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Physica E 41 (2009) 1025–1028

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Corr

Couches

Cedex, F

E-m

journal homepage: www.elsevier.com/locate/physe

Mott and Efros-Shklovskii hopping conductions in poroussilicon nanostructures

Md. Nazrul Islam a, Sanjay K. Ram b,c,�, Satyendra Kumar c

a QAED-SRG, Space Application Centre (ISRO), Ahmedabad 380015, Indiab Laboratoire de Physique des Interfaces et des Couches Minces (UMR 7647 du CNRS), Ecole Polytechnique, 91128 Palaiseau Cedex, Francec Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

a r t i c l e i n f o

Available online 22 August 2008

PACS:

81.05.Rm

72.20.Ee

73.63.–b

72.80.Cw

Keywords:

Porous silicon

Electrical and electronic properties

Hopping transport

77/$ - see front matter & 2008 Elsevier B.V. A

016/j.physe.2008.08.047

esponding author at: Laboratoire de Phys

Minces (UMR 7647 du CNRS), Ecole Poly

rance. Tel.: +33169 33 43 21.

ail address: sanjayk.ram@gmail.com (S.K. Ram

a b s t r a c t

Electrical conductivity of electrochemically etched porous silicon was studied over a wide temperature

range from 15 to 450 K. Applicability of various electrical transport mechanisms has been critically

analyzed. While the conductivity data above room temperature shows extended state conduction,

lowering the temperature leads to Berthelot-type conduction (180–280 K). Further, Mott’s hopping

(in the range 140–180 K) and Efros-Shklovskii hopping (below 120 K) conduction are found to be

operating in lower temperature ranges. A clear crossover from Mott to Efros-Shklovskii variable range

hopping transport is observed at low temperatures.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The potential applications of porous silicon (PS), which isbasically an ensemble of silicon nanocrystallites, include solarcells [1], chemical and thermal microsensing devices [2], andother optoelectronic devices. For optimization and improvementof device applications, a deeper understanding of the carriertransport behavior and mechanisms in PS is imperative [3–8]. Inorder to investigate the coplanar current transport mechanisms inPS layers, the problems of complex and inhomogeneous micro-structure and the distribution of crystallite sizes in PS need to beaddressed [9,10]. An appraisal of the reported research shows thatdifferent transport mechanisms were interpreted on the basis ofeither structure or disorder-induced localization [11–13].

Previous studies involving measurements carried out over alimited temperature range have suggested that Mott hoppingtransport can be considered at low temperatures, in which case aT�0.25 dependence is followed [4,13]. Berthelot-type conductionwas also reported and attributed to the fractal structure of PS[3,5]. It is therefore obvious that no clear transport mechanism inPS layers has been deduced so far. In the present work, we reportthe results of coplanar dc conductivity measurement of thick

ll rights reserved.

ique des Interfaces et des

technique, 91128 Palaiseau

).

PS layers over a wide temperature range from 15 to 450 K.Applicability of various transport mechanisms has been criticallyanalyzed.

2. Experimental details

Thick PS layers were prepared by electrochemical anodizationof boron-doped (10 0) Si wafers having resistivity of 6–10O cmusing HF (40–48 wt%) and C2H5OH (99.9%) (1:1 by volume). Thedetails of procedure and conditions of sample preparation andelectrical measurements are described in our previous reportedwork [14,15].

The average crystallite size in the PS layers determined fromX-ray diffraction (XRD) analyses, was found to be E25–30 A in thedirection of the (4 0 0) plane. Scanning electron microscopy (SEM)showed a sponge-like microstructure with columnar siliconskeletons, and a porous layer thickness E20–40mm. Room-temperature photoluminescence [16] and micro-Raman measure-ments [9,10] on the same sample spot confirmed the presence ofsilicon nanocrystallites. Micro-Raman measurements demon-strated a bimodal crystallite size distribution in the PS network[9,10].

