Modeling carrier transport and electric field evolution in Gaussian disordered organicfield-effect transistorsFei Liu, Jack Lin, Takaaki Manaka, and Mitsumasa Iwamoto

Citation: Journal of Applied Physics 109, 104512 (2011); doi: 10.1063/1.3590154 View online: http://dx.doi.org/10.1063/1.3590154 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of the carrier density in disordered organics with Gaussian density of states on organic field-effecttransistors J. Appl. Phys. 115, 044507 (2014); 10.1063/1.4863180 Master equation model for Gaussian disordered organic field-effect transistors J. Appl. Phys. 114, 074502 (2013); 10.1063/1.4818497 The Maxwell-Wagner model for charge transport in ambipolar organic field-effect transistors: The role of zero-potential position Appl. Phys. Lett. 101, 243302 (2012); 10.1063/1.4771989 Carrier mobility in organic field-effect transistors J. Appl. Phys. 110, 104513 (2011); 10.1063/1.3662955 Modeling of static electrical properties in organic field-effect transistors J. Appl. Phys. 110, 014510 (2011); 10.1063/1.3602997

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Modeling carrier transport and electric field evolution in Gaussiandisordered organic field-effect transistors

Fei Liu,1,2,a) Jack Lin,3 Takaaki Manaka,3 and Mitsumasa Iwamoto3,b)

1School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China2Center for Advanced Study, Tsinghua University, Beijing, 100084, China3Department of Physical Electronics, Tokyo Institute of Technology, 2-12-1 O-okayama,Meguro-ku, Tokyo 152-8552, Japan

(Received 18 January 2011; accepted 12 April 2011; published online 31 May 2011)

Stimulated by the time resolved microscopic optical second-harmonic generation (TRM-SHG)

experiment, we model the carriers transport and electric field evolution in the channel of

three-dimensional Gaussian disordered organic field-effect transistors (OFETs) by the coupled

time-dependent master equation and Poisson equation. We show that this model with the

Miller-Abrahams rate can satisfactorily account for the experimental observations that include

different profiles of the electric field in the channel and the diffusionlike migration of the field

peaks with respect to time. Particularly, we find that the dynamic mobility proposed by us earlier is

distinct from the standard one in the presence of a typical disorder, which is attributed to the

uncompleted energy relaxation of the carriers in the transport process. VC 2011 American Instituteof Physics. [doi:10.1063/1.3590154]

I. INTRODUCTION

In the past two decades, organic field effect transistors

(OFET)1 have attracted intensive interest due to their great

potential in practical applications. They are considered to be

competitive candidates for existing thin film transistor appli-

cations requiring large area coverage.2 Besides technical

advantages, these organic semiconductor devices are also

very intriguing from the academic viewpoint, from which we

can explore fundamental carrier transport mechanism in or-

ganic materials. For instance, on the basis of transfer charac-

teristics of OFET, one can extract important carrier mobility

or density of trapped states.3,4

On the experimental side, there are few approaches that

can investigate carrier transport in the OFETs. Measuring the

transfer characteristics at steady-state is used most frequently

in laboratories. Very recently, several groups including ours

are devoted to developing transient approaches.5–8 Different

from the others, which mainly extended the time-of-flight

(TOF) technique9 into OFET structures, we developed a time-

resolved microscopic optical second-harmonic generation

(TRM-SHG) technique8 to directly visualize the SHG inten-

sity evolution in the OFETs’ channel by a two-photon process.

Because the local SHG intensity is almost proportional to

square of the local electric field and the latter is determined by

the carrier distribution, this technique possesses distinctive

advantages in exploring carrier dynamics in the OFETs.

The TRM-SHG experiment revealed many interesting

observations10 about the electric field along the channel

direction of the pentacene OFETs. Particularly, two of them

attract our interest. The first is that the peak position of the

field migrates in a diffusionlike way, i.e., the square of the

position is proportional to time.10 The second is that the field

profiles are very distinct for the devices with different insula-

tors. In the device with polymethyl methacrylate (PMMA)

insulator, there is a sharp peak at the front edge of the field,

while the profiles in the device with SiO2 insulator are broad

and change smoothly;10 see Fig. 1(a). It is worthy pointing

out that the time required for the front edge of the field to

reach a certain position in the device with PMMA insulator

is much larger than that in the device with SiO2 insulator,

though the local electric field of the former is stronger than

the field of the latter.

