Scaling analysis of field-enhanced bandtail hopping transport in amorphous carbon nitride

phys. stat. sol. (b) 244, No. 6, 2081–2099 (2007) / DOI 10.1002/pssb.200642043

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

Scaling analysis of field-enhanced bandtail hopping transport

in amorphous carbon nitride

C. Godet*, 1, J. P. Kleider 2, and A. S. Gudovskikh2

1 Laboratoire de Physique des Interfaces et des Couches Minces,

UMR 7647 CNRS – Ecole Polytechnique, 91128 Palaiseau Cedex, France 2 Laboratoire de Génie Electrique de Paris, CNRS, UMR 8507; Supélec; Université Paris-Sud;

Université Pierre et Marie Curie – Paris 6; 11 rue Joliot-Curie, Plateau de Moulon,

91192 Gif-sur-Yvette Cedex, France

Received 25 January 2006, revised 14 December 2006, accepted 3 January 2007

Published online 16 February 2007

PACS 72.20.Ee, 73.50.Bk, 73.50.Dn, 73.50.Fq, 73.61.Jc

Hopping transport within a bandtail distribution of localized electronic states has been investigated in

amorphous carbon nitride (a-C1–xNx :H, x = 0.23) as a function of temperature T and electric field F. The

conductivity σ follows Mott’s law in the ohmic regime, i.e. ln (σohmic

) varies linearly with T –1/4, while at

higher field, a scaling law, ln (σ/σohmic

) = ϕ [FS/T] with S = 0.67 (±0.05), is found. Data are fully consistent

with a field-enhanced bandtail hopping (FBTH) model in which the effective temperature concept de-

scribes the non-equilibrium occupation probability of tail states. A “filling rate” method, considering for-

ward non-activated hopping transitions, is developed to analyze the high field regime of FBTH. For an

exponential distribution with disorder energy E0, increasing F shifts the transport energy EDL

towards shal-

lower tail states, with a density of states N(EDL

) ∼ (F)3 (E0)–4. In this model, FBTH is parametrized using

ln σ(T, F) vs T –1/4 plots, which provide field-dependent apparent values of prefactor (ln σ 00) and slope

(T0

1/4). As F increases, both parameters strongly decrease. This behavior (observed in a-C1–xNx :H,

x = 0.23, for F > 5 × 104 V cm–1) is a signature of band tail hopping transport. Our FBTH model predicts a

minimum value 00

minσ of σ

00(F), which is indeed observed in a-C1–xNx :H (x = 0.23) for T < 70 K

( 00

minσ ≈ 10–6 S cm–1 at F

min ≈ 3 × 105 V cm–1). Near Fmin, kT*

eff = (1/kT

eff – 1/E 0)–1 is parametrized by

kT*eff

∼ F q. The value of q that best reproduces the experimental results, q = 0.7 ± 0.1, is consistent with

the scaling exponent S = 0.67 and with the density of states parameters deduced from the Ohmic regime.

Hence, this “filling rate” method applied to FBTH transport appears to be very useful to analyze the ap-

parent prefactor σ 00(F) and to derive the effective temperature Teff

(T, F) which governs bandtail states oc-

cupation and FBTH conductivity.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Super-linear current voltage characteristics have frequently been observed in disordered semiconductors.

Non-linear transport phenomena are generally analyzed using the dependence of the current density J on

the electric field strength F as ln (J/F) ≈ F m, with m ≈ 1/2 being the most commonly found value [1–3].

However, in some cases, the Poole–Frenkel mechanism is not consistent with the calculated number of

charged traps or independently measured dielectric constant values.

A general alternative explanation for field-enhanced conductivity (FEC) is closely related to the exis-

tence of band tail states. Within a strong energy dependent distribution, increasing the electric

field aligns an increasing number of empty and accessible states to the occupied states which allows the

* Corresponding author: e-mail: [email protected]

2082 C. Godet et al.: Scaling analysis of field-enhanced bandtail hopping transport

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

charge carriers to proceed via phonon-assisted tunneling (hopping) transitions (Fig. 1) [4]. The energy

distribution of localized electronic states, usually described by exponential [5] or Gaussian [6] disorder,

is thus crucial in the modeling of FEC data.

Following early predictions by Mott [7] and Shklovskii [8] on field-enhanced hopping transport, a

number of models have been proposed to describe both steady-state and transient FEC in different den-

sity of states (DOS) distributions. A satisfactory agreement has been obtained using either energy de-

pendent mobility µ(E) [9, 10] or percolation theory [5, 11–15], giving an activationless F –1/4 dependence

of ln σ of the form

ln σ = ln σ* – (F0/F)1/4 , (1)

in the very high field regime [8, 9, 13]. However, in the intermediate to strong field regime, some discrep-

ancies appear in the field exponent [11, 12]. Furthermore, some fundamental problems still remain to be

solved, such as a reasonable approximation of non-equilibrium occupation of localized states, at high fields.

In this work, a combination of experimental σ(T, F) data obtained in a-C1–xNx :H (x = 0.23) films and

modeling results is discussed. Our modeling uses an extension of the “filling rate method” which was

previously applied only to the Ohmic regime [16, 17]. We show that some scaling properties are ex-

pected for field-assisted bandtail hopping (FBTH) within exponential distributions of localized states,

which can be used as a signature of bandtail hopping. Using the effective temperature concept to de-

scribe the tail states occupation probability function, it is suggested that Teff can be estimated from the

experimental field dependence of the apparent slope and prefactor values obtained from ln σ(T, F) vs

T –1/4 plots.

The background of this work, namely the basis of the “filling rate method” along with the need for

further modelling, is detailed in Section 2. Experimental details are given in Section 3. In Section 4, we

describe the dependence of the conductivity σ(T, F) in a-C1–xNx :H with special emphasis on the high

F – low T regime. The results are analysed and discussed in detail in Section 5 in order to clarify the

origin of the observed scaling properties. Finally, Section 6 provides a summary of this work along with

the main conclusions.

2 Background and need for further modeling

For Variable Range Hopping (VRH) in a three-dimensional (3D) space (d = 3), the linear (Ohmic) re-

gime is described by a T –1/(d+1) Mott law:

ln σ = ln σ 00 – (T0 /T)1/4 . (2)

Ene

rgy

Distance R

EF

e F Rh

Ei

Ej

ln N(E )

FIELDENERGY LOSS

ENERGY GAINNON ACTIVATED

ENERGY GAINACTIVATED

Ri

Rj

Fig. 1 Schematic representation of a field-assisted forward (against the field) hopping transition within

an exponential band tail distribution N(E). Due to the large electric field, a higher concentration of shal-

lower empty localized states is available at site (j) for an electron initially sitting at site (i).

phys. stat. sol. (b) 244, No. 6 (2007) 2083

www.pss-b.com © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

The latter expression was established for a uniform energy distribution of electronic states, which

leads to a dependence of the apparent activation energy as (ET – EF) ∼ T 3/4 and to a weak increase of

σ00

values with increasing DOS value, N(EF), at the Fermi level. In order to overcome the limitations of

percolation models for describing the prefactor dependence in bandtail hopping (BTH) transport [5], we

have developed a “filling rate method” for the Ohmic regime [16, 17]. Since the aim of this work is to

extend this method to the analysis of the high field BTH regime, let us recall briefly the main hypothesis

and useful results.

The filling rate of an empty electronic state at energy E near the dominant transport path at energy ET

in the bandtail is computed assuming that: (a) the transition rate (from site i to site j separated by a dis-

tance Rh) has the usual Miller–Abrahams dependence on energy jump and tunneling distance

νij = ν0 exp [–2γ Rh – max ((Ej – Ei)/kT, 0)] , (3)

where ν0 is a phonon frequency (typically 1013 s–1) and T is the phonon bath temperature, (b) the localiza-

tion length γ –1 is independent of state energy, (c) the electric field value F is small, i.e. (eFRh ) � kT, (d)

the strong energy dependence of the bandtail DOS is described by an exponential distribution N(E) =

N(EF) exp [(E – EF)/E0], (e) the occupation probability at equilibrium, f(E), is a Fermi–Dirac function,

( f ) owing to the exponential decay of transition rates with energy and distance, the “useful” volume,

typically ∼Rh3(E), for filling a state at energy E, does not contain any other empty state, i.e. Rh

3(E) E0N(EF)

(exp [(E – EF)/E0] – 1) = 1.

