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Conductance of island and granular metal films
Solid State Communications 180 (2014) 39–43
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Conductance of island and granular metal films A.P. Boltaev, F.A. Pudonin, I.A. Sherstnev n Lebedev Physical Institute of the Russian Academy of Sciences, Leninskiy prospekt, 53, 119991 Moscow, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 31 March 2013 Received in revised form 1 October 2013 Accepted 18 November 2013 by F. Peeters Available online 23 November 2013
Results of measurements of the specific surface conductance of island metal films at different temperatures are presented. The study of conduction allowed us to establish processes, which determine the transfer of charge carriers in granular and island metal structures. These processes determine the excess charge carriers concentration in film and, on the other hand, characterize the transfer speed of excess charge carriers from one island to another (i.e. mobility). Moreover, these processes occur independently from each other. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Thin film A. Island film C. Percolation D. Conductance
1. Introduction Conductance of island and granular metal films in insulating state (conductance increases with the temperature growth) is widely discussed in the literature [1–16]. It is established that transfer of charge carriers in granular and island films is caused by tunnelling of electrons between islands. Tunnelling of electrons from one neutral island to another changes energy of the system by the value approximately equal to the island charge energy E e2 =2C, where e is the electron charge, C is the island capacitance. Under such charge transfer conditions the conductance should be described by activation dependence (Arrhenius law) s ¼ s0 expð E=kTÞ, where k is the Boltzmann constant, and T is the temperature [15]. However, it has been experimentally found that conductance of the island metal structures and nanocomposites often depends on the temperature according to the expression [1,7]: s ¼ s0 exp½ ðT 0 =TÞ0:5 ;
ð1Þ
where s0 is the film conductance at high temperatures and T0 is material dependent parameter. Thus, most of the experimental data are described by the “1/2 law” [1–3]. Different theoretical models have been used to explain the “1/2 law” [4–8,15]. In general, these models are a modification of the theory of hopping conduction in semiconductors, where the “1/2 law” is interpreted as the appearance of Coulomb gap in the density of electronic states near the Fermi level. Papers [4,5] point to the important role of the Coulomb interaction between charged particles. There are also other models explaining the “1/2 law”, for example, in the paper [7] conductance n
Corresponding author. Tel.: þ 7 9191027160. E-mail address: [email protected] (I.A. Sherstnev).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.11.022
of granular structures is associated with the variation in sizes of metal granules. Several studies have shown that degree (x) in expression describing nanocomposite conductance is not always equal to 1/2. In the paper [8] it is shown that, when the hopping length is less than size of islands and close to the distance between them, the standard theory of hopping conduction with the variable hopping length is not applicable. In the paper [9] it was found that x¼ 0.75, in the paper [10] x ¼0.72, in the paper [11] x ¼1. In the paper [11], where the mechanism of conduction in island metal films of Au, Ni, and Pt was studied, authors experimentally have shown that conductance of 1 nm thick film changes with temperature according to the Arrhenius law. Also, it was noted that with the increase of film thickness up to 3 nm deviation from the activation dependence has been observed. The reasons of such deviations from the dependence with x ¼1/2 still remain the subject of debate. The nature itself of the conductance dependence of island and granular films in accordance to s ¼ s0 exp½ ðT 0 =TÞ0:5 in wide temperature range (kT o E e2 =2C) remains mystery [17]. In this paper we will show that change of conductance in granular and island films in wide temperature range is defined by generally accepted activation and tunnel processes, but, what is very important, these processes take place independently. This approach to the problem of charge carriers transfer in granular and island films allows us to explain the conductance dependence on the temperature with any degree (x) in the expression.
2. Problem formulation and experiment We shall consider, as in the paper [11], that current flow in the island metal films and granular systems is caused by two
40
A.P. Boltaev et al. / Solid State Communications 180 (2014) 39–43
consecutive processes. The first process determines the excess charge carriers concentration in the island or granular film and it is associated with the excitation of electrons from the traps to the neutral islands with the charged island formation or with the tunnelling of electrons from one neutral island to another with the formation of positively and negatively charged islands. This process occurs with system energy change on the value E e2 =2C. At low temperatures positively and negatively charged islands can be separated by neutral islands. Type of electron transfer process from one neutral island to another (e.g. virtual hopping process [15,16]) has an impact only on the relaxation time of the equilibrium concentration of charged islands at given temperature. This first process determines the concentration of islands with excess charge carriers. The greater the concentration of charged islands, the higher the system conductance. Second process characterizes the transition speed of the excess charge carriers between the islands (i.e. mobility) and it is caused by tunnelling of charge carriers from charged island to the neutral one. This process adds term in conductance that takes barrier properties dependence into account s p expð L=λÞ, where L is the hopping length, λ ¼ ℏ=ðmWÞ0:5 is the length of electron wave function decay in dielectric that separates metal islands, m is the electron mass, W is the tunnelling barrier height (practically equal to a half-width of the dielectric band gap). This kind of tunnelling changes the energy of the system by ΔE e2 ð1=C 1 1=C 2 Þ, where C1 and C2 are the capacitances of different size islands. If islands are identical then the energy of the system does not change. We must note that charge energy E e2 =2C b ΔE e2 ð1=C 1 1=C 2 Þ, so later we neglect ΔE. Therefore the conductance of island film in insulator state can be defined as follows: s ¼ s0 expð L=λ E=kTÞ;
ð2Þ
We assume that this equation can describe any kind of experimental conductance dependence in island structures, including Eq. (1). We assume that experimental dependence of the granular and island structures conductance, which is described by Eq. (1), is, firstly, a consequence of activation energy E magnitude variation. Secondly, the dependence can be caused by change of the hopping length L with the structure temperature change. Moreover, these two processes occur independently from each other. The first process is responsible for the excess charge carriers concentration in the film. Variation of the activation energy magnitude in granular and island systems may be associated with the presence of charged defects in dielectric matrix of the structure [16] or with the variation of granule size. For example, in the structures with insulating conductance the island size can vary from D ¼ 10 nm to D ¼ 200 nm. Charge energy of islands ðE e2 =ɛDÞ in its turn can vary from 0.005 eV to 0.1 eV [1,11,12]. In this case, the conductance dependence on the temperature differs from the activation dependence s ¼ s0 expð E=kTÞ, in which the activation energy is constant. For example, let us review island metal film, which contains two types of islands. The first type with D1 island size and N1 island concentration, and the second type with D2 island size and N2 island concentration. Charge energy of these islands equals, E1 e2 =ɛD1 and E2 e2 =ɛD2 (E1 ⪡E2 or D1 ⪢D2 ). At low temperatures (T E1 =k) due to tunnelling transitions of electrons between neutral islands or due to electron excitation from the traps to the neutral islands part of the islands becomes positively or negatively charged. The excess electrons or holes are captured by islands with minimum charge energy in the first place. In our case, it is the islands with the size D1, charge energy E1, and charged islands concentration n1 ¼ N 1 expð E1 =kTÞ [7,11].