The Gaussian disorder model (GDM) credited to Bassler

et al.11 has achieved considerable success in understanding

carriers behaviors in disordered organic solid materials. We

naturally ask whether the same model can account for these

two observations in the OFETs. We must emphasize that the

GDM was usually studied in simple diode structures.11,12 To

our best knowledge, the GDM combined with relatively

complicated OFET structures was fewer, particularly as we

are concerned about the transient behavior. In this work, we

develop the coupled time-dependent master equation and

Poisson equation to numerically solve the carriers transport

and electric field evolution in a three-dimensional (3D)

Gaussian disordered OFET. Our aims are to check whether

the GDM model can self-consistently explain the TRM-SHG

observations, including the various electric field profiles and

diffusionlike migration of the field peaks in the channel, and

whether the dynamic mobility we proposed earlier10 agrees

with the common mobility.

II. THREE-DIMENSIONAL MODEL OF THEDISORDERED OFET

The TRM-SHG experiment is not repeated here. The

details may be found in our previous work.8,10 We model the

OFET as a 3D cubic Nx � Ny � Nz lattice with lattice dis-

tance a. The upper part of the device is the organic layer in

which the carriers are transported. The lattice site is denoted

a)Author to whom correspondence should be addressed. Electronic mail:

liufei@mail.tsinghua.edu.cn.b)Electronic mail: iwamoto@pe.titech.ac.jp.

0021-8979/2011/109(10)/104512/4/$30.00 VC 2011 American Institute of Physics109, 104512-1

JOURNAL OF APPLIED PHYSICS 109, 104512 (2011)

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by a vector n ¼ ðnx; ny; nzÞ. Only hole carriers are injected

from the source electrode, and the drain electrode is

neglected. This simplification is rational, because the carriers

do not reach at the drain electrode in the time interval of the

TRM-SHG experiment. We model the source electrode as a

sheet on the surface of the organic layer, the left and right

edges of which are at positions ðnls; Ny; nzÞ and ðnr

s; Ny; nzÞ,nz¼ 1, …, Nz. To account for the fact that the sizes of the

real devices along z-direction are far larger than the other

two dimensions, a periodic boundary condition is imposed

along the z-direction. Figure 1(b) is a schematic diagram of

the xy-cross section of the device used in calculation.

We assume carriers transport in the disordered organic

layer by nearest-neighbor hopping by thermally assisted tun-

neling11 and the hopping rate from site n to m has the

Miller-Abrahams form13

knm ¼ k0 exp½�ðEn � Em þ jEn � EmjÞ=2kBT�; (1)

where the prefactor k0 is a constant with dimension of

inverse of time, kB is the Boltzmann constant, T is the tem-

perature, and En is the overall energy including the intrinsic

energy en and electrostatic energy wðnÞ on site n; the former

is randomly drawn from a Gaussian density function with

variance r. In this work we do not take positional disorder

into account for simplicity. Additionally, we also assume the

hopping rates from the electrode to organic sites and vice

versa are given by the same expression as the rates between

organic sites. The carrier transport and electric field evolu-

tion are obtained by numerically solving the coupled 3D

time-dependent master equation and 2D Poisson equation

dpn

dt¼X

n6¼m

½knmð1� pnÞpm � kmnð1� pmÞpn�; (2)

ð@2x þ @2

y ÞwðnÞ ¼e

e0er

1

Nza3

X

mz¼nz

pm; (3)

where pn is the occupational probability of site n, e is the

hole charge, e0er is the permittivity of the materials, and the

sum in Eq. (3) is overall sites m with the same components

nz, which indicates that we approximate the carrier density

along z-direction to be uniform when evaluating the electro-

static potential. These two equations can be numerically

solved by iterative method.14

The parameters are as follows: a¼ 1.6 nm, Nx¼ 401,

Ny¼ 26, Nz¼ 40, nls ¼ 51, nr

s ¼ 61, T¼ 300 K, r¼ 0.05 as

weak disorder and 0.1 eV as typical disorder. We have no

special intension to select these lattice parameters except for

computational convenience. As in the real TRM-SHG experi-

ment, the gate-source voltage Vgs is stepped from zero to –2 V

at initial time in calculation. We also set the injection barrier

to be zero10 to mimic the gold electrode in practice. We must

admit that the calculated device size is far smaller than that of

the real device. However, because the magnitude of calculated

electric field is close to the real case, we do not think that

increasing size would drastically change our results.