The temperature dependence of electrical transport in planarwas studied over a wide temperature range from 15 to 450 K. Thecurrent transport mechanism presented here corresponds to ourcurrent–temperature (I–T) data measured at a fixed bias voltage of1.5 V well within the Ohmic range of I–V characteristics over the

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-4.6

M.N. Islam et al. / Physica E 41 (2009) 1025–10281026

entire temperature range. Measured currents are directly relatedto the material conductivity through usual geometrical para-meters.

-5.34.0 4.4 4.8 5.2 5.6 6.0

-5.2

-5.1

-5.0

-4.9

-4.8

-4.7

m = -0.86

m = 0.26

m = 0.55

ln{d

[ln (I

)]/d

[ln (T

)]}

ln (T)

Fig. 2. Plot of ln [d{ln(I)}/d{ln(T)}] vs. ln(T), open circles are experimental data and

lines are least-squares fit. Values of exponent m are also shown in figure.

3. Results

The current measured from 450 K down to 15 K is plottedagainst the reciprocal of T in Fig. 1. An Arrhenius-type activatedbehavior is observed for the dark conductivity in the I–T datameasured above room temperature (inset, Fig. 1). A single currentconduction mechanism, I ¼ I0 exp(�Ea/kT), can be fitted to thewhole data. Here k is the Boltzmann constant and Ea is theactivation energy for conduction. Ea is found to be E0.51 eVconsistent with the results obtained by Ben-Chorin et al. [6]. Thisvalue is comparable to the activation energy in intrinsic c-Si. Threedistinct regions having different slopes can be seen in Fig. 1.Region-A shows relatively fast varying current with temperaturefor T4170 K. The value of Ea is found to be around 50 meV in thistemperature range. Region-B is weakly temperature dependentand results in a very small value of Ea (E8 meV) for 40–120 K. InRegion-C (To40 K) the current becomes almost independent of T

and remains constant down to the lowest measurement tempera-ture (15 K). These values of Ea are very low compared with thevalues obtained from high temperature I–T data. Such acontinuous lowering of Ea as a result of cooling indicates thatthe tunneling of carriers may be a dominant form of currentconduction at lower temperatures.

In a disordered system, the carrier conduction at lowtemperatures may be assisted by variable range hopping (VRH)given as [17]:

InðIÞ / ðTx=TÞm, (1)

The value of m depends on the grain sizes and the temperatureregion. The exponent m gives information on the type of carrierconduction mechanism. In a bulk material, m ¼ 1

4 is attributed toMott VRH conduction where electron–electron (e–e) interactionsare neglected. On the other hand, for Efros-Shklovskii (ES)hopping mechanism including long-range e-e interactions, m ¼ 1

2

10-11

10-12

102.4 2.7 3.0 3.3

-11

10-10

10 -9

10 -8

Ea =8 meV

E a=50 meV

(C)

(B)

(A)

Cur

rent

(A)

1000/T (K-1)0 10 20 30 40 50 60 70

Ea =0.51 eV

Cur

rent

(A)

1000/T (K-1)

Fig. 1. Semi logarithmic plot of dark current against reciprocal temperature from

15 to 450 K. Inset shows the Arrhenius behavior in I–T data above room

temperature. Symbols represent experimental data and the lines are the least-

squares fit.

is expected [18]. Further, m ¼ �1 is expected for Berthelot-typeconduction. In this equation, Tx is a constant for the material. Inthe following text and equations, ‘x’ is replaced with ‘M’ for Mott’shopping, ‘ES’ for ES hopping and ‘B’ for Berthelot-type conduction.

On fitting the I–T data of Fig. 1 to Mott’s hopping model andBerthelot-type current conduction, we observed that I–T data of180–300 K range are fitted equally well to both, resulting in anambiguous interpretation of current transport mechanisms. Onthe other hand, none of these two models were able to explain thelower temperature (o170 K) results, as they resulted in absurdmaterial parameters. These are further discussed in the nextsection.