Charge carriers transfer in the electric field will be carried out due to tunnelling transitions between charged and neutral islands of size D1. Energy of the system does not change, because islands have the same size, hence, the tunnelling probability does not depend on the temperature. Conductance dependence on the temperature in the proximity of T E1 =k will be of the activation type with the activation energy close to E1. With the temperature growth the excess charge carriers will be excited on smaller islands (of D2 size) with charge energy E2 e2 =ɛD2 . Charged islands of size D2 will now participate in tunnelling transition of charge carriers. Activation type of conductance still remains with temperature growth. As two types of islands take part in the charge carriers transfer, the process will be characterized by the aggregate activation energy (ES). The ES value will depend on the temperature and concentration of charged islands n1 ¼ N 1 expð E1 =kTÞ and n2 ¼ N 2 expð E2 =kTÞ. Activation energy ES in this case can be represented by the following expression: ES ¼
E1 N 1 expð E1 =kTÞ þ E2 N 2 expð E2 =kTÞ ; N 1 expð E1 =kTÞ þ N 2 expð E2 =kTÞ
where E1 N 1 expð E1 =kTÞ and E2 N 2 expð E2 =kTÞ are changes in energy of the systems, in which n1 and n2 charged islands are exited. The activation energy ES will increase with the temperature growth from ES E1 to ES -E2 (E1 ⪡E2 ). The dependence of energy ES on the temperature and, consequently, the conductance dependence on the temperature is determined by the distribution of island sizes. If islands are identical then the activation energy is constant and the conductance dependence on the temperature obeys the Arrhenius law s ¼ s0 expð E=kTÞ [11]. On the other hand, in paper [16] authors report that they have created periodic granulated structures and the island size was controlled with an accuracy of few percent. However, the “1/2 law” has been observed in these structures. As suggested by the authors, such conductance dependence on the temperature in the periodic granular systems can be connected with the presence of charged defects in dielectric matrix [16], which creates a random potential in the structure and leads to the disorder of charge energy. In general case of arbitrary island size distribution the activation energy ES can be presented as follows: ES ¼
∑n1 Ei N i expð Ei =kTÞ ; ∑n1 N i expð Ei =kTÞ
ð3Þ
where i ¼ 1; 2; 3; …; n characterizes specific size of the islands and their concentration. It should be emphasized that with the temperature growth it is becoming possible for one, two, three, and more excess electrons to transit on larger islands. The charge energy, for example, of size D island will have a value of E e2 =ɛD, E ð2eÞ2 =ɛD, E ð3eÞ2 =ɛD, etc. The second process determines the charge carriers transfer in the electric field and caused by tunnelling transitions between charged and neutral islands, which are characterized by the hopping length L. Obviously, any kind of conductance dependencies for the island and granular films s ¼ s0 exp½ ðT 0 =TÞx , where x¼ 0.5 [1–3], x¼ 0.75 [9], x ¼0.72 [10], x ¼1 [11], can be explained by the dependence s ¼ s0 expð L=λ ES =kTÞ, where the activation energy ES and the hopping length L depend on the temperature, but do not depend on each other. To verify these assumptions we have created island metal films of Tungsten (W) from 0.63 to 2 nm thick. The topography of the film surface was studied by the atomic force microscopy (AFM) method. The differential conductance dependence of metallic films on the temperature was measured. Tungsten films were grown by the RF-sputtering method. At the beginning of the process a vacuum chamber with the sample was kept at 2 10 6 mbar pressure. Then Tungsten was sputtered
A.P. Boltaev et al. / Solid State Communications 180 (2014) 39–43
41
Fig. 1. AFM-images of Tungsten films on dielectric substrate. (a) 0.63 nm; (b) 0.8 nm.
Table 1 Experimentally defined and calculated parameters of island films. No. of Film type T 0 ; s0 sample and thickness
Fig. 2. Specific surface conductance dependence on temperature for Tungsten films of different effective thicknesses (sample 1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm; 5 – 2.0 nm). Solid lines correspond to “1/2 law” fitting.
on a cold glass substrate (temperature did not exceed 70 1C) in an argon atmosphere (a pressure of 10 3 mbar); RF-power on the target was at 700 W. Effective thickness of metal layers was determined from the deposition time and well-known deposition rate. AFM measurements of film topography show that films with a thickness of 0.9 nm and less have island structure. Islands in each film have a wide range of sizes. Longitudinal dimensions of the islands in 0.63 nm thick film [Fig. 1(a)] vary from 5 nm to 30 nm, and in the 0.8 nm film [Fig. 1(b)] vary from 8 nm to 100 nm. Samples with obtained structures (of 0.5–1.5 mm width and of 2–3.5 mm length) were made for the differential conductance measurements. Measurements were carried out at the 77–300 K temperature range. Frequency dependence of the conductance for all samples remained unchanged until f ¼ 10 kHz. Differential conductance measurements were carried out at frequency f ¼ ω=2π ¼ 2:5 kHz. In this study voltage U ¼ U 1 cos ðωtÞ (U 1 ¼ 10 2 V) was applied to the sample through Indium Ohmic contacts created of the film surface. The error of measurement did not exceed 5%. The specific surface differential conductance dependence on temperature of Tungsten films on dielectric substrate is shown in Fig. 2 for five structures with various effective thicknesses. As can be seen in the figure, conductance of the Tungsten film changes by seven orders of magnitude for different films. Conductance increases with the temperature growth.