Partial Ohmic conductivity values δσ(E) = eN(E) f(E) µ(E) δE, with µ(E) ≈ (4eν 0/3E0) γRh3(E)

× exp [–2γRh(E)] derived from the above hypothesis, are integrated to obtain σ(T) values as a function of

the disorder parameter, E0, and the localization parameter Lp = N(EF) γ –3.

In contrast with previous studies [5, 10, 15, 18, 19], our modeling does not necessarily consider

strongly localized systems, e.g. for the simulations shown in Fig. 2, Lp values range from 10–4 to 1 eV–1.

For Lp � 1 eV–1 [case A in Ref. [16]], the transport energy for BTH is (ET – EF) ∼ ln T [5, 10, 15, 18, 19];

this weak temperature dependence has been taken as a proof that BTH conductivity cannot follow Eq. (2)

[15]. However, owing to the fact that δσ(ET) contains not only the product N(ET) f(ET) but also a factor

µ(ET) which strongly increases with increasing temperature, our results show that ln σ is still reasonably

described by a linear T –1/4 dependence over 25 decades, for Lp = 10–4 eV–1 (Fig. 2). For Lp ≤ 1 eV–1

(case B in Ref. [16]), the transport energy is close to EF and (ET – EF) ∼ T. Equation (2) still holds, with

smaller slopes and prefactor values, some deviations occuring only at very low temperatures. Using Lp as

10-20

10-15

10-10

10-5

100

105

0 0.1 0.2 0.3 0.4 0.5

6.4x10-1

8.0x10-2

1.0x10-2

1.2x10-3

1.5x10-4

Con

duct

ivity

σ OH

MIC

(S.c

m-1

)

T -1/4 (K -1/4)

LP

(eV-1)

E° = 100 meVBandtail Hopping

Fig. 2 Linear dependence of ln σ (F ≈ 0, T ) vs. T –1/4, for different values of the localization parameter

Lp = N(EF) γ –3, using E 0 = 0.10 eV. The fits (solid lines) of Ohmic hopping conductivity simulations were

performed in the range 100–400 K.



a parameter, a linear relationship is obtained between apparent slopes T01/4 and prefactors ln σ 00. The

filling rate method shows that pre-exponential factor σ 00 (F = 0) values are highly sensitive to the shape

of the DOS distribution, in contrast with the slope T01/4 [16, 17].

Hydrogenated amorphous carbon nitride (a-C1–xNx :H) has been used to investigate the BTH conduc-

tivity. Disordered carbons are interesting as model materials to investigate VRH electrical transport [20]

because variations in the dihedral angle localize all π and π* electronic states within the σ–σ* gap [21]

so that the contribution of extended states transport within delocalized states is negligible over a wide

range of temperatures.

Transport properties of a-C1–xNx :H alloys had been found in good agreement with the “filling rate

method” for modeling BTH conductivity at low field. First, in the Ohmic regime, the temperature de-

pendence of hopping conductivity follows the T –1/4 law (Eq. (2)). Second, by changing synthesis condi-

tions or carbon alloy stoichiometry [16, 17, 20, 22], a linear correlation (with positive slope) is observed

between apparent slope T01/4 and prefactor ln σ 00, obtained in the Ohmic regime. Third, for a-C:H and

a-C1–xNx :H films, the F 2 dependence of ln (σ /σohmic) observed in the low-field regime [23] is qualita-

tively consistent with hopping models where both transitions along the electric field and backwards are

allowed [9, 13].

However, at higher field, two puzzling results remained unexplained and motivated further modeling

of FBTH transport. The first one is related to the scaling properties. Indeed, for electric fields larger than

5 × 104 V cm–1, enhancement over 10 orders of magnitude in conductivity (above the Ohmic value σohmic)

is achieved at low temperatures [24] with a scaling behavior as

ln (σ /σohmic) = ϕ[F S/T] . (4)

The scaling exponent S = 2/3 found in a-C1–xNx :H is not compatible with values expected from most

FEC models [25–27] including some hopping transport models using a function ϕ(b) of a single reduced

field parameter b = (eF/2γ kT) thus corresponding to a value of S = 1 [9, 28].

The second puzzling point appearing at high field concerns the effective temperature Teff. The problem

of non linear electronic transport in energetically disordered localized systems is complicated because

the electric field enters the field-enhanced conductivity not only explicitly through the field-induced

shifts of electron energies (Fig. 1), but also implicitly via the non-equilibrium distribution of charge

carriers. Within an energy distribution N(E) of localized electronic states, under high electric fields, a

charge redistribution occurs under the combined action of thermally-activated and field-assisted upwards

hopping transitions, which is only partially compensated by downwards (phonon emission) hopping

transitions.

We adopt the well-known effective temperature concept [28–30] to parametrize the non-equilibrium

occupation function f(E) of localized tail states, using a Boltzmann approximation with an effective

temperature Teff. A quadratic relationship,

T 2eff ≈ T 2 + [C(F) γ –1eF/k ]2 , (5)

with a constant value of C ≈ 0.69 ± 0.03, has been found either by solving linear balance equations [28]

or by performing Monte Carlo simulations [30]. However, it has been shown experimentally, using

σ(T, F) = σ(Teff, F = 0), that C(F) γ –1 cannot be considered as field-independent in carbon alloys [31]. It

rather follows a F –1/3 dependence (consistent with the observed scaling exponent S = 2/3), rather similar

to previous experimental results [4, 31–33].

The modeling based on the extension of the “filling rate method” to the high field BTH developed in

Section 5 will allow to clarify these puzzling results.

3 Experimental details

Symmetrical (Al/TiW/a-C1–xNx :H/TiW/Al) devices were obtained at the intersection of perpendicular

bottom and top electrodes [23]. Dense hydrogenated amorphous carbon nitride (a-C1–xNx :H) films were

deposited on patterned TiW electrodes by using the decomposition of acetylene and nitrogen in an elec-

phys. stat. sol. (b) 244, No. 6 (2007) 2085


Original

Paper

tron cyclotron resonance plasma [34]. The a-C1–xNx :H film (50 nm-thick as obtained by spectroscopic

ellipsometry) was grown near room-temperature on the RF-biased substrate holder with a typical ion

energy Eion = 110 ± 20 eV. For a flow-rate ratio (N2/C2H2) = 2, the stoichiometry N/(C + N) = 23 at% and

the hydrogen content H/(H + C + N) ≈ 30 at% were obtained from Nuclear Reaction Analysis and Elastic

Recoil Detection, respectively.

Conductivity values in sandwich MCM devices were similar, within a factor of 2, to those obtained in

a coplanar geometry using (Al/a-C1–-xNx :H/Al) structures on glass substrates [23]. The current density

was independent of the device area (in the range 5 × 5 to 200 × 200 µm2) showing negligible pinhole

effects, and symmetrical J–V characteristics were found with respect to voltage polarity, indicating bulk-

controlled transport.

The current density J(F) was measured in the range 250–330 K under primary vacuum, and down to

80 K or 34 K using two different turbomolecular evacuated cryostat chambers. Measurements of very

low ohmic currents were possible down to 56 K whereas at lower temperatures, σohmic was estimated by

extrapolation using Eq. (2). The carbon film temperature was measured by attaching a Pt resistor to the

glass substrate using silver paste; this temperature is slightly larger than that of a Pt probe pasted on the

copper substrate holder (equal to the temperature given by the thermocouple located inside the holder):

the largest difference (5 K at 34 K) is mainly responsible for the error bar on the slope T01/4 (Eq. (2))

which is estimated to be ±20%.

4 Experimental results

The J(F) characteristics at different temperatures, shown in Fig. 3, reveal a strong field enhancement of

conductivity σ (defined as J/F). In the ohmic regime, the conductivity σohmic(T) does not follow an

Arrhenius behavior with a single activation energy but rather a T –1/4 law (Fig. 4) characteristic of

3D hopping transport. The very high value of the Ohmic prefactor σ00 = 1.2 × 109 S cm–1 (Eq. (2))

is not consistent with a uniform DOS distribution but rather reveals a strong energy dependence of

N(E) [16, 17]. Using previous modeling of hopping within an exponential bandtail distribution

N(E) = N(EF) exp [(E – EF)/E0], the ohmic conductivity in a-C1–xNx :H was accurately described with a

disorder parameter E0 = 0.10–0.15 eV and a localization parameter Lp = N(EF) γ–3 = 10 ± 2 × 10–3 eV–1

as deduced from the slope T01/4 = 137.5 K1/4 [23]. This Lp value corresponds to a transport energy

(ET – EF) ≈ 0.20 eV at T = 300 K.