III. RESULT AND DISCUSSION

A. The profiles of the electric field

We first examine the profiles of the calculated electric

field along the x-direction Ex in the device’s channel given

two disorder configurations with the weak and typical disor-

ders mentioned above; see the red thin lines in Fig. 2. The

times selected are those when these profiles’ positions are

around 190 nm. We see that, in the weak disorder case the

profile is relatively smooth and broad, while in the typical

disorder case an obvious peak is present. Additionally, we

also note that the spreading velocity of the former is faster

than that of the latter by 7 times. Because in both cases the

device structures and applied gate voltages are the same, the

field differences are of course attributed to the underlying

different carrier occupational probabilities; see the red thin

lines in Figs. 2(c) and 2(d). One may find that, although both

probabilities vary dramatically along the channel, the fluctu-

ation amplitude in the typical disorder case is obviously

FIG. 2. (Color online) The calculated electric field profiles (a and b) and

carriers occupational probability (c and d) near the interface between the in-

sulator and organic layer at different times in unit t0¼ k�10 . The red thin lines

and the black solid lines are from typical disorder configurations and aver-

ages, respectively.

FIG. 1. (a) The profiles of the electric field along the channel direction

measured in two typical OFETs with PMMA (dash line) and SiO2 (solid

line) insulators. The zero point indicates the edge position of the source elec-

trode. (b) A schematic diagram of xy-cross section of the device used in calcu-

lation. The dashed region represents the insulator substrate. The gray circles

represent hole carriers. Vs and Vg are the source and gate voltages, respec-

tively. The relative dielectric constants for different parts are also indicated.

104512-2 Liu et al. J. Appl. Phys. 109, 104512 (2011)

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larger than that in the weak disorder case. Particularly, this

fluctuation induces a sharp boundary near the front edge of

nonzero occupational probability in the former, which is re-

sponsible for the remarkable peaks. In turn, these peak posi-

tions also precisely indicate the boundaries of the injected

carriers. In the weak disorder case, on the contrary, such a

boundary is very ambiguous. We must point out that these

characteristics remain true even after an average over the

disorder configurations; see the black solid lines in Fig. 2.

Compared with Fig. 1(a), we may regard the pentacene

OFETs with SiO2 and PMMA insulators as the weak and

typical disorder devices, respectively.

The above findings could be qualitatively understood by

the GDM. The possibility of a site having a deeper intrinsic

energy is higher for a device with larger disorder, e.g., the

typical disorder here. The carriers in such sites are more dif-

ficult to hop into neighbor sites in the time interval of our

calculation or the TRM-SHG experiment. Namely, they are

the “traps” in carrier transport.10 Because all sites are identi-

cal, a sheet of trapped charges with nearly uniform density is

formed as the injected carriers move along the channel. They

induce the peaks in the field profiles or SHG intensity.

B. Peak positions versus time

We now study the migration of the peak position �xðtÞ of

the calculated electric field. It is interesting to check whether

they follow simple power functions of time t (/ ta) as pre-

sented in the experiment. The results are shown in Figs. 3(a)

and 3(b) for the respective disorders. To compare with the ex-

perimental data, we scale these times and positions by the last

instant times and corresponding positions in each data sets,

respectively. Because the ultimate aim of this study is to cal-

culate the mobility, we also present the scaled field at the peak

position as function of time in Figs. 3(c) and 3(d). Noted that

we do not carry out an average over disorder configurations;

these data are not sensitive to the detailed configurations.

The first impression is that power functions can indeed

describe the peak position and the corresponding field

(/ t�b) as functions of time. Regardless of the disordered

extent, both exponents for �x are about 0.5, i.e., diffusionlike

as we mentioned at the beginning. However, the exponents

for the field are sensitive to the concrete value of r. b in the

weakly disordered device is –0.42, while in the typical disor-

der device it is only –0.27. Additionally, we also note that

aþb ’ 1 in Figs. 3(a) and 3(c). This identity indicates that

the decay of the electric field at the peak position accompa-

nies a slow-down of the peak’s migration, and especially

they are linearly dependent. In a physical aspect, such a sit-

uation is analogous to the motion of single carriers with con-

stant mobility under a time-variable electric field.

Intriguingly, this simple result is not available in the device

with the typical disorder, because the slower decay of the

field does not results in a faster migration of the peak as

expected in the weak disorder case; see Figs. 3(b) and 3(d).

Hence, we get a conclusion from the transient aspect that sig-

nificant energetic disorder obstructs the formation of con-

ducting channel in the OFETs. According to the previous

classification, we plot the scaled experimental data from the

devices with SiO2 and PMMA insulators as the weak and

typical disorder cases, respectively; see the crosses in Fig. 3.

We see that they agree well.