3. Results and discussion According to our model the value of conductance is proportional to the product of probability of charge carriers tunnelling
Island size, defined from AFM-images
Island size, defined from conductance dependence DS ¼ 6…15 nm
1
Tungsten, T 0 ¼ 9300 K s0 ¼ 1:1 105 Ω 1 0.63 nm
D ¼ 5…30 nm
2
Tungsten, T 0 ¼ 2100 K s0 ¼ 4:7 105 Ω 1 0.8 nm
D ¼ 8…100 nm DS ¼ 10…40 nm
3
Tungsten, T 0 ¼ 72 K s0 ¼ 8:11 105 Ω 1 0.95 nm
4
Tungsten 1.16 nm
T 0 ¼ 25 K s0 ¼ 1:5 104 Ω 1
6
Gold [16]
T 0 ¼ 4000 K
DS ¼ 80…180 nm DS ¼ 120…300 nm D ¼ 5:5 nm
ɛ ¼ 4, DS ¼ 12…16 nm ɛ ¼ 6, DS ¼ 8…10 nm
between the islands on the probability of activation [see Eq. (2)] [5,6,11]. Presented in Fig. 2 measured temperature dependencies of the samples 1–5 conductance can be easily approximated by the “1/2 law”. For the convenience Fig. 2 contains only several experimental points. Theoretical dependencies presented as solid lines in Fig. 2. The values of parameters T0 and s0 for samples 1–4 are presented in Table 1. As can be seen in Fig. 2, in our case, the experimental conductance dependence on the temperature almost coincides with the “1/2 law” dependence. However, as noted by some authors, the “1/2 law” not always explains the conduction in disordered systems. Probably, the mechanism of current flow in the island and granular systems due to two consecutive processes mentioned above will explain more precisely the nature of transfer of charge carriers in disordered systems. According to above considerations and specific surface conductance measurements in the metal films we will show that mechanism of charge carriers transfer in the island and granular films in wide temperature range is mainly determined by the activation process. Activation energy ES, which determines the excess electrons and holes concentration, depends on the island size dispersion and the temperature. Activation energy ES we will determine from the temperature dependence of the film conductance (Fig. 2). Then from the temperature dependence of the energy ES we will calculate the island size (D e2 =ɛES ) and compare it with island size, which have been identified from the AFM studies. If these two sizes would be in satisfactory agreement then proposed above activation model that describes
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the conductance dependence on the temperature of island films can be considered fair. In order to find the temperature dependencies of ES and L=λ from the measured conductivity values we are using Eqs. (1) and (2). We take into consideration several values of temperature, for which ES and L=λ values will be defined. We assume that in the vicinity of these temperatures (5% of the value) ES and L=λ values do not depend on the temperature (hence temperature dependencies for whole temperature range are smooth). Experimental data for this small temperature range still contains several points, so it can be approximated by s ¼ s0 expð L=λ ES =kTÞ, where L=λ and ES are constant parameters. Repeating this procedure for several temperatures we obtain temperature dependencies for ES and L=λ. Values of activation energy ES and ratio L=λ for four samples (1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm) versus the temperature are presented in Figs. 3 and 4 respectively. Fig. 3 also shows the energy ES dependence on the temperature for a threelayer periodic gold structure [16], which was determined from conductance dependence on the temperature. From the energy ES dependence on the temperature we calculated the island size in the films. It was assumed that ES e2 =2C S , where CS is the average charged island capacitance at given temperature. The isolated island capacitance was taken equal to C S ¼ ɛDS =2. It was assumed that close neutral islands in the film have weak influence on charged island capacitance. The island sizes calculated from the energy ES dependence on the temperature and measured from the AFM images presented in Table 1. Relative permittivity of dielectric between the islands in single layer films 1–4 was assumed to equal 3 in these calculations. In Table 1 the size DS of granules in a three-layer periodic gold structure [16] is presented. For the granule size calculation of the three-layer sample value of permittivity was taken equal to 4, as the authors of the work [16] stated. But if the value of permittivity were taken equal to 6 the calculated size of the granules would be closer to the size measured from the AFM images D¼ 5.5 nm [16]. While comparing sizes D and DS it is necessary to remember that D is the real size of the island. On the other hand, DS is the
Fig. 3. Activation energy ES dependence on temperature for Tungsten films of different effective thicknesses (sample 1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm) and (6) for three-layer gold film [16].
Fig. 4. Temperature dependence of L=λ parameter for Tungsten films of different effective thicknesses (sample 1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm).