10-14

10-12

10-10

10-8

10-6

103 104 105 106

103.2 K87.0 K79.3 K70.6 K56.2 K42.6 K34.0 K

Field (V.cm-1)

Con

duct

ivity

(J/

F)

(S.c

m-1

)

Device area50 µm x 50 µm

F

σOHMIC

Fig. 3 Experimental field-enhanced hopping conductivity σ (T, F) at different temperatures (34–103 K)

in a 50 nm thick sandwich a-C1–xNx:H device.



10-13

10-11

10-9

10-7

10-5

10-3

0.3 0.35 0.4 0.45 0.5

5.0 x 105

4.4 x 105

4.0 x 105

3.0 x 105

2.0 x 105

1.4 x 105

1.0 x 105

6.0 x 104

4.0 x 104

2.0 x 104

1.0 x 104

4.0 x 103

T -1/4 (K -1/4)

Con

duct

ivity

(J/

F)

(S.c

m-1

)

σOHMIC

ACTIVATIONLESSREGIME

SlopeT

01/4 (F )

Field (V.cm-1)

T (K)120 4070 30

Fig. 4 Ohmic and field-dependent conductivity σ (T, F) vs. T –1/4. The slope T01/4 = 137.5 K1/4 and the

prefactor σ00 = 1.2 × 109 S cm–1 have been obtained from a least-squares fit of Eq. (2) in the Ohmic regime

(F ≤ 1 × 104 V cm–1). Apparent slopes and prefactors were deduced from the T –1/4 dependence at low tem-

peratures (dashed line).

At low-to-intermediate electric fields (b ≤ 0.3), the F

2-dependence of ln (σ /σohmic) is a further confir-

mation of hopping within a 3D space (Fig. 5). This result confirms experiments performed at higher

temperatures [23, 24]. Using the 3D Apsley–Hughes (AH) model for a uniform DOS [9], the F

2 depend-

ence at intermediate field strength has been used to extract the decay length γ –1 of localized π state wave

functions in a-C1–xNx :H (2.4 ± 0.4 nm) [23, 24]. Together with the Ohmic regime slope, this γ –1 value

provides an estimate of N(EF) = 5 – 12 × 1017 cm–3 eV–1. However, since analytical solutions for hopping

in exponential tails are not available, it has not been possible to estimate the error which may arise from

the assumptions of a uniform DOS.

As shown in Fig. 5 (log–log scale), for a-C1–xNx :H devices ln (σ /σohmic) is unexpectedly scaling as

(F 2/3/T) rather than as (F/T) obtained previously for a-C:H devices in a narrower temperature range [23].

10-2

10-1

100

101

102

10 102 103 104 105

103.2 K87.0 K79.3 K70.6 K56.2 K42.6 K34.0 K

F / T (V.cm-1.K-1)

a-C1-x

Nx:H Slope

(1/2)

2

REGIMEF

ln(

σ/

σO

HM

IC)

(a)

10-2

10-1

100

101

102

100 101 102 103 104

103.2 K87.0 K79.3 K70.6 K56.2 K42.6 K34.0 K

F / T 3/2 (V.cm-1.K-3/2)

ln(

σ/

σ OH

MIC

)

2

REGIMEF


Slope(1/2)

a-C1-x

Nx:H

(b)

Fig. 5 Dependence of ln (σ /σohmic) on parameters: a) (FT –1); b) (FT –3/2). A scaling law is observed over 3 decades in

(FT –3/2), with a F 2 dependence at low-to-intermediate fields, and a slope smaller than 1/2 at very high fields (activa-

tionless regime). High temperature data (333 K) for the same device are shown as bold lines.

phys. stat. sol. (b) 244, No. 6 (2007) 2087


Original

Paper

10-13

10-11

10-9

10-7

10-5

0.03 0.05 0.07 0.09

103.2 K87.0 K79.3 K70.6 K56.2 K42.6 K34.0 K

Cur

rent

(A)

F -1/4 (V.cm-1)-1/4

Slope

F0

1/4 = 744 (V.cm-1)1/4

Device area50 µm x 50 µm

Fig. 6 Plot of the current I(T, F) vs. F–1/4, at high fields near the activationless regime. A linear depend-

ence (Eq. (1)) is found at low T for F ≥ 3 × 105 V cm–1.

In Fig. 5(a), the data at 333 K (continuous line) plotted as a function of (FT –1) do not overlap the data

taken at low temperatures, which also show a large scattering; this demonstrates that S = 1 is not an ac-

ceptable value. Similarly, using a systematic variation of the scaling exponent to plot the experimental

data measured in the experimental range of 34–333 K, and considering that a deviation by 30% of this

scaling law can be detected, one obtains a validity range for the scaling parameter S = 0.67 ± 0.05. The

transition from a F

2-behavior at intermediate field strength towards a F

m dependence, with m decreasing

as F increases, is consistent with the 3D hopping characteristics found with a uniform DOS distribution

[9]. In this work, new data at low T (34–100 K) confirm a dependence of ln (σ /σohmic) close to F

1/2 up to

high field values, and reveal a slope m ≈ 0.35 (smaller than 1/2) for values of the scaled field 10 ≤ b ≤ 23.

Concerning the activationless regime, at the lowest temperatures (34–50 K), an apparent activation

energy of 3 meV (close to kT) is observed in a standard Arrhenius plot. However, a reasonable linearity

of ln σ vs F –1/4 (Eq. 1) is obtained only within a limited range of electric field values (Fig. 6). Apparent

values of σ* = 3.4 × 106 S cm–1 and F01/4

= 744 V1/4 cm–1/4 (F0 = 3.1 × 1011 V cm–1) have been obtained for

F > 3 × 105 V cm–1. A typical length scale for field-assisted hopping is given by (kT0/eF0) = 1.0 ± 0.1 nm.

It is also emphasized that the prefactor σ* found in the activationless regime (Eq. (1)) is smaller than the

prefactor σ00 in the Ohmic regime (Eq. (2)) by three orders of magnitude.

10-8

10-4

100

104

108

1012

0 50 100 150


F 2

REGIME

OHMIC

σ°°

(F)

(S.c

m-1

)

T0

1/4 (F ) (K1/4)

a-C1-x

Nx:H

increasingfield F

Fig. 7 Correlated decrease of both apparent slope T 01/4

and prefactor σ 00 for increasing field strength, obtained

from ln σ (T, F) vs. T –1/4 plots in Fig. 4 (34 < T < 70 K).



0

10

20

30

40

50

60

70

105 106

T < 80 K (5)T < 71 K (4)T < 56 K (3)T < 43 K (2)

Slo

peT

01/

4(K

1/4 )

Field F (V.cm-1)

(a)

10-6

10-5

10-4

10-3

10-2

10-1

100

105 106

T < 80 K (5)T < 71 K (4)T < 56 K (3)T < 43 K (2)

Field F (V.cm-1)

σ°°(

F)

(S.c

m-1

)

(b)

Fig. 8 a) Experimental apparent slope T 01/4(F) and b) prefactor σ 00(F) values at high fields, obtained from conduc-

tivity ln σ (T, F) vs. T –1/4 plots, using different T ranges for linear fitting.

Although Eq. (2) is not expected to be strictly fulfilled for high field strengths, the apparent slope

T01/4(F) and prefactor ln σ00(F) values can be obtained within any arbitrarily narrow T range (Fig. 4). At

high T (250–330 K), both parameters decrease as a function of increasing electric field with a linear

correlation (slope λ = 0.21 K–1/4) (not shown), as also observed in unalloyed a-C:H [24]. Here, additional

data in the low T range (34–70 K) give access to very low T01/4(F) values close to the activationless re-

gime (Fig. 7). A linear correlation (dashed line) with a similar slope is observed for moderate field

strengths F < 2 × 105 V cm–1.

