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Incoherent single electron effect in a disordered tunneling barrier
13June 1994 PHYSICS
LETTERS
A
Physics Letters A 189 ( 1994) 237-242
ELSEVIER
Incohekent single electron effect in a disordered tunneling barrier S.-T. Yau School cefElectrica1and ElectronicEngineering, Nanyang TechnologicalUniversity,NanyangAvenue, 2263, Singapore
Received 24 January 1994; revised manuscript received 16 March 1994;accepted for publication 5 April 1994 Communicated by J. Plouquet
Insight on electron transport through a disordered tunneling barrier is obtained using a scanning tunneling microscope operated at room temperature. We interpret the experimental results within the frame of incoherent sequential charging of localized states in the tunneling barrier by single electrons. This kind of single electron transport mechanism is somewhat different from the one described by the conventional theory of single electron tunneling. We explain the difference and present experimental evidence.
The understanding of single electron transport in metals and semiconductors is of fundamental importance in the electron transport mechanism at the mesoscopic level, Single electron charging, an effect associated with the phenomenon of Coulomb blockade, has been theorized and experimentally observed recently in various systems at temperatures ranging from millikelvins to above room temperature (for a general review, see Ref. [ 1 ] ) . The most common system which allows single electron charging to be observed, is a voltage-driven array of ultra-small capao itors with capacitance C in the range of 10T8 F connected in series. The small capacitance is required in order that the charging energy of an electron e*/2Cis greater than the thermal noise kBT. Steps in the current-voltage (Z-V) curves or equally spaced peaks in the conductance curves manifest the discrete nature of the charging mechanism. The small capacitor can be made artificially with the present fabrication techniques. However, one can also use “unintentionally made” capacitors to observe single-electron charging. In a one-dimensional Si device, Meirav et al. [ 2 ] observed periodic oscilRlsevier Science B.V. SS’DIO375-9601(94)00267-S
lations in the conductance curve, which are attributed to the presence of impurities in the semiconductor. Recently Chandrasekhar et al. [ 31 observed the single electron charging effect in disordered indium oxide wires. It is suggested that series of localized states (LS) in the wire behave as arrays of serially connected capacitors. In these works, the concept of capacitance is used in the interpretation of the experimental results. In this Letter, we present the observation of features in the Z-V curves of a nanometer size metalinsulator-semiconductor tunneling junction at room temperature. The junction is formed between the tip of a scanning tunneling microscope (STM) and a Si substrate, whose surface is chemically passivated. The features in the Z-V curves reflect correlated electrostatic interactions between occupied localized states in the tunneling barrier. We interpret our results in terms of incoherent sequential tunneling via chains of localized states. In our formalism, the localized states are in some way treated as capacitors, since they can accommodate electrons. However, the concept of capacitance is not utilized. In the conventional treat-
238
S.-T. Yau /Physics
LettersA 189 (1994) 237-242
ment of single electron tunneling [ 4,5], in which electrons tunnel coherently through a central electrode of a continuous energy spectrum, the steps in the I-Vcurve are evenly spaced due to the constant values of the junction capacitances. In our formalism, the steps are not necessarily evenly spaced. The spacing depends on the distance between neighboring localized states. We show that this effect occurs at zero temperature and elevated temperatures. In this experiment, an STM was used at room temperature. When taking images, the STM was operated in the constant-current mode. I- I’ curves were generated by disabling the feedback loop and ramping the biasing voltage between the tip and the sample while recording the tunneling current. Each point of the curves shown in this Letter is averaged over ten data points. The n-type Si sample has an orientation of ( 111) and a resistivity of 1 Q cm. It was degreased in a solution of HzS04 and HzOz (4 : 1 by volume), rinsed in de-ionized water, etched in 49% HF, and finally rinsed with de-ionized water. Mechanically cut gold and iridium-platinum wires were used as tunneling tips. The polarities in the I- T/curves indicate the tip bias polarities. As shown in the STM image in Fig. 1, the surface layer of the passivated Si consists of clusters. These clusters are organic (hydrocarbon) species as determined by X-ray photoelectron spectroscopy [ 61. Therefore the insulating tunneling barrier consists of the air-gap and the organic layer. Z-V curves are generated with the tip positioned at random locations on the surface. Since the tunneling barrier is a Schottky barrier with an inter-facial layer [ 7 1, its I- V curves as the one shown in Fig. 2, show the typical rectifying characteristics: in the reverse-bias region, the current remains almost constant at a low level; in the forward-bias region, the current rises exponentially. This suggests that in the forward-bias region the dominant mechanism of electron transfer across the barrier is thermionic emission of electrons from the semiconductor to the metallic tip. The curve in Fig. 2 is typical of the many curves, which show the rectifying characteristics. However, sometimes we obtained Z-V curves which have small features superimposed upon them in the forward-bias region. The features appear in the forms of steps and peaks. Fig. 3a shows eleven steps in the Z-V curve, while in Fig. 3b there is also a peak in ad-
Fig. 1. STM image ( 1000 A x 1000 8, ) of the passivated Si surface. The image was obtained with a tunneling current of 0.86 nA and a tip biasing voltage of 1.2 V. Organic species appear as clusters with a dimension of 100 A x 100 A.The span of the grey scale is 17A.