average size of the islands, which contain excess charge at given temperature. Moreover, at any temperature of the sample the condition Dmin o DS o Dmax , where Dmin (Dmax ) is the minimum (maximum) island size in the particular structure, should be satisfied. Indeed, at low temperatures excess charge carriers are induced not only on large islands but also on smaller ones. Therefore, the average size of the islands, which have excess charge carriers, DS o Dmax . On the other hand, at high temperature both small and large islands will be partially filled. In this case, the condition Dmin o DS is satisfied. As it can be seen from the results presented in Table 1, the condition Dmin oDS oDmax is satisfied for the samples 1, 2 and 6, for which the real island size is known. Thus, we established that temperature dependence of conductance of island and granular systems is indeed of the activation type and the deviation from the Arrhenius law is determined by the variation of activation energy in magnitude. Variation in magnitude of activation energy can be associated not only with the variation in island sizes but also, perhaps, with the influence of a random potential. This is clearly seen from the structure 6 data (Table 1), where real island size equal to D ¼ 5:5 nm and size obtained from the analysis of the conductance dependence on the temperature ranges from DS ¼ 12 nm to DS ¼ 16 nm when ɛ ¼ 4. Hopping length of an electron presented in Fig. 4 turned out to be unexpectedly small. Indeed, if we take into account that the length of electron wave function decay is approximately equal to λ ¼ ℏ=ðmWÞ0:5 0:2 nm, the highest hopping length in the structure 1 approximately equal to L 1:2 nm (Fig. 4). In the rest of the structures hopping length is less. This hopping length is significantly less than minimum island size in the structure 1. Secondly, to make a jump from one neutral island to another it is necessary to spend energy value equal to ES e2 =2ɛL 0:5 eV. The probability for electron to have such energy at temperature T 100 K is very low. Considering the above we can conclude that tunnelling to length L 1:2 nm and less can be carried out only between neighboring islands without any loss of energy. Jump on the neighboring island in this case possible only between charged and neutral islands of about the same size. Hopping length decreases with the temperature growth. With the temperature growth smaller islands begin to take part in charge carriers transfer; the distance between them is less, so that explains the decrease. According to the mentioned above mechanism of charge carriers transfer in island and granular systems tunnelling to nearby islands without loss of energy determines transfer (i.e. mobility) of charge carriers. This transfer occurs between charged and neutral islands and it is characterized by hopping length L. It should be noted that in the case of L λ approximate equation determining hopping length L should be replaced with more precise one. However, in our case it is not necessary because calculations of length L, which is smaller than island size, have shown that electrons jump between neighboring islands. Thus, the mechanism of current flow in island and granular systems due to two consecutive processes fully explains the nature of charge carriers transfer in structures 1–4 and in the sample 6 [16]. Conductance of the sample 5 (Fig. 2) equals to s ¼ 7 1 10 4 Ω and practically does not depend on the temperature. The thickness of the film equals to 2.0 nm, hence, resistivity of the film 5 is ρ ¼ 3:8 10 4 Ω cm. If the resistivity value of the film lies within ρ ð2…3Þ 10 4 Ω cm, that film, according to the Ioffe–Regel–Mott criterion, is on the border between insulator and metal. Since the resistivity value of the sample 5 is close to the criterion and practically does not depend on the temperature, we can assume that this film defines the border of metal–insulator transition.
A.P. Boltaev et al. / Solid State Communications 180 (2014) 39–43
4. Conclusion The current flow mechanism in island and granular systems due to two consecutive processes fully explains the charge carriers transfer in the films with conductance close to the threshold of metal–insulator transition (see Fig. 2, sample 4), and in the films, in which conductance is several orders of magnitude less than the threshold (Fig. 2, samples 1–3). Main difference between conductance dependencies of the films is in the value of activation energy. In sample 1 the activation energy ES ⪢kT (Fig. 3) at all temperatures and conductance changes by two orders of magnitude with temperature change in three times (Fig. 2). In sample 4 ES ⪡kT (Fig. 3). These films can be described by power-law dependence [18]. Performed studies of the conductance allowed us to establish the processes that determine the charge carriers transfer in granular and island structures. These processes determine the excess charge carriers concentration in the islands and, on the other hand, characterize the speed of excess charge carriers transfer from one island to another (i.e. mobility). Moreover, these processes occur independently from each other. The process that determines the excess charge carriers concentration in the islands is mainly related to the electron transfer from neutral island to the other neutral island and it is of the activation type; the deviation of the conductance dependence from the Arrhenius law determined by the variation of the activation energy in magnitude. Activation energy increases with the temperature growth. It should be emphasized that electron transfer from one neutral island to another (e.g. virtual hopping process [15,16]) may have an impact only on the time of establishing the equilibrium concentration of charged islands at given temperature. The transfer of excess charge carriers from one island to another is caused by tunnelling of electrons or holes from charged
43
islands to neutral islands. Moreover, the excess charge carriers tunnel to the nearest neutral island without changing the energy value. The average hopping length decreases with temperature growth.
Acknowledgments This work was supported by RFBR Grant no. 11-02-00748-a and contract no. 16.513.11.3143 of Ministry of Education and Science of the Russian Federation. References [1] B. Abeles, P. Sheng, M.D. Coutts, Y. Arie, Adv. Phys. 24 (1975) 407. [2] B.A. Aronzon, D.Y. Kovalev, A.E. Varfolomeev, A.A. Likal'ter, V.V. Ryl'kov, M.A. Sedova, Phys. Solid State 41 (1999) 857. [3] D.A. Zakheim, I.V. Rozhansky, I.P. Smirnova, S.A. Gurevich, J. Exp. Theor. Phys. 91 (2000) 553. [4] J. Klafter, P. Sheng, J. Phys. C 17 (1984) L93. [5] S.T. Chui, Phys. Rev. B 43 (1991) 14274. [6] J. Zhang, B.I. Shklovskii, Phys. Rev. B 70 (2004) 115317. [7] E.Z. Meilikhov, J. Exp. Theor. Phys. 88 (1999) 819. [8] I.P. Zvaygin, R. Keiper, Phys. Stat. Sol. B (2002) 151. [9] N. Markovic, C. Christiansen, D.E. Grupp, A.M. Mack, G. Martinez-Arizala, A.M. Goldman, Phys. Rev. B 62 (2000) 2195. [10] C.J. Adkins, E.G. Astrakharchik, J. Phys.: Condens. Matter 10 (1998) 6651. [11] C.A. Neugebauer, M.B. Webb, J. Appl. Phys. 33 (1962) 74. [12] A.P. Boltaev, N.A. Penin, A.O. Pogosov, F.A. Pudonin, J. Exp. Theor. Phys. 99 (2004) 827. [13] A.P. Boltaev, N.A. Penin, A.O. Pogosov, F.A. Pudonin, J. Exp. Theor. Phys. 96 (2003) 940. [14] A.P. Boltaev, F.A. Pudonin, J. Exp. Theor. Phys. 103 (2006) 436. [15] I.S. Beloborodov, A.V. Lopatin, V.M. Vinokur, K.B. Efetov, Rev. Mod. Phys. 79 (2007) 469. [16] T.B. Tran, I.S. Beloborodov, X.M. Lin, T.P. Bigioni, V.M. Vinokur, H.M. Jaeger, Phys. Rev. Lett. 95 (2005) 076806. [17] K.B. Efetov, A. Tschersich, Phys. Rev. B 67 (2003) 174205. [18] A.S. Ioselevich, D.S. Lyubshin, JETP Lett. 90 (2009) 672.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Conductance of island and granular metal films A.P. Boltaev, F.A. Pudonin, I.A. Sherstnev n Lebedev Physical Institute of the Russian Academy of Sciences, Leninskiy prospekt, 53, 119991 Moscow, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 31 March 2013 Received in revised form 1 October 2013 Accepted 18 November 2013 by F. Peeters Available online 23 November 2013