In order to differentiate between the Mott and Berthelot typeof conductions and also to get an insight into the possibletransport mechanism below 150 K, we have plotted ln[d{ln(I)}/

d{ln(T)}] vs. ln(T) in Fig. 2, where the exponent m Eq. (1) can beobtained from slope [19,20]. The lines in Fig. 2 are least-squares fitto the experimental data (open circles). Three different values ofm are observed in different temperature regimes. In particular,m ¼ 0.55 for 50 KoTo120 K, m ¼ 0.26 for 140 KoTo180 K andm ¼ �0.86 for 180 KoTo290 K are obtained. The first value ofm ¼ 0.55 suggests existence of ES hopping within the Coulombgap near the Fermi energy. The later two slopes of m indicate theMott hopping and Berthelot-type conduction, respectively.

4. Discussion

Having obtained an idea about the different possible conduc-tion mechanisms acting in the different temperature regions, wenow proceed to take up each in detail, to determine the goodnessof fit, and whether the fit results in physically plausible materialparameters.

A fitting of the low-temperature data (temperature range180–290 K) to the Berthelot-type conductivity expression, I ¼ I0

exp(T/TB) yields TB ¼ 90 K, as demonstrated in Fig. 3. This value ofTB lies well within the reported range E20–100 K [3,5]. Accordingto the expression of barrier width (a) given in Ref. [21] usingBerthelot type of conduction, we found the value of aE2.3 nm(for TB ¼ 90 K and carrier effective mass ¼ 0.1me) in our system.

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2.5x10-11

7.5x10-12

5x10-12

2.5x10-12

10-12

10-11

150 200 250 300

TB = 90 K

Cur

rent

(A)

T (K)

Fig. 3. Logarithmic dependence of dark current with temperature T, showing

Berthelot-type behavior. Symbols represent experimental data and the line is

least-squares fit.

4.0x10-12

3.5x10-12

3.0x10-12

2.5x10-12

0.270 0.275T-0.25 (K-0.25)

100/T0.5 (K-0.5)

TM= 7x104K

0.280 0.285 0.290 0.295

-11.6710 12 14 16

-11.65

-11.64

-11.63

-11.61

-11.60

log

[Cur

rent

(A)]

Cur

rent

(A)

TES=8 K

Fig. 4. Low-temperature I(T) data as a function of T�1/4 in the temperature range

(140–180 K). The inset shows a least-squares fit to the log (I) vs. 100/T1/2 data in the

temperature range from 40 to 110 K. Symbols represent experimental data and the

lines are the least-squares fit.

M.N. Islam et al. / Physica E 41 (2009) 1025–1028 1027

This value of a is of the same order of crystallite sizes in the PSlayer, which supports the applicability of Berthelot-type behaviorin this temperature range. The possibility of Berthelot-typeconduction behavior can be thought of as a physical consequenceof the tunneling of carriers between thermally vibrating sitesplaced on the surface of PS [22,23].

Now, we take up the Mott’s hopping model for the temperaturerange 140–180 K in Fig. 4. The value of TM is found to be 7�104,where TM is related to the density of states (DOS) at the Fermienergy (Nf) by expression TM ¼ bMa3/(kBNf), where a is the decayconstant of localized wave function and bM is a constant. Weestimated the value of NfE1021 eV�1 cm�3 for bM ¼ 16 anda�1¼ 10 A in our case. There maybe some variations in the value

of bM, as discussed by Rosenbaum et al. [24]. The parameters likelocalization length (a�1) and hopping distance (Rhop) are related inMott’s case as Rhop ¼ 0.4(TM/T)(1/4)/a. We obtained RhopE17 A(T ¼ 200 K) and E19 A (T ¼ 150 K) for the values of TM ¼ 7�104

and a�1¼10 A. Since the hopping length increases as the

temperature decreases, the effective diameter of silicon columnd/R is also reduced (d being the distance between the pores).Using relation R(T)/dE1, the mean diameter of the siliconquantum wires comes out to be E20–25 A. This value isconsistent with the value we obtained from SEM. Our Ramanscattering and XRD results also suggest similar mean sizes of thecrystallites [9,10].

