5)tocalculatethe capacitanceofthedispersivemedium.Fig.4comparesthecalculationresultswiththeexperimentalnegativecapacitancedatareportedin Ref.[2]atlowfrequencies.Inthecalculation,thefollowingparametersareused:

μ

0=10−10m2V−1s−1;A/L=8;andp=2× 1020cm−3. FromFig.4,itcanbeseenthat ourcalculationresultsagreewellwiththereporteddata,exceptfortheonewithapplied voltageof3V at

ω

=16rads−1.Thisdiscrepancyisbelievedtobeduetothefluctuationintheexperimentaldataatextremelylowfrequencies[2].

Fig. 4. Comparison of the simulated negative capacitance under different applied voltages (2 V and 3 V) using the proposed model and the reported experimental data given in Ref. [2] .

4. Conclusion

Inthiswork,we havedevelopedamodelbasedonDrude theorytodescribe negativecapacitance indispersiveorganicpolymers.In themodel,weusedrealcarriermobility,andintroducedaphaseangledifferencebetweentheaveragepositionofthechargesandapplied anelectricfield,insteadofutilizingcomplexmobility.Whenasuitablephaseangledifferenceisgiven,ourmodelcanaccuratelydescribe thedispersivenature,aswellasthenegativecapacitanceeffect,oforganicpolymerswithagoodmatchinkeyparameters,suchascarrier mobility,carrier concentration,etc. Moreimportantly, theconcept ofphase angledifference used inthismodelprovides a moredirect physicalinsightintonegativecapacitancethantheoneusingcomplexmobility.

DeclarationofCompetingInterest None.

CRediTauthorshipcontributionstatement

You-LinWu:Conceptualization,Methodology,Fundingacquisition,Writing-originaldraft.Jing-JennLin:Datacuration,Visualization, Formalanalysis.H.L.Kwok:Supervision,Investigation,Resources.

Acknowledgment

TheauthorswouldliketoexpresstheirappreciationforfinancialsupportfromtheMinistryofScienceandTechnology,Taiwan,R.O.C. undercontractno.MOST107-2221-E-260-013.

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