Results of measurements of the specific surface conductance of island metal films at different temperatures are presented. The study of conduction allowed us to establish processes, which determine the transfer of charge carriers in granular and island metal structures. These processes determine the excess charge carriers concentration in film and, on the other hand, characterize the transfer speed of excess charge carriers from one island to another (i.e. mobility). Moreover, these processes occur independently from each other. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Thin film A. Island film C. Percolation D. Conductance
1. Introduction Conductance of island and granular metal films in insulating state (conductance increases with the temperature growth) is widely discussed in the literature [1–16]. It is established that transfer of charge carriers in granular and island films is caused by tunnelling of electrons between islands. Tunnelling of electrons from one neutral island to another changes energy of the system by the value approximately equal to the island charge energy E e2 =2C, where e is the electron charge, C is the island capacitance. Under such charge transfer conditions the conductance should be described by activation dependence (Arrhenius law) s ¼ s0 expð E=kTÞ, where k is the Boltzmann constant, and T is the temperature [15]. However, it has been experimentally found that conductance of the island metal structures and nanocomposites often depends on the temperature according to the expression [1,7]: s ¼ s0 exp½ ðT 0 =TÞ0:5 ;
ð1Þ
where s0 is the film conductance at high temperatures and T0 is material dependent parameter. Thus, most of the experimental data are described by the “1/2 law” [1–3]. Different theoretical models have been used to explain the “1/2 law” [4–8,15]. In general, these models are a modification of the theory of hopping conduction in semiconductors, where the “1/2 law” is interpreted as the appearance of Coulomb gap in the density of electronic states near the Fermi level. Papers [4,5] point to the important role of the Coulomb interaction between charged particles. There are also other models explaining the “1/2 law”, for example, in the paper [7] conductance n
Corresponding author. Tel.: þ 7 9191027160. E-mail address: [email protected] (I.A. Sherstnev).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.11.022
of granular structures is associated with the variation in sizes of metal granules. Several studies have shown that degree (x) in expression describing nanocomposite conductance is not always equal to 1/2. In the paper [8] it is shown that, when the hopping length is less than size of islands and close to the distance between them, the standard theory of hopping conduction with the variable hopping length is not applicable. In the paper [9] it was found that x¼ 0.75, in the paper [10] x ¼0.72, in the paper [11] x ¼1. In the paper [11], where the mechanism of conduction in island metal films of Au, Ni, and Pt was studied, authors experimentally have shown that conductance of 1 nm thick film changes with temperature according to the Arrhenius law. Also, it was noted that with the increase of film thickness up to 3 nm deviation from the activation dependence has been observed. The reasons of such deviations from the dependence with x ¼1/2 still remain the subject of debate. The nature itself of the conductance dependence of island and granular films in accordance to s ¼ s0 exp½ ðT 0 =TÞ0:5 in wide temperature range (kT o E e2 =2C) remains mystery [17]. In this paper we will show that change of conductance in granular and island films in wide temperature range is defined by generally accepted activation and tunnel processes, but, what is very important, these processes take place independently. This approach to the problem of charge carriers transfer in granular and island films allows us to explain the conductance dependence on the temperature with any degree (x) in the expression.
2. Problem formulation and experiment We shall consider, as in the paper [11], that current flow in the island metal films and granular systems is caused by two
40
A.P. Boltaev et al. / Solid State Communications 180 (2014) 39–43
consecutive processes. The first process determines the excess charge carriers concentration in the island or granular film and it is associated with the excitation of electrons from the traps to the neutral islands with the charged island formation or with the tunnelling of electrons from one neutral island to another with the formation of positively and negatively charged islands. This process occurs with system energy change on the value E e2 =2C. At low temperatures positively and negatively charged islands can be separated by neutral islands. Type of electron transfer process from one neutral island to another (e.g. virtual hopping process [15,16]) has an impact only on the relaxation time of the equilibrium concentration of charged islands at given temperature. This first process determines the concentration of islands with excess charge carriers. The greater the concentration of charged islands, the higher the system conductance. Second process characterizes the transition speed of the excess charge carriers between the islands (i.e. mobility) and it is caused by tunnelling of charge carriers from charged island to the neutral one. This process adds term in conductance that takes barrier properties dependence into account s p expð L=λÞ, where L is the hopping length, λ ¼ ℏ=ðmWÞ0:5 is the length of electron wave function decay in dielectric that separates metal islands, m is the electron mass, W is the tunnelling barrier height (practically equal to a half-width of the dielectric band gap). This kind of tunnelling changes the energy of the system by ΔE e2 ð1=C 1 1=C 2 Þ, where C1 and C2 are the capacitances of different size islands. If islands are identical then the energy of the system does not change. We must note that charge energy E e2 =2C b ΔE e2 ð1=C 1 1=C 2 Þ, so later we neglect ΔE. Therefore the conductance of island film in insulator state can be defined as follows: s ¼ s0 expð L=λ E=kTÞ;
ð2Þ
We assume that this equation can describe any kind of experimental conductance dependence in island structures, including Eq. (1). We assume that experimental dependence of the granular and island structures conductance, which is described by Eq. (1), is, firstly, a consequence of activation energy E magnitude variation. Secondly, the dependence can be caused by change of the hopping length L with the structure temperature change. Moreover, these two processes occur independently from each other. The first process is responsible for the excess charge carriers concentration in the film. Variation of the activation energy magnitude in granular and island systems may be associated with the presence of charged defects in dielectric matrix of the structure [16] or with the variation of granule size. For example, in the structures with insulating conductance the island size can vary from D ¼ 10 nm to D ¼ 200 nm. Charge energy of islands ðE e2 =ɛDÞ in its turn can vary from 0.005 eV to 0.1 eV [1,11,12]. In this case, the conductance dependence on the temperature differs from the activation dependence s ¼ s0 expð E=kTÞ, in which the activation energy is constant. For example, let us review island metal film, which contains two types of islands. The first type with D1 island size and N1 island concentration, and the second type with D2 island size and N2 island concentration. Charge energy of these islands equals, E1 e2 =ɛD1 and E2 e2 =ɛD2 (E1 ⪡E2 or D1 ⪢D2 ). At low temperatures (T E1 =k) due to tunnelling transitions of electrons between neutral islands or due to electron excitation from the traps to the neutral islands part of the islands becomes positively or negatively charged. The excess electrons or holes are captured by islands with minimum charge energy in the first place. In our case, it is the islands with the size D1, charge energy E1, and charged islands concentration n1 ¼ N 1 expð E1 =kTÞ [7,11].