C. Dynamic mobility

We have proposed a dynamic mobility on the basis of

the TRM-SHG experimental data:10

l ¼ 1

Ex½�xðtÞ�d�xðtÞ

dt’ a

Ex½�xðtÞ��xðtÞ

t: (4)

The last identity uses the diffusionlike migration of the peak

position. The current model provides us an opportunity to

check whether it is consistent with the conventional mobility

that is calculated by Monte Carlo simulation11 or steady-

state master equation method in a diode structure.15 The

results are shown in Fig. 4(a). We immediately see that, the

dynamic mobility is inconsistent with the other two mobili-

ties given the same carrier density and applied electric field,

particularly in the typical disorder case. Interestingly, we

also find that over larger electric field range, the dynamic

mobility shows a certain Poole-Frenkel-like16 behavior, i.e.,

l / exp½cffiffiffiffiffiEx

p�), where c is a constant with a dimension of

the inverse of square root of electric field. When we apply

Eq. (4) to the experimental data of a device with PMMA in-

sulator obtained under three different gate voltages, similar

behavior is also presented; see Fig. 4(b).

A very plausible cause of this discrepancy between the

dynamic mobility and the standard one is that the former is

based on the transient motion of the carriers, of which

dynamic equilibrium or relaxation process in energy space is

not established or completed in the time interval of our cal-

culation or the TRM-SHG experiment. Two evidences in

Fig. 4(a) support this conjecture. First, as time increases

accompanying a decay of the electric field, the dynamic mo-

bility decreases fast toward the standard mobility. Second, in

FIG. 3. Scaled peak positions and field at the same positions versus scaled

time in log10 – log10 plot. The open symbols are from calculation, and the

crosses are from the TRM-SHG experiment on the OFETs with PMMA and

SiO2 insulators. The numbers in these figures are the slopes of the linear fits.

104512-3 Liu et al. J. Appl. Phys. 109, 104512 (2011)

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the weak disorder device the dynamic mobility is very close

to the standard one. In fact, previous work11 has revealed

that, in the absence of carrier-carrier interaction and electric

field, the carrier relaxation time with the typical disorder is

about 108 t0, which is 6 orders of magnitude larger than that

of the weak disorder and is also beyond our whole calcula-

tion time. Although this result is not precise under current

circumstance with high field and high carrier density, it

roughly provides us some cues about this discrepancy. Of

course, further quantitative analysis or calculation are needed

to give a definite answer. On the other hand, if the dynamic

mobility is a transient consequence, one might be surprised

why it is still Poole-Frenkel-like, which was only reported in

steady-state mobility measurement or calculation.11 We think

that this observation is only superficial. Considering that �x and

Exð�xÞ are power functions of time, Eq. (4) can be rewritten as

ln l ’ 2ð1� a� bÞb

lnffiffiffiffiffiEx

pþ C; (5)

where C is a constant. Hence, ln l is seemingly a linear function

offfiffiffiffiffiEx

pdue to the logarithm function and we may regard the

slope of Eq. (5) as c. Substituting the values of a and b in Figs.

3(b) and 3(d), and the typical field strength 2:5� 105 V=cm in

Fig. 4(a), we obtain c � 4 � 10�3 ðV=cmÞ�1=2. Analogously,

one can calculate c in Fig. 4(b) is its half since the typical experi-

mental field strength therein is four time larger than the field

strength calculated.

IV. SUMMARY

In this work, we use the coupled time-dependent master

equation and Poisson equation to evaluate the carrier trans-

port and electric field evolution in the channel of three-

dimensional Gaussian disordered OFETs. Our calculation

shows that the GDM combining with the OFET structure can

well account for the observations in the TRM-SHG experi-

ment. In addition, we also numerically demonstrate that the

dynamic mobility does not equal to the common steady-state

mobility in the presence of significant disorder. This finding

reminds us that the conventional drift-diffusion equation

using the standard mobility formula is unsuitable in studying

the transient responses of disordered devices.

ACKNOWLEDGMENTS

We thank Dr. Martin Weis for generously providing us

Fig. 1(a). F.L. is supported by the JSPS’s Fellowship Program

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FIG. 4. (Color online) (a) Comparison between the dynamic and standard mobilities as functions of square root of the electric field: The open symbols are

obtained by applying Eq. (3) to our calculated data in Fig. (3), the crosses are the results of the Monte Carlo simulation programed by ourselves, and the black

solid lines are the mobility evaluated by the parametrization scheme proposed by Pasveer et al. (see Refs. 15 and 17). The carrier densities of the latter two

cases are 0.01/site, because the calculation shows that the occupational probabilities at the peak position near the interface remains around this number. (b)

The dynamic mobility as function of square root of the electric field obtained from a real device with PMMA insulator, the field-effect mobility of which is

0.01 cm2/Vs. These data were collected from the TRM-SHG experiment conducted under three gate voltages (see Ref. 18). Notice that the dimension of the pa-

rameter c is ðV=cmÞ�1=2.

104512-4 Liu et al. J. Appl. Phys. 109, 104512 (2011)

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