However, a non-monotonic behavior is revealed at very high fields, with a mimimum in σ 00(F) near

F = 3 × 105 V cm–1. The field value at the minimum, Fmin, and the prefactor σ 00(Fmin) both decrease,

showing a deeper minimum, when the fitting is performed at smaller T values (Fig. 8). To our best

knowledge, a mimimum in σ 00(F) has not been reported before. In addition, for electric field strength

increasing above 2 × 104 V cm–1, the large decrease of the slope T01/4 (F), along with that of the apparent

activation energy Eact (Fig. 3 in Ref. [24]), reveals a weaker contribution of thermally activated transi-

tions to the hopping rate. Hence, in the following Sections, we will focus on modeling the high-field and

low temperature regime.

5 Analysis and discussion

Our analysis is focused on a range of reduced field parameter values b = (eF/2γkT) > 1 where field-

assisted transitions are expected to dominate over thermally-activated transitions [9, 31]. In this regime,

we show that hopping transport depends on both T and Teff(T, F). We introduce a new scaling parameter

q to parametrize the power-law dependence of kT*eff ∼ F q related to the effective temperature as

kT*eff = (1/kTeff – 1/E0)–1; rather than guessing a correct value, we will show that parameter q can be de-

rived from high field experiments, in particular from the dependence of the prefactor σ 00 on electric

field. In Section 5.1, we explain why parameters q and S have very similar values in the high-field limit.

In Section 5.2, the concept of a demarcation level EDL, as previously derived by Monroe [35, 36] for both

transient and steady-state phenomena in band tails, is extended to the case of FBTH. FEC data are inter-

preted in a field-enhanced bandtail hopping model, where field-excited carriers reach asymptotically a

demarcation level EDL(T, F) defined by equal average values of energy gain and energy loss.

Our approach to obtain the forward conductivity σF (T, F) is based on the calculation of the typical

filling rate ΓF(Ej, T, F) of a given empty electronic state near the demarcation level, integrating all non-

thermally activated forward transitions from neighbor tail states. An analytic approximation of FBTH

conductivity σF(T, F) is derived in Section 5.3. The scaling properties are quantitatively related to the

phys. stat. sol. (b) 244, No. 6 (2007) 2089


Original

Paper

shift of the preferred transport path at energy EDL(T, F) away from the Fermi level (upper in the band

tail) producing a decrease of the typical hopping length Rh (EDL) as F increases.

In Section 5.4, we present a method which allows a precise derivation of some DOS parameters and

effective temperature Teff values using low T – high F experiments.

5.1 Scaling analysis

The existence of some scaling properties between field and temperature in FBTH transport is impor-

tant as a signature of bandtail hopping. In this Section, we first show that the scaling exponent,

S = 0.67 ± 0.05, observed for FEC in a-C1–xNx :H films is not consistent with previous models reported in

the literature, although this result is rather similar to previous experimental S values. In a second step, we

indicate a possible source of inaccurate calculations in previous field-enhanced conductivity models,

namely the role of the non-equilibrium population of bandtail states.

In a-C1–xNx :H, FEC data show a scaling behavior given by a function ϕ (Eq. (4)) which is characteris-

tic of the transport mechanism. A similar approach was used for transport in a-Ge with 0.5 ≤ S ≤ 0.7

(Eq. (1) in [37]). The field enhancement parameter ln (σ /σohmic) being a function of (F 2/3/T), the scaling

exponent S = 2/3 found in a-C1–xNx :H is not compatible with values expected from most FEC models:

– S = 1/2: a (F 1/2/T) scaling was observed in a-C sandwich devices [25] as expected for the Poole–

Frenkel mechanism for charge carrier emission from a Coulombic trap [26].

– S = 1: scaling as (F/T) has been predicted by some microscopic models based on phonon-assisted

tunneling within a distribution of localized electronic states, where VRH conductivity or Teff(T, F) are

described by a single dimensionless parameter b = (eF/2γ kT) [9, 27, 28].

– S = 4/5: however, it has been concluded [7, 26] that if eFRh(T) � kT, the argument in the expo-

nential function of the hopping conductivity should be corrected by a term which is linear in

F :J = σF exp [eFRh(T)/kT] = σF exp [eFγ –1T0 1/4/2kT 5/4] providing a scaling parameter S = 4/5 = 0.8.

– S = 0.68: in a percolation approach, a dependence of (J/σF) as exp [(eFγ –1/kT)1/1+ν (T0/T)1/4] has

been derived by Shklovskii [11] by considering the characteristic correlation radius L0 in the critical

hopping subnetwork rather than the length of a typical jump Rh; a value of S = 0.68 is obtained from a

correlation radius index ν = 0.9. This shows that different scaling exponents can be expected for the

microscopic mechanism (described by b = eF/2γkT, S = 1) and for the macroscopic observable conduc-

tivity. In spite of the scaling exponent being close to S = 2/3, Shklovskii’s model is restricted to low field

strength (eF/2γ kT) < 1, and a F

m dependence with a slope m < 1/2 would require unphysical values of

the percolation critical exponent ν > 3.

Although previous hopping models considered the same rate equations, and in some cases exponen-

tial DOS distributions, a crucial parameter remains difficult to model, namely the occupation probability

of bandtail states. In the high-field limit, the quadratic field dependence approximation (Eq. 5) reduces

to [Teff /T] ∼ [F/T]. In this work, a new scaling parameter, q, is introduced, based on a parametriza-

tion of kT*eff = (1/kTeff – 1/E0)–1 ∼ F q. Rather than guessing a correct value, q is derived from the

analysis of prefactor σ 00(F) data. For intermediate and high fields such that kTeff � E0, one obtains

[T*eff /T] ∼ [Teff /T] ∼ [F q/ T] and q = 0.71. This q value is very close to the S value derived from the analy-

sis of ln (σ /σohmic) using Eq. (4). Indeed, if the effective temperature governs both high-field conductivity

and non-equilibrium electron distribution, one would expect that [Teff /T] ∼ [F S/T] ∼ [F q/T] in the high-

field limit.

5.2 Demarcation energy for hopping in a high electric field

In this Section, the well known transport level concept is recalled in the Ohmic regime of bandtail hop-

ping and a demarcation level EDL(T, F) is derived for the low T – high F regime, following Monroe’s

derivation of ET(T).

At low electric fields, a particular energy level in the band tail of a disordered semiconductor, called

the transport energy ET, plays a crucial role in the hopping transport under both equilibrium and non-



equilibrium conditions and for both transient and steady-state phenomena [5, 35, 38]. The fastest

thermally activated hop of an electron sitting at an energy Ei occurs normally towards the transport

energy ET irrespective of its starting energy, provided that Ei is deeper than ET. For Ei > ET, on average

the fastest rate is for hopping down towards states nearer to ET. For an exponential energy distribution

N(E) = N(EF) exp [(E – EF)/E0], which is commonly used as a model of band tail in disordered semicon-

ductors, if field-assisted hopping transitions are neglected, the temperature-dependent transport energy

ET defined by

N(ET)/N(EF) = (2kT/3E0)3 (E0N(EF) γ–3)–1 , (6)

is a function of both localization parameter Lp = N(EF) γ –3 and bandtail disorder energy E0 [5, 10, 15, 16,

18, 38]. The transport energy plays the role of a mobility edge, which becomes significantly deeper as

the temperature decreases.

In a strong electric field, an electron in a deep state at energy Ei will typically hop against the field

direction (forward direction) due to the large increase in the density of available states (Fig. 1). From

numerical simulations, Cleve et al. [30] concluded that, starting from a deep state close to EF, an electron

will hop upwards in energy, making successive hops so that each next hop is on average shorter than the

previous one and corresponds to a smaller energy difference between two states involved. In contrast,

carriers located in shallow states predominantly loose energy in each hop because the DOS is very high

and the energy gained by hopping against the field is therefore small (eF ∆Rh < E0). A balance must

occur when the energy gain per hop, related to the typical hop length Rh (Fig. 1), is equal to the energy

loss, related to the number of accessible sites for down-hopping.