Charge carriers transfer in the electric field will be carried out due to tunnelling transitions between charged and neutral islands of size D1. Energy of the system does not change, because islands have the same size, hence, the tunnelling probability does not depend on the temperature. Conductance dependence on the temperature in the proximity of T E1 =k will be of the activation type with the activation energy close to E1. With the temperature growth the excess charge carriers will be excited on smaller islands (of D2 size) with charge energy E2 e2 =ɛD2 . Charged islands of size D2 will now participate in tunnelling transition of charge carriers. Activation type of conductance still remains with temperature growth. As two types of islands take part in the charge carriers transfer, the process will be characterized by the aggregate activation energy (ES). The ES value will depend on the temperature and concentration of charged islands n1 ¼ N 1 expð E1 =kTÞ and n2 ¼ N 2 expð E2 =kTÞ. Activation energy ES in this case can be represented by the following expression: ES ¼
E1 N 1 expð E1 =kTÞ þ E2 N 2 expð E2 =kTÞ ; N 1 expð E1 =kTÞ þ N 2 expð E2 =kTÞ
where E1 N 1 expð E1 =kTÞ and E2 N 2 expð E2 =kTÞ are changes in energy of the systems, in which n1 and n2 charged islands are exited. The activation energy ES will increase with the temperature growth from ES E1 to ES -E2 (E1 ⪡E2 ). The dependence of energy ES on the temperature and, consequently, the conductance dependence on the temperature is determined by the distribution of island sizes. If islands are identical then the activation energy is constant and the conductance dependence on the temperature obeys the Arrhenius law s ¼ s0 expð E=kTÞ [11]. On the other hand, in paper [16] authors report that they have created periodic granulated structures and the island size was controlled with an accuracy of few percent. However, the “1/2 law” has been observed in these structures. As suggested by the authors, such conductance dependence on the temperature in the periodic granular systems can be connected with the presence of charged defects in dielectric matrix [16], which creates a random potential in the structure and leads to the disorder of charge energy. In general case of arbitrary island size distribution the activation energy ES can be presented as follows: ES ¼
∑n1 Ei N i expð Ei =kTÞ ; ∑n1 N i expð Ei =kTÞ
ð3Þ
where i ¼ 1; 2; 3; …; n characterizes specific size of the islands and their concentration. It should be emphasized that with the temperature growth it is becoming possible for one, two, three, and more excess electrons to transit on larger islands. The charge energy, for example, of size D island will have a value of E e2 =ɛD, E ð2eÞ2 =ɛD, E ð3eÞ2 =ɛD, etc. The second process determines the charge carriers transfer in the electric field and caused by tunnelling transitions between charged and neutral islands, which are characterized by the hopping length L. Obviously, any kind of conductance dependencies for the island and granular films s ¼ s0 exp½ ðT 0 =TÞx , where x¼ 0.5 [1–3], x¼ 0.75 [9], x ¼0.72 [10], x ¼1 [11], can be explained by the dependence s ¼ s0 expð L=λ ES =kTÞ, where the activation energy ES and the hopping length L depend on the temperature, but do not depend on each other. To verify these assumptions we have created island metal films of Tungsten (W) from 0.63 to 2 nm thick. The topography of the film surface was studied by the atomic force microscopy (AFM) method. The differential conductance dependence of metallic films on the temperature was measured. Tungsten films were grown by the RF-sputtering method. At the beginning of the process a vacuum chamber with the sample was kept at 2 10 6 mbar pressure. Then Tungsten was sputtered
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Fig. 1. AFM-images of Tungsten films on dielectric substrate. (a) 0.63 nm; (b) 0.8 nm.
Table 1 Experimentally defined and calculated parameters of island films. No. of Film type T 0 ; s0 sample and thickness
Fig. 2. Specific surface conductance dependence on temperature for Tungsten films of different effective thicknesses (sample 1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm; 5 – 2.0 nm). Solid lines correspond to “1/2 law” fitting.
on a cold glass substrate (temperature did not exceed 70 1C) in an argon atmosphere (a pressure of 10 3 mbar); RF-power on the target was at 700 W. Effective thickness of metal layers was determined from the deposition time and well-known deposition rate. AFM measurements of film topography show that films with a thickness of 0.9 nm and less have island structure. Islands in each film have a wide range of sizes. Longitudinal dimensions of the islands in 0.63 nm thick film [Fig. 1(a)] vary from 5 nm to 30 nm, and in the 0.8 nm film [Fig. 1(b)] vary from 8 nm to 100 nm. Samples with obtained structures (of 0.5–1.5 mm width and of 2–3.5 mm length) were made for the differential conductance measurements. Measurements were carried out at the 77–300 K temperature range. Frequency dependence of the conductance for all samples remained unchanged until f ¼ 10 kHz. Differential conductance measurements were carried out at frequency f ¼ ω=2π ¼ 2:5 kHz. In this study voltage U ¼ U 1 cos ðωtÞ (U 1 ¼ 10 2 V) was applied to the sample through Indium Ohmic contacts created of the film surface. The error of measurement did not exceed 5%. The specific surface differential conductance dependence on temperature of Tungsten films on dielectric substrate is shown in Fig. 2 for five structures with various effective thicknesses. As can be seen in the figure, conductance of the Tungsten film changes by seven orders of magnitude for different films. Conductance increases with the temperature growth.