n
0.9 I
0.8
I
v 0.1
Ow --_______-___
-0.1 r -1.2
-0.5
I I /I
-_____----_______
1
i
0 VOLTAGE
0.5
1.2
(V)
Fig. 2. Current-voltage characteristic of the passivated Si surface. The shape of the curve suggests that the electron transfer mechanism is thermionic emission of electrons.
dition to the ten steps. In all of the curves that show features, we observe a threshold of 0.7 V, below which no features appears. The features are not equally spaced along the biasing voltage axis. This phenomenon can readily be discerned by observing the spacing separating the peaks in the numerically derived conductance curve (dI/dV versus V) in Fig. 3a. In general, the spacings of a curve range between 30 and 60 mV. The range of the biasing voltage is N 1 V. Within this range, the number of peaks in the con-
S.-T. Yau /Physics LettersA 189 (1994) 237-242
641
-0.11 -1.1
I -0.5
0
0.5
1.1
VOLTAGE (V)
Fig. 3. Current-vpltage characteristics of the passivated Si surface. (a) Curve shpws several steps, which are not equally spaced. Inset: conductance curve numerically derived from the I-Vcurve. (b) Curve shows several steps as well as a peak.
ductance curve is approximately the same and the peaks are approximately located at the same positions. Sometimes curves with identical peak characteristics were obtained. The tip-sample separation was varied by varying the set-point tunneling current to obtain I- Vctuves. The set-point current and hence the junction resistance was varied over two orders of magnitude. This change corresponds to a change of N 2 A in the di$tance between the tip and the sample. However, we did not observe changes in the manner the spacings appear. The fact that ismall features appear in the I- Vcurves suggest that there are some kind of central electrodes
239
existing between the tip and the bulk Si. We believe that these electrodes are in fact impurity species imbedded in the passivated Si layer. The impurities are very likely some transition metals, which are contaminants in the chemicals used to prepare the Si surface. These impurity species induce localized states in the tunneling barrier. There are several publications on single electron charging at room temperature [ 8-101. These works all show equally spaced steps in the I-Vcurve, which result from the constant capacitances between the central electrode and the two outer electrodes. In our curves, although the spacings are on the same order, the steps are not equally spaced. Therefore our room temperature measurements cannot be explained by the conventional theory of single electron charging [ 431. Also according to the conventional theory, changing the junction capacitance will change the step spacing and if the change in capacitance is large enough, steps will disappear [ 111. We changed the tip-sample distance and hence the junction capacitance as described above, but we did not observe these phenomena. We believe that our measurements are the results of the incoherent sequential tunneling via chains of localized states in the tunneling barrier as recently predicted by Raikh and Asenov [ 12 1. We first consider the effect of two localized states. Consider the energy diagram of a tunneling junction in Fig. 4a. Two LSs, LSl and LS2, are present in the barrier between the semiconductor and the metallic electrodes. The bias-dependent energies of the LSs are denoted by El ( V) and E2( V). When the temperature is high enough so that thermionic emission is the dominant electron transport mechanism, the thermally emitted electrons tunnel via the top of the barrier Es from the semiconductor to LSl (S+LSl). The S4LSl and LS2+M transitions are elastic processes. The LSl +LS2 transition is an inelastic process, in which the electron localized at LSl hops into LS2 accompanied by the emission of a phonon. Each LS can be occupied by only one electron since the strong repulsive Hubbard energy forbids two electrons to be on the same impurity site simultaneously. Therefore each LS can accommodate only one electron. When a LS is occupied by an electron, the energy of neighboring LSs is shifted due to Coulomb interaction as shown
240
S.-T. Yau /Physics Letters A 189 (I 994) 237-242
=EF-E2(V)-@z,
_____
%m
Semiconductor
Met&II
Semiconductor
Fig. 4. Energy diagrams of a metal-insulator-semiconductor tunneling junction. There are two localized states, LSl and IS2, in a chain in the insulating layer. (a) The two localized states are not occupied by electrons. (b) LS2 is occupied by an electron and the energy level of L!31 is raised by U,.