In this dynamical equilibrium context, the asymptotic energy level (called Εt in [4] and ε f in [30]) is

defined here by a field-dependent demarcation energy EDL(T, F) in the band tail, corresponding to the

balance between electronic energy loss (down hopping) and field-assisted energy gain. By analogy with

the temperature-dependent transport energy ET(T) in the ohmic regime (Eq. (6)), electronic states near

EDL(T, F) behave as a preferred transport path.

To obtain EDL(T, F), we explicitly consider the fact that both temperature and electric field affect the

forward transition rates, in order to calculate the integrated rates of hopping transitions, respectively

towards shallower and towards deeper energy levels (Appendix A1). In Eqs. (A1)–(A2), the dimen-

sionless reduced field parameter b = eF/2γkT is apparently consistent with a scaling exponent S = 1 in

Eq. (4). The demarcation level EDL(T, F) being defined by equal energy gain and energy loss probabili-

ties, at the optimum hopping length Rh(EDL), one obtains:

exp [eFRh(EDL)/E0)] – 1 = 2eFRh(EDL)/(E0 + kT*) . (7)

10-2

10-1

100

100

101

102

0.04 0.07 0.1 0.4 0.7

eF

Rh

E°

/

eFR

hkT

*/

kT* / E°

slope- 4/3

TRANSPORTENERGY E

DL(T, F )

0.2

slope- 1.0

T (K)50 20010030 350

Fig. 9 Dependence of (eFRh/E0) and (eFR

h/kT*) on

the parameter kT*/E0 at the demarcation energy (EDL

)

showing a quasi-constant value of (eFRh) ≈ E0 for

(kT*/E0) ≤ 0.2. The corresponding temperature axis

(upper scale) was obtained using E0 = 0.1 eV.

phys. stat. sol. (b) 244, No. 6 (2007) 2091


Original

Paper

In contrast with Eq. (6) for ET(T) in the ohmic regime, this result demonstrates that the demarcation level

EDL(T, F) is independent of the localization parameter N(EF) γ –3 and strongly depends on the disorder

parameter E0; it corresponds to a scaled hopping energy X = eFRh(EDL)/E0 ≈ 1, as found previously [30].

While the T dependence of X (Eq. (7)) remains small as long as kT* � E0, X decreases as a power law

T* –1/3 near (kT*/E0) ≈ 0.2, and more strongly when (kT*/E0) exceeds ≈0.4 (Fig. 9). Using Eq. (7) and a

typical disorder energy E0 = 0.1 eV, a decrease of [eFRh(EDL)/kT*] by one decade (from 20 to 2.0) is

obtained for an increase in temperature from 70 K to 450 K. To illustrate the validity range of the low T

approximation, eFRh(EDL) ≈ E0 holds below ≈200 K, if one assumes E0 = 0.1 eV.

Here, T* is related to the phonon bath temperature which enters the hopping transition rate between an

occupied site and an empty site (Eq. (3)) rather than to the effective temperature Teff describing a Boltz-

mann occupation function of bandtail states.

For each temperature, the value of parameter X = (eFRh/E0) at the demarcation level is given by

Eq. (7) and the density of states N(EDL) is obtained in the next section.

5.3 Field enhancement of band tail hopping

In this section, in order to obtain analytically the field-enhanced bandtail hopping conductivity σF(T, F),

we compute the filling rate ΓF(Ej) of an empty bandtail state near EDL, by first deriving the field-

dependent average hopping distance Rh(Ej) for a fixed energy level Ej in an exponential band tail. In a

second step, to reproduce the approximate T

–1/4 dependence of experimental FEC data, the analytic ex-

pression for FBTH conductivity σF(T, F) is expressed in the form: ln σF(T, F) = ln Π(F) – [T0/T]1/4 Σ(F).

In a third step, the F-dependence of the prefactor ln Π(F) is analyzed in detail using the effective tem-

perature concept, by assuming a power-law parametrization for kT*eff ∼ F q; the existence of a minimum in

Π(F) at a high field strength Fmin allows the derivation of both q and kTeff values.

In Appendix 2, due to the exponential dependence of tunneling transition rates, hopping towards the

second nearest neighbor is neglected. Equation (A6) is applied to the case Ej = EDL. For usual values of

kT*/E0, 0.8 ≤ X ≤ 1.2 is found (Fig. 9) so that (eFRh) ≈ E0. Using Eq. (7), one obtains a constant value

for:

N(EDL) (E0)

4 (eF)–3 = 1.4 ± 0.1 . (8)

The F

3 dependence of N(EDL) and the occupation probability approximately given by:

ln f(EDL) = –(E0/kTeff) ln [(eF)3/((E0)4 N(EF))] , (9)

will be used in the following discussion of the relationship between ln σ 00(F) and T01/4 (F) values.

In contrast with this directional approach (Eq. (A6)), some previous works [39] have considered an

isotropic hypothesis (Eq. (A8)). In the weak-field limit, both directional Eq. (A6) and isotropic Eq. (A8)

equations are approximated by N(Ej) E0Rh

3 = 1. The comparison of Rh(Ej) at three values of the density of

states in the range 1019–1021 cm–3 eV–1 (Fig. 10) indicates that the field-induced decrease of the average

hopping distance is stronger for states lying closer to the Fermi level, i.e. smaller N(Ej) values. In addi-

tion, a stronger decrease of Rh(Ej) vs F is found with the isotropic model (Eq. (A8)), which will obvi-

ously influence the computation of the FEC values.

In order to describe the filling rate and the partial conductivity for states near the demarcation level, in

the high F – low T regime, we consider non-activated hopping transitions in the forward direction

(against the field for electrons) from all states (Ei) within a distance Rh(Ej). The effective temperature

concept is introduced by assuming that the occupancy probability of bandtail states is given by a Boltz-

mann function f(Ei) = exp [–Ei/kTeff]. In Appendix A3, using (kT*eff)–1 = (kTeff)

–1 – (E0)–1, the filling rate for

the transport path near EDL:

ΓF(EDL) = 2π (ln 2) ν0γ –1(kT*eff)

2 (eF)–1 Rh N(EDL) f(EDL) exp (–2γRh) [exp (eFRh/kT*eff) – 1] , (10)



0.4

0.6

0.8

1

103 104 105 106

1019

1020

1021

1019

1020

1021

Field F (V.cm-1)

Hop

ping

leng

thR

h(F)

/R

h(F=

0)

N (Ej) (cm

-3eV

-1)

Isotropic Directional

is used to obtain the calculated forward conductivity:

σF(T, F) ≈ 6π(ln 2)2 e2

ν0γ –2(E0)–6 (kT*eff)

2 (eF)3 f(EDL) exp (– 2γRh) exp (eFRh/kT*eff) . (11)

In this model, by analogy with Eq. (2), the FBTH conductivity is parametrized using ln σ(T, F) vs T –1/4

plots, which provide field-dependent apparent values of prefactor (ln σ 00) and slope (T01/4). Hence, ln σF

(T, F) is decomposed into two F-dependent terms:

ln σF(T, F) = ln Π(F) – [T0/T]1/4 Σ(F) , (12)

with

Σ(F) = Rh(F)/Rh(F = 0) , (13)

ln Π(F) ≈ ln [6π(ln 2)2 e2

ν0/γ –1E0] + ln [(E0)–5 (kT*eff)2 (eFγ

–1)3] + [eFRh(F)/kT*eff]

– (E0/kTeff) ln [(E0)–4 N(EF)–1 (eF)3] . (14)

Σ and Π are the mathematical expressions for apparent values of relative slope T01/4(F)/T0

1/4(F = 0) and

prefactor σ 00(F), respectively. Equation (13) describes a decrease of the apparent slope Σ(F) towards

arbitrarily small values, proportional to Rh(F) as F increases. The last term in Eq. (14) dominates the

F-dependence of Π(F), with a large influence of the parameter N(EF) (E0)4.

In a ln Π(F) vs ln (F) plot, as F increases, the decrease of the apparent prefactor is given by the de-

rivative:

d ln Π(F)/d ln (F) = [d ln (kT*eff)/d ln (F)] [2 – (E0/kT*eff) + (E0/kT*eff) ln [(E0)–4 N(EF)–1(eF)3]]

– (3E0/kT*eff) . (15)

This derivative is specific of the exponential band tail distribution and mainly results from the decreasing

value of f(EDL) = exp (–EDL/kTeff) as the demarcation level moves away from the Fermi level EF. As F

increases, Eqs. (8) and (12)–(14) show that a steeper bandtail (smaller E0) gives a stronger increase of

N(EDL) which is compensated by the decrease of f(EDL).