3. Results and discussion According to our model the value of conductance is proportional to the product of probability of charge carriers tunnelling
Island size, defined from AFM-images
Island size, defined from conductance dependence DS ¼ 6…15 nm
1
Tungsten, T 0 ¼ 9300 K s0 ¼ 1:1 105 Ω 1 0.63 nm
D ¼ 5…30 nm
2
Tungsten, T 0 ¼ 2100 K s0 ¼ 4:7 105 Ω 1 0.8 nm
D ¼ 8…100 nm DS ¼ 10…40 nm
3
Tungsten, T 0 ¼ 72 K s0 ¼ 8:11 105 Ω 1 0.95 nm
4
Tungsten 1.16 nm
T 0 ¼ 25 K s0 ¼ 1:5 104 Ω 1
6
Gold [16]
T 0 ¼ 4000 K
DS ¼ 80…180 nm DS ¼ 120…300 nm D ¼ 5:5 nm
ɛ ¼ 4, DS ¼ 12…16 nm ɛ ¼ 6, DS ¼ 8…10 nm
between the islands on the probability of activation [see Eq. (2)] [5,6,11]. Presented in Fig. 2 measured temperature dependencies of the samples 1–5 conductance can be easily approximated by the “1/2 law”. For the convenience Fig. 2 contains only several experimental points. Theoretical dependencies presented as solid lines in Fig. 2. The values of parameters T0 and s0 for samples 1–4 are presented in Table 1. As can be seen in Fig. 2, in our case, the experimental conductance dependence on the temperature almost coincides with the “1/2 law” dependence. However, as noted by some authors, the “1/2 law” not always explains the conduction in disordered systems. Probably, the mechanism of current flow in the island and granular systems due to two consecutive processes mentioned above will explain more precisely the nature of transfer of charge carriers in disordered systems. According to above considerations and specific surface conductance measurements in the metal films we will show that mechanism of charge carriers transfer in the island and granular films in wide temperature range is mainly determined by the activation process. Activation energy ES, which determines the excess electrons and holes concentration, depends on the island size dispersion and the temperature. Activation energy ES we will determine from the temperature dependence of the film conductance (Fig. 2). Then from the temperature dependence of the energy ES we will calculate the island size (D e2 =ɛES ) and compare it with island size, which have been identified from the AFM studies. If these two sizes would be in satisfactory agreement then proposed above activation model that describes
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the conductance dependence on the temperature of island films can be considered fair. In order to find the temperature dependencies of ES and L=λ from the measured conductivity values we are using Eqs. (1) and (2). We take into consideration several values of temperature, for which ES and L=λ values will be defined. We assume that in the vicinity of these temperatures (5% of the value) ES and L=λ values do not depend on the temperature (hence temperature dependencies for whole temperature range are smooth). Experimental data for this small temperature range still contains several points, so it can be approximated by s ¼ s0 expð L=λ ES =kTÞ, where L=λ and ES are constant parameters. Repeating this procedure for several temperatures we obtain temperature dependencies for ES and L=λ. Values of activation energy ES and ratio L=λ for four samples (1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm) versus the temperature are presented in Figs. 3 and 4 respectively. Fig. 3 also shows the energy ES dependence on the temperature for a threelayer periodic gold structure [16], which was determined from conductance dependence on the temperature. From the energy ES dependence on the temperature we calculated the island size in the films. It was assumed that ES e2 =2C S , where CS is the average charged island capacitance at given temperature. The isolated island capacitance was taken equal to C S ¼ ɛDS =2. It was assumed that close neutral islands in the film have weak influence on charged island capacitance. The island sizes calculated from the energy ES dependence on the temperature and measured from the AFM images presented in Table 1. Relative permittivity of dielectric between the islands in single layer films 1–4 was assumed to equal 3 in these calculations. In Table 1 the size DS of granules in a three-layer periodic gold structure [16] is presented. For the granule size calculation of the three-layer sample value of permittivity was taken equal to 4, as the authors of the work [16] stated. But if the value of permittivity were taken equal to 6 the calculated size of the granules would be closer to the size measured from the AFM images D¼ 5.5 nm [16]. While comparing sizes D and DS it is necessary to remember that D is the real size of the island. On the other hand, DS is the
Fig. 3. Activation energy ES dependence on temperature for Tungsten films of different effective thicknesses (sample 1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm) and (6) for three-layer gold film [16].
Fig. 4. Temperature dependence of L=λ parameter for Tungsten films of different effective thicknesses (sample 1 – 0.63 nm; 2 – 0.8 nm; 3 – 0.95 nm; 4 – 1.16 nm).