Now, a least-squares fit to the log (I) vs. 100/T0.5 data in thetemperature range from 40–110 K is shown in the inset of Fig. 4,where TES is estimated to be E8 K. In understanding ES hoppingmechanism, the long-range Coulomb interaction between thelocalized state electrons is known to cause the DOS to tend to zeroin the vicinity of Fermi energy (Ef), which results in a soft energygap, i.e., Coulomb gap corresponding to the parabolic DOS. Theconstant TES is related to Coulomb gap energy (D) and DOS (g0) as:D ¼ (k/2)(TESTr)

1/2 and g0 ¼ 38p2k2er3eo

3/(29e6) (TESTr), respectively[25–27]. Here, er is the relative dielectric constant, e the electroniccharge and Tr the transition temperature between Mott and EShopping conduction [25–27]. As seen in Fig. 2, the transition fromMott to ES hopping is quite clear and Tr is found to be E150 K.

Using this value of TES, we find DE1.5 meV and g0E1015�1016

eV�1 cm�3 (for er ¼ 2–4) [13]. We note that the DOS thus obtainedis lower than that obtained from Mott hopping conduction at

higher temperatures. Theoretically, the DOS from ES conduction isexpected to be lower than that from Mott conduction due to theCoulomb interaction, which should reduce the DOS near the Fermilevel [18,26].

Similar to the Mott’s hopping model, in the ES hopping case,we calculate RhopE[0.25(TES/T)(1/2)/a] and aE(TESkBere0)/bESe2,where bES ¼ 0.83, is a constant [24,26,27]. The value of relativedielectric constant varies in an inhomogeneous system as PS, andthus assuming the value of er of PS to lie between the values of c-Siand amorphous silicon, the possible values of Rhop may beE1050 A (for er ¼ 10) [6] and 5300 A (for er ¼ 2) at 100 K [13]. Itis known that for ES hopping to be valid, the energy involved inthe hopping should be smaller than the width of the Coulomb gap[18,26]. Therefore, the temperature-dependant hopping distancein ES hopping mechanism should be longer than the temperature-dependent hopping distance of Mott [26]. Accordingly, the valuesof Rhop obtained by us for ES hopping mechanism are higher thanthose for Mott VRH. Likewise, higher values are found for thelocalization length (a�1)E1.5–3.7 m (for er ¼ 10 to 2) as well.Similar behavior in the values of a�1 and Rhop are also reported byseveral researchers [24,25,28]. The variations in the obtainedphysical parameters with different temperatures also emphasizethe inhomogeneous fractal nature of PS microstructure. Therefore,TES predicts reasonable values for the Coulomb gap and DOS in oursystem and ES hopping is a possible process that can lead to aparabolic DOS around Ef at low temperatures.

5. Conclusions

We have studied the temperature-dependent electrical con-ductivity of porous Si nanostructures over a wide range oftemperature (15–450 K). Different conduction mechanisms arefound to be predominant in different temperature zones in theform of exp(const� T–m). Berthelot type of conduction behaviorwas interpreted for the value of mE�0.86 in the high-temperaturezone (180–290 K). On the other hand, the I–T data at the lowesttemperature range (o120 K), fits well for the value of m E0.55,indicating Efros-Shklovskii hopping, i.e., carrier conduction in the

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defect states near the Fermi level by variable range hoppingwithin the Coulomb gap. In addition, the middle zone of thetemperature range (140–180 K) indicates Mott’s variable rangehopping. A crossover from ES hopping to Mott hopping can beclearly seen at TrE150 K.

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