Whereas a monotonic decrease of Σ(F) vs F is expected, according to Eq. (15) this model predicts

a minimum value of the prefactor Π(F) at a field Fmin. This prediction is indeed observed using the

low T data (34 < T < 70 K) (Fig. 8): a minimum value of σ00 ≈ 0.3–5.8 × 10–5 S cm–1 is found at

Fmin = 2.65–3.70 × 105 V cm–1, corresponding to a reduced field value b ≈ 8 ± 1. We show below that the

product (eFmin)3 (E0)–4 N(EF)

–1 = N(EDL)/N(EF) at Fmin is more relevant than parameter b = (eF/2γkT) to

Fig. 10 Field dependence of the typical hopping

length value (Rh) at several energy levels above EF

with respective DOS values of 1019 to 1021 cm–3 eV–1,

using γ–1 = 1 nm, E0 = 0.10 eV and T = 100 K. A

stronger decrease of (Rh) is found in the isotropic

hypothesis (Eq. (A8)) as compared to the anisotropic

case (Eq. (A6)).

phys. stat. sol. (b) 244, No. 6 (2007) 2093


Original

Paper

describe the minimum σ 00(F) behavior. Using a power-law parametrization for kT*eff ∼ F q (q > 0), and

setting kT*eff(F) = kT*eff(Fmin) [F/Fmin]q in Eq. (15), the effective temperature kTeff(Fmin) at the minimum

value of the prefactor Π(F) occurring at F = Fmin is given by:

2kT*eff(Fmin)/E0 = (3/q) + 1 – ln [(E0)–4 N(EF)

–1 (eFmin)3] (16)

and the calculated prefactor near Fmin writes:

ln Π(F) ≈ ln [6π(ln 2)2 e2ν0/γ –1

E0] + 2q ln (F/Fmin) + ln [Lp(kT*eff(Fmin))

2/E0]

+ [E0/kT*eff(Fmin)] [F/Fmin]-q [1 – ln [(E0)–4 N(EF)

–1 (eF)3] . (17)

5.4 Derivation of DOS parameters and effective temperature

Using this filling rate model for FBTH conductivity, in the following we present a method to estimate

the effective temperature Teff and the density of states parameters, N(EF) and E0, from high field transport

experiments and analytic expressions of prefactor σ 00(F) and effective temperature Teff(F). We simply

use the value of the localization parameter Lp as deduced from the ohmic regime (equal to 10–2 eV–1

in the particular case of a-C1–xNx :H) and set the phonon frequency from Eq. (3) to a physical value

(ν0 = 1013 s–1 in amorphous carbon) with no further adjustable parameters.

In order to check for the robustness of this method to derive the q-exponent which describes the effec-

tive electron distribution temperature Teff(F), we have systematically analyzed the apparent σ 00(F) values

obtained at low lattice temperatures using different T ranges (Fig. 8). The simulations have been per-

formed in order to optimize the value of σ 00(Fmin) and the curvature of the plot σ 00(F) near Fmin. We

discuss below the simulations of the data obtained in the lowest T range, which provide the deepest mini-

mum in σ 00(F); quantitatively similar results were found with other experimental data from Fig. 8.

First, we estimate the range of physically permitted q values. In Eq. (16), the condition kT*eff(Fmin) > 0

imposes a maximum value of q, qmax, which depends on the material’s characteristics, N(EF) and E0. As

shown in Fig. 11, using a fixed value of N(EF) = 1 × 1018 cm–3 eV–1 (near the physical value obtained in

Section 3) and disorder parameter variable in a broad range 20 meV ≤ E0 ≤ 200 meV, qmax is found in

the range 0.30 ≤ qmax ≤ 2. On the other hand, a typical range of variation for the effective temperature,

kT*eff ≤ E0, gives a lower boundary 0.25 ≤ qmin ≤ 0.90.

Next, Eqs. (16) and (17) have been solved at Fmin to obtain optimum q values, qopt, defined by Π(Fmin)

equal to the experimental σ 00(Fmin). As expected, qmin ≤ qopt ≤ qmax (Fig. 11). Figure 12 gives a further

indication about the sensitivity of Eqs. (16) and (17) to obtain q values which reproduce the shape of the

minimum in σ 00(F). Using fixed values of N(EF) = 1 × 1018 cm–3 eV–1 and E0 ≈ 120 meV, a variation of q

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2

qMAX

qMIN

qOPT

q

(eV)E°

Fig. 11 Calculations of q, defined in the parametri-

zation of the field dependence of kT*eff ∼ F q, as a

function of bandtail disorder parameter E0, using

N(EF) = 1 × 1018 cm–3 eV–1. The extremum values corre-

spond to kT*eff (Fmin) = 0 (qmax, dashed line) and to

kT*eff (Fmin) = E0 (qmin, dash-dotted line). The optimum

values (qopt, full line) are derived from Π (Fmin) = σ00(Fmin).



10-8

10-6

10-4

10-2

100

102

105 106

Experimentq = 0.65q = 0.715 = q

OPTq = 0.75

Field F (V.cm-1)

Pre

fact

orσ

°°(S

.cm

-1)

(a)

10-7

10-5

10-3

10-1

101

105 106

Experiment

E°=0.07 eV; qOPT

=0.448

E°=0.12 eV; qOPT

=0.715

E°=0.17 eV; qOPT

=1.171

(b)

Field F (V.cm-1)

Pre

fact

orσ

°°(S

.cm

-1)

Fig. 12 Sensitivity of simulations of the experimental σ00(F) (for 34–43 K data in Fig. 8 and assuming

N(EF) = 1 × 1018 cm–3 eV–1) to: a) q values near qopt = 0.715, assuming E0 = 0.12 eV; b) changes in E 0 values near the

best estimate of 0.12 eV.

by a few percent away from the optimum (qopt = 0.715) leads to a change in Π(Fmin) by several decades

(Fig. 12a). Hence qopt is unambiguously determined by the choice of N(EF) and E0. Next, the curvature of

Π(F) near Fmin is strongly dependent on the choice of E0 (Fig. 12b). Hence E0 is strongly constrained by

the choice of N(EF). As a matter of fact, Fig. 13a shows that the relevant parameter which describes q(F)

near Fmin is the product (eFmin)3 (E0)–4 N(EF)

–1 = N(EDL)/N(EF), independent of the particular value chosen

for N(EF). This parameter arises from the expression of f(EDL), as shown by Eqs. (9) and (14).

Figure 13b represents the dependence of kT*eff, at Fmin, on the “universal” parameter N(EDL)/N(EF)

at Fmin, using E0 as the variable parameter. From a systematic change of the input parameters N(EF)

and E0, it appears that the best simulations of experimental σ 00(F) data are obtained for [N(EF), E0]

couples which correspond to the maximum in kT*eff; for our experimental results, the latter is found at

N(EDL)/N(EF) ≈ 80, independent of N(EF), which corresponds to qopt = 0.715 (Fig. 13a).

In conclusion, a single unambiguous value of N(EF) cannot be derived from the high field regime

alone; in our high field experiments, the most probable value of the Fermi level density of states is close

to N(EF) = 1 × 1018 cm–3 eV–1 (consistent with Ohmic regime characteristics); this value determines an

0

0.5

1

1.5

2

100 101 102 103 104 105

N(EF) = 2x1017 cm-3 eV-1

N(EF) = 1x1018 cm-3 eV-1

N(EF) = 5x1018 cm-3 eV-1

qO

PT

(a)

N(EDL

) / N(EF

)

0.01

0.03

0.05

0.07

100 101 102 103 104 105

(eV

)kT

EF

F*

N(EDL

) / N(EF

)

2x1017 cm-3 eV-1

1x1018 cm-3 eV-1

5x1018 cm-3 eV-1

(b)

Fig. 13 Dependence of qopt (a) and kT*eff (b), at Fmin, on the “universal” parameter N(EDL)/N(EF) at Fmin, using E 0 as

the variable parameter. The best simulations of experimental σ 00(F) data are obtained for [N(EF), E0] couples which

correspond to the maximum in kT*eff, found at N(EDL)/N(EF) = 80. Hence, qopt = 0.715 is deduced.

phys. stat. sol. (b) 244, No. 6 (2007) 2095


Original

Paper

Table 1 Best simulations of σ00(F) in a-C1–xNx:H (x = 0.23): for each N(EF) value, the best E 0

opt value

was derived, leading to the determination of the electronic temperature Teff and its power-law exponent

qopt, near Fmin.