average size of the islands, which contain excess charge at given temperature. Moreover, at any temperature of the sample the condition Dmin o DS o Dmax , where Dmin (Dmax ) is the minimum (maximum) island size in the particular structure, should be satisfied. Indeed, at low temperatures excess charge carriers are induced not only on large islands but also on smaller ones. Therefore, the average size of the islands, which have excess charge carriers, DS o Dmax . On the other hand, at high temperature both small and large islands will be partially filled. In this case, the condition Dmin o DS is satisfied. As it can be seen from the results presented in Table 1, the condition Dmin oDS oDmax is satisfied for the samples 1, 2 and 6, for which the real island size is known. Thus, we established that temperature dependence of conductance of island and granular systems is indeed of the activation type and the deviation from the Arrhenius law is determined by the variation of activation energy in magnitude. Variation in magnitude of activation energy can be associated not only with the variation in island sizes but also, perhaps, with the influence of a random potential. This is clearly seen from the structure 6 data (Table 1), where real island size equal to D ¼ 5:5 nm and size obtained from the analysis of the conductance dependence on the temperature ranges from DS ¼ 12 nm to DS ¼ 16 nm when ɛ ¼ 4. Hopping length of an electron presented in Fig. 4 turned out to be unexpectedly small. Indeed, if we take into account that the length of electron wave function decay is approximately equal to λ ¼ ℏ=ðmWÞ0:5 0:2 nm, the highest hopping length in the structure 1 approximately equal to L 1:2 nm (Fig. 4). In the rest of the structures hopping length is less. This hopping length is significantly less than minimum island size in the structure 1. Secondly, to make a jump from one neutral island to another it is necessary to spend energy value equal to ES e2 =2ɛL 0:5 eV. The probability for electron to have such energy at temperature T 100 K is very low. Considering the above we can conclude that tunnelling to length L 1:2 nm and less can be carried out only between neighboring islands without any loss of energy. Jump on the neighboring island in this case possible only between charged and neutral islands of about the same size. Hopping length decreases with the temperature growth. With the temperature growth smaller islands begin to take part in charge carriers transfer; the distance between them is less, so that explains the decrease. According to the mentioned above mechanism of charge carriers transfer in island and granular systems tunnelling to nearby islands without loss of energy determines transfer (i.e. mobility) of charge carriers. This transfer occurs between charged and neutral islands and it is characterized by hopping length L. It should be noted that in the case of L λ approximate equation determining hopping length L should be replaced with more precise one. However, in our case it is not necessary because calculations of length L, which is smaller than island size, have shown that electrons jump between neighboring islands. Thus, the mechanism of current flow in island and granular systems due to two consecutive processes fully explains the nature of charge carriers transfer in structures 1–4 and in the sample 6 [16]. Conductance of the sample 5 (Fig. 2) equals to s ¼ 7 1 10 4 Ω and practically does not depend on the temperature. The thickness of the film equals to 2.0 nm, hence, resistivity of the film 5 is ρ ¼ 3:8 10 4 Ω cm. If the resistivity value of the film lies within ρ ð2…3Þ 10 4 Ω cm, that film, according to the Ioffe–Regel–Mott criterion, is on the border between insulator and metal. Since the resistivity value of the sample 5 is close to the criterion and practically does not depend on the temperature, we can assume that this film defines the border of metal–insulator transition.
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4. Conclusion The current flow mechanism in island and granular systems due to two consecutive processes fully explains the charge carriers transfer in the films with conductance close to the threshold of metal–insulator transition (see Fig. 2, sample 4), and in the films, in which conductance is several orders of magnitude less than the threshold (Fig. 2, samples 1–3). Main difference between conductance dependencies of the films is in the value of activation energy. In sample 1 the activation energy ES ⪢kT (Fig. 3) at all temperatures and conductance changes by two orders of magnitude with temperature change in three times (Fig. 2). In sample 4 ES ⪡kT (Fig. 3). These films can be described by power-law dependence [18]. Performed studies of the conductance allowed us to establish the processes that determine the charge carriers transfer in granular and island structures. These processes determine the excess charge carriers concentration in the islands and, on the other hand, characterize the speed of excess charge carriers transfer from one island to another (i.e. mobility). Moreover, these processes occur independently from each other. The process that determines the excess charge carriers concentration in the islands is mainly related to the electron transfer from neutral island to the other neutral island and it is of the activation type; the deviation of the conductance dependence from the Arrhenius law determined by the variation of the activation energy in magnitude. Activation energy increases with the temperature growth. It should be emphasized that electron transfer from one neutral island to another (e.g. virtual hopping process [15,16]) may have an impact only on the time of establishing the equilibrium concentration of charged islands at given temperature. The transfer of excess charge carriers from one island to another is caused by tunnelling of electrons or holes from charged
43
islands to neutral islands. Moreover, the excess charge carriers tunnel to the nearest neutral island without changing the energy value. The average hopping length decreases with temperature growth.
Acknowledgments This work was supported by RFBR Grant no. 11-02-00748-a and contract no. 16.513.11.3143 of Ministry of Education and Science of the Russian Federation. References [1] B. Abeles, P. Sheng, M.D. Coutts, Y. Arie, Adv. Phys. 24 (1975) 407. [2] B.A. Aronzon, D.Y. Kovalev, A.E. Varfolomeev, A.A. Likal'ter, V.V. Ryl'kov, M.A. Sedova, Phys. Solid State 41 (1999) 857. [3] D.A. Zakheim, I.V. Rozhansky, I.P. Smirnova, S.A. Gurevich, J. Exp. Theor. Phys. 91 (2000) 553. [4] J. Klafter, P. Sheng, J. Phys. C 17 (1984) L93. [5] S.T. Chui, Phys. Rev. B 43 (1991) 14274. [6] J. Zhang, B.I. Shklovskii, Phys. Rev. B 70 (2004) 115317. [7] E.Z. Meilikhov, J. Exp. Theor. Phys. 88 (1999) 819. [8] I.P. Zvaygin, R. Keiper, Phys. Stat. Sol. B (2002) 151. [9] N. Markovic, C. Christiansen, D.E. Grupp, A.M. Mack, G. Martinez-Arizala, A.M. Goldman, Phys. Rev. B 62 (2000) 2195. [10] C.J. Adkins, E.G. Astrakharchik, J. Phys.: Condens. Matter 10 (1998) 6651. [11] C.A. Neugebauer, M.B. Webb, J. Appl. Phys. 33 (1962) 74. [12] A.P. Boltaev, N.A. Penin, A.O. Pogosov, F.A. Pudonin, J. Exp. Theor. Phys. 99 (2004) 827. [13] A.P. Boltaev, N.A. Penin, A.O. Pogosov, F.A. Pudonin, J. Exp. Theor. Phys. 96 (2003) 940. [14] A.P. Boltaev, F.A. Pudonin, J. Exp. Theor. Phys. 103 (2006) 436. [15] I.S. Beloborodov, A.V. Lopatin, V.M. Vinokur, K.B. Efetov, Rev. Mod. Phys. 79 (2007) 469. [16] T.B. Tran, I.S. Beloborodov, X.M. Lin, T.P. Bigioni, V.M. Vinokur, H.M. Jaeger, Phys. Rev. Lett. 95 (2005) 076806. [17] K.B. Efetov, A. Tschersich, Phys. Rev. B 67 (2003) 174205. [18] A.S. Ioselevich, D.S. Lyubshin, JETP Lett. 90 (2009) 672.