N(EF)

(cm–3 eV–1)

E 0opt

(eV)

qopt kT*eff

(eV)

Teff

(K)

2 × 1017 0.18 0.7 0.066 560

1 × 1018 0.12 0.7 0.042 360

5 × 1018 0.08 0.7 0.027 230

optimal E0 = 0.12 ± 0.02 eV and qopt = 0.715. The uncertainty on N(EF) leads to some uncertainties on E0

and kTeff values which have been estimated in Table 1. However, the result qopt = 0.715 remains valid,

independent on N(EF) variations. In addition, for all curves in Fig. 8, i.e. independent of the fitted T

range, the best q values are close to q = 0.7 ± 0.1. Finally, the values of Teff at high fields, derived from

this filling rate method, are found in the range 500–600 K (depending on the fitting range in Fig. 8) at

the highest F values of 5.5 × 105 V cm–1, corresponding to a density of states N(EDL) = 8 × 1020 cm–3 eV–1

at the demarcation level and to a typical ratio (Teff /T) ≈ 10 ± 2.

An important result of high field transport modeling of FEC data is the insight gained in the disorder

of the DOS distribution. It is found that the material parameters N(EF) and E0 cannot be chosen inde-

pendently since the relevant parameter is the product N(EF) (E0)4, arising from the expression of f(EDL),

as shown by Eqs. (9) and (14). The value of E0 = 0.12 ± 0.02 eV derived from FEC data is in good agree-

ment with the estimate of E0 ≈ 0.10–0.15 eV, derived from modeling of σohmic(T) [23]; the latter was

obtained with a reasonable accuracy because the prefactor ln σ 00 is extremely sensitive to changes in the

disorder parameter E0 of the exponential band tail [17].

The good quantitative agreement between the field dependence of slope and prefactor obtained ex-

perimentally in a-C1–xNx :H and the computed values derived from this bandtail hopping model relies

upon the calculation of field-assisted forward filling rates for empty localized states located in an expo-

nential bandtail. In contrast with percolation models, this approach provides an accurate estimate of both

pre-exponential and exponential factors.

Our analytic results can also be used as a signature of the exponential character of localized bandtails.

In the Apsley–Hughes (AH) model [9] for field-assisted hopping within a uniform DOS, using the same

hopping transition rates (Eq. (3)), apparent values of (Π) and (Σ) both decrease as F –1/4 (i.e. Π and Σ are

nearly proportional at high fields). In contrast with the weak decrease of ln σ 00 ≈ –1/4 ln F for increasing

F, predicted in the AH model, our modeling of FBTH predicts a very strong decrease of ln σ 00 vs ln F,

with a typical slope given by Eq. (15). The latter behavior is more consistent with experimental high

field transport in a number of amorphous alloys, including a-C1–xNx :H films. Hence, the correlation be-

tween the apparent slope T01/4 and prefactor ln σ 00 values clearly discriminates hopping within a uniform

DOS from band tail hopping.

Interestingly, the nearly constant apparent σ 00 value, expected at all fields for a uniform DOS, also

holds for the activationless regime of bandtail hopping, at small phonon bath (or lattice) temperature T,

as a result of the large effective electronic temperature.

6 Summary and conclusions

Hopping transport within a bandtail distribution of localized electronic states has been investigated in

a-C1–xNx :H (x = 0.23) sandwich devices both in the low field (ohmic) and high field (field-enhanced)

regimes. Conductivity data at high electric field are fully consistent with a field-enhanced bandtail hop-

ping (FBTH) model, in which field-excited hopping carriers reach asymptotically a demarcation energy

EDL(T, F), where energy gain and energy loss probabilities equilibrate, on average. In this work, the ef-

fective temperature concept has been used to describe the non-equilibrium occupation probability of

exponentially distributed tail states. A “filling rate” method, considering forward non-activated hopping



transitions, has been developed to determine EDL(T, F) and to analyze the high field regime of FBTH.

Since in the Ohmic regime, consistent parameters for hopping transport were obtained for a wide variety

of a-C based materials [22], we believe that this high field transport model can be used to investigate

other carbon alloys as long as the transport level lies within the distribution of localized bandtail states.

To assess our model, FBTH transport characteristics are parametrized using ln σ (T, F) vs. T –1/4 plots,

which provide field-dependent apparent values of prefactor (ln σ 00) and slope (T01/4). As F increases, both

computed parameters strongly decrease, as observed experimentally. Being proportional to f(EDL),

σ00(T, F) is responsible for the decreasing “efficiency” of increasing field strength, in the intermediate

field regime, to promote electronic transitions forbidden in the Ohmic regime. This trend is reflected by

the decrease of the apparent exponent m, observed in Fig. 5b, for increasing field strength. It is empha-

sized that the field dependence of conductivity as ln σ ≈ F

m, with an apparent exponent m ≈ 1/2, com-

monly reported for many disordered semiconductors, is compatible with FBTH transport, without

assuming Coulombic potential wells for electron traps. Experimentally, the F

1/2 dependence of the field

enhancement factor ln (σ /σohmic) has been observed in carbon nitride films, up to high values of

b = (eF/2γ kT) ≤ 10, while low T data reveal a slope m ≈ 0.35 (i.e. smaller than 1/2) for scaled fields

10 ≤ b ≤ 23 (F/T 3/2 > 10

3) in the activationless regime.

In the very high field region, our models predicts the existence of a minimum value 00

minσ of σ 00(F) at

F = Fmin, which is indeed observed in a-C1–xNx :H (x = 0.23) for T < 70 K (00

minσ ≈ 10–6 S cm–1 at Fmin ≈

3 × 105 V cm–1). We are not aware of other literature reports concerning this low temperature behavior.

Near Fmin, the effective temperature Teff describing the bandtail occupation probability, has been ap-

proximated analytically, using kT*eff = (1/kTeff – 1/E0)–1 ∼ F q. The value of q that best reproduces experi-

mental results is q = 0.7 ± 0.1. This value is consistent with that of the exponent S = 0.67 ± 0.05 describ-

ing the experimentally observed scaling law, ln (σ /σohmic) = ϕ[FS/T]. Hence, our “filling rate” method

applied to FBTH transport appears to be useful to analyze the apparent prefactor σ 00(F) and to derive the

effective temperature Teff(T, F) which governs bandtail states occupation and FBTH conductivity σ(T, F)

at low T (Eq. (11)).

Finally, this work shows that both Ohmic and nonlinear regimes are complementary to understand

hopping mechanisms in localized electronic systems, and to derive quantitative parameters for the energy

distribution of bandtail states. Consider first the low field Ohmic regime. For an exponential energy

distribution characterized by a disorder energy E0, we have shown that the linear dependence of ln σ vs.

T –1/4, established for a uniform DOS distribution, still holds; the slope T 01/4 is related to the localization

parameter N(EF) γ –3 (i.e. strongly dependent on the localization radius) while the magnitude of the

prefactor is also related to the disorder energy E0. As shown previously [17], the strong increase in

prefactor values as the slope T 01/4 increases (as found in carbon films and alloys) is a signature of bandtail

hopping, while the opposite correlation is expected for hopping within a uniform DOS distribution. In

contrast, at high field, the demarcation level EDL (T, F) is independent of γ –1 and strongly depends on the

disorder parameter E0, as N(EF) (E0)4. With increasing field, the transport path at energy EDL shifts

towards shallower bandtail states, with a density of states N(EDL) ∼ (eF)3 (E0)–4 in the low T limit

(kT � E0). It has to be stressed that the density of states parameters deduced from the Ohmic regime are

in good agreement with the typical values of N(EF) ≈ 1 × 1018 cm–3 eV–1, E0 ≈ 0.12 eV and (Teff/T) ≤ 10

(Table 1) derived from this high field study.

Acknowledgments The authors are grateful to Dr Virginia Chu (INESC-MN, Lisbon, Portugal) for the processing

of sandwich devices and to Dr Sushil Kumar (NPL, New Delhi) for some conductivity measurements.

Appendix

A1 – Demarcation energy

In the derivation of the demarcation level EDL (T, F), we first write the hopping range as a function of the

polar angle θ = (F, Rij) between electric field and hopping vector. We then integrate over all forward

phys. stat. sol. (b) 244, No. 6 (2007) 2097


Original

Paper

(π/2 < θ < π) electron tunneling transitions in an exponential distribution of band tail states, the respec-

tive probabilities of hopping transitions from a state Ei towards a state Ej which contribute to energy gain

and to energy loss, according to three cases depicted in Fig. 1.

In a transition of a charge (–e) from an occupied site (i) to an unoccupied site (j), separated in distance

by Rij, the electric field modifies the actual energy of a hop from (Ej – Ei) to (Ej – Ei) – (–e) FRij cos θ ,

where θ is the polar angle θ = (F, Rij) between field F and jump vector Rij directions. Using the reduced

coordinates R′ = 2γ R and E′ = E/kT, the hopping range Rij in Eq. (3) becomes a function of b = eF/2γ kT:

2γ Rij → (1 + b cos θ) R′ij + (E′j – E′i) for E′j > E′i – bR′ij cos θ , (A1)

2γ Rij → R′ij for E′j < E′i – bR′ij cos θ . (A2)

Taking into account the distribution of polar angles θ = (F, Rij), corresponding to forward

(π/2 < θ < π) electron tunneling transitions in an exponential distribution of band tail states, the respec-

tive probabilities of hopping transitions from Ei towards Ej which contribute to energy gain and to energy

loss (Fig. 1) are integrated over θ and Ej values. However, due to the exponential dependence of hopping

rates on Rh, the relevant hopping events are restricted to a narrow range (Rh – ∆Rh) < Rij < (Rh + ∆Rh).

Hence, one obtains the following weighting factors W for the different forward hopping transitions:

E gain (T-activated): Ei – eFR cos θ < Ej , W ≈ (1/kT – 1/E0)–1 [exp (–eFR cos θ /E0)] ,

E gain (non-activated): Ei < Ej < Ei – eFR cosθ , W ≈ E0 [exp (–eFR cos θ /E0) – 1] ,

E loss (non-activated): Ej < Ei , W ≈ E0 .

It can be noticed that both non-activated and activated transitions contributing to energy gain have simi-

lar expressions by changing E0 to kT* = (1/kT – 1/E0)–1. For the optimum hopping distance, integrating

over θ angles (0 < θ < π), the equality between energy gain and energy loss probabilities defines implicitly

the demarcation level EDL (T, F) through the value of the optimum hopping characteristic length Rh (EDL):

exp [eFRh(EDL)/E0)] – 1 = 2eFRh(EDL)/(E0 + kT*) . (A3)

A2 – Hopping length at high field

Due to the exponential dependence of tunneling transition rates, hopping towards the second nearest

neighbor in the energy-distance space, will be neglected. Hence, Rh(Ej) is defined by the condition that

the number of states located below Ej (given by Ei + eFR cos θ < Ej), at a distance smaller than Rh, is

equal to unity:

2π � dRR2 � dθ sin θ � dEiN(Ei) = 1 with (Ei + eFR cos θ) < Ej and R < Rh(Ej) . (A4)

Integration of the resulting expression:

(2E02/eF) � dRRN(Ej) sinh (eFRh/E0) = 1, (A5)

provides a relationship between the density of states N(Ej) and the ratio X = (eFRh/E0):

N(Ej) E0(Rh/X)3 [2X cosh (X) – 2 sinh (X)] = 1. (A6)

In contrast with this directional approach (Eq. (A6)), a different expression for Rh(Ej) in an exponential

band tail is obtained from the isotropic integration of:

(4π/3) Rh3 � dEiN(Ei) = 1 with (Ei + eFRh) < Ej , (A7)

which gives a field-dependent hopping length:

(4π/3) N(Ej) E0Rh

3 exp [–eFRh/E0] = 1 . (A8)



A3 – Filling rate and conductivity at the demarcation energy

The filling rate of a given unoccupied electronic site at energy (Ej) is calculated by considering non-

activated hopping transitions from all other states (Ei) within a distance Rh(Ej). In first approximation, we

assume that the occupancy probability is given by a Boltzmann function f(Ei) = exp [–Ei/kTeff] with an

effective temperature Teff for bandtail electrons and setting EF = 0. Restricting the electronic transitions to

be against the field (–1 < u = cos θ < 0), the non-activated forward filling rate ΓF(Ej) of an empty state

located in the vicinity of EDL(T, F) is given by:

ΓF(Ej) = 2π � dRR2 � dθ sin θ � dEiN(Ei) f(Ei) ν0 exp (–2γR) (Ej + eFR cos θ) < Ei

and

Rh (Ej) – ∆Rh < R < Rh (Ej) + ∆Rh . (A9)

Using (kT*eff)–1 = (kTeff)

–1 – (E0)–1,

ΓF(Ej) = 2πν0N(EF) kT*eff � dRR2 exp (–2γR) � du exp [–(Ej + eFRu)/kT*eff) ,

with

Rh(Ej) – ∆Rh < R < Rh (Ej) + ∆Rh . (A10)

The product N(EF) exp [– Ej/kT*eff] = N(Ej) f(Ej) is recognized in the filling rate, which writes:

ΓF(Ej) = 2πν0(kT*eff)2 (eF)–1 N(Ej) f(Ej) � dRR exp (–2γR) [exp (eFR/kT*eff) –1] ,

with

Rh(Ej) – ∆Rh < R < Rh (Ej) + ∆Rh . (A11)

An approximate value of the integral can be obtained by deriving the half width at half maximum of the

integrand function near Rh(EDL): ∆Rh ≈ (ln 2/2) γ –1. Hence, for the transport path near EDL:

ΓF(EDL) = 2π ln 2ν0γ –1(kT*eff)

2 (eF)–1 RhN(EDL) f(EDL) exp (– 2γRh) [exp (eFRh/kT*eff) –1] . (A12)

Using N(EDL) ≈ (eF)3/(E0)4 and (eFRh) ≈ E0, a F – dependent pre-exponential factor, proportional to (E0)–3

(kT*eff)2 (eF) f(EDL), appears in ΓF(EDL). Since the forward partial conductivity is given by

(dσ F/dE) = (eRh /F) N(EDL) [1 – f(EDL)] ΓF(EDL) , (A13)

and f(EDL) � 1 near the transport level, one obtains from Eq. (A12):

(dσ F/dE) = 2π ln 2e2ν0 γ –1(kT*eff)

2 Rh 2(eF)–2 exp (– 2γRh) exp (eFRh/kT*eff) N(EDL)2 f(EDL) , (A14)

which is integrated using ∆E ≈ 3E0(∆R/Rh) = (3 ln 2/2) eFγ –1, to obtain the calculated forward conductiv-

ity:

σF(T, F) ≈ 6π(ln 2)2 e2ν0γ

–2(E0)–6 (kT*eff)2 (eF)3 f(EDL) exp (–2γRh) exp (eFRh/kT*eff) . (A15)

This result is to be compared to the thermally activated partial conductivity value:

(dσohmic/dE) = [4e2ν0γ(Rh0)3 exp (–2γRh

0)/3E0] N(E) f(E) with Rh0 = Rh(F = 0) , (A16)

σohmic(T) = [4 ln 2e2ν 0(Rh0)2 exp (–2γRh

0)] N(ET) f(ET) , (A17)

obtained in the Ohmic regime from the net flow of charge carriers along the direction of a small applied

electric field, where ET is the transport level (Eq. (6)) and the expression under brackets in (Eq. A16) is a

partial apparent mobility µ(E) for electronic states at energy E in the band tail. Although µ(E) is inde-

pendent of T, the shift of the transport level towards shallower electronic states (i.e. with larger N(ET))

provides a temperature-activated µ(ET) with a decrease of Eact (µ) as N(EF) γ–3 increases [20].

phys. stat. sol. (b) 244, No. 6 (2007) 2099


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Paper

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Documents

Scaling analysis of field-enhanced bandtail hopping transport in amorphous carbon nitride