in Fig. 4b. The Hamiltonian of the pair of LSs can be expressed as [ 13 ] H(n19 n,)=Jff,(V)n,
+&(vn2+Qn,n2,
(1)
where n, and n, are the occupation numbers of the LSs, which takes on values of either 1 or 0. Uc is the Coulomb shift; it is equal to e’/C,,E,r where r is the distance between the two LSs. In general the S+LSl transition is allowed if As-LSI =H( 1,
~2)
-H(O,
n2)
=E,(V)+Ucn2-Es
-Es
(2)
The corresponding conditions for the LS 1+LS2 and LS2+ M transitions are hs1432
=H(O,
=E2(V)-&(V
and
1)
-H(
1,O)
(3)
(4)
When LS2 is not occupied ( n2= 0 ) and if V is increased such that Eq. (2) gives Es 2 El ( V), then the S+ LS 1 transition will occur. If LS2 is occupied by an electron ( n2= 1 ), then at the previous value of V the S-+LSl transition cannot occur since Eq. (2) does not hold due to the presence of the Coulomb interaction term Uc. Here we see that the energy of LSl depends on the occupation status of LS2. Thus the S+LSI transition is blocked. The blockade is lifted when the electron occupying LS2 tunnels into the metal electrode. However, if V is further increased such that Eq. (2) holds, the S+LSl transition becomes allowed independently of the occupation of LS2. Thus, a time correlation of tunneling processes between S-+LSl and LS2+M is established and it produces a step in the I- T/curve. For appropriate values of V, El ( V) and E2( V), the steps can also be caused by the blockade of the LSZ-+M transition usingEq. (4). At lower temperatures, when thermionic emission via the top of the barrier Es is suppressed, thermionic-field emission [ 71 of electrons below Es becomes the dominant transport mechanism for moderately doped semiconductors. Thus the voltage position of a given step in the I- V curve may shift as temperature varies. The shift may also occur at temperatures high enough such that thermionic emission above Es is possible. In this case the inequality sign in Eq. (2) is reversed. Fig. 5 shows the results of a Monte Carlo simulation of a metal-insulator-metal junction at zero and finite temperatures [ 121. If we assume that the thickness of the surface layer is 30 8, and that there are five LSs aligned spatially in a chain, the average distance (r) between two nearest LSs is 6 A. For a relative dielectric constant E, of 5, the formula Uc= e2/4neoq( r) yields a value of 7.7x 10e20 J. The actual numerical value of UC should be about three times less because of the screening provided by the presence of the metallic tip. Therefore UC is approximately 3x 10m20J and the ratio kT/U, is 0.14 at room temperature. According to the estimation, steps can readily be observed at room temperature as shown in Fig. 5. In the above, we considered the situation in
S.-T. Yau /Physics Letters A 189 (1994) 237-242
‘\
I 0.4 0.2 1 0
---a 2
d) TIUp6.7 r
I
I
,
1
1
2
3
4
I 6
V/UC Fig. 5. Monte Carlo simulations at zero and finite temperatures (reprinted from Ref. [ 121). The tunneling junction consists of two metallic electrodes denoted by P and r. There are two localized states locatgd in the insulating region between the eleo trodes. The biasi@ voltage is normalized to the Coulomb shift Uc and the tunneiling current is normalized to a maximum current I-. (SeeRef. [ 121 for the definition of I-. )
which two LSs are in a chain connecting the two electrodes. Using the same arguments, it can be shown that the Z-V curve of chains consisting of more than two LSs exhibits more than one step. Each step occurs at the voltage at which it is energetically favorable for an additional electron to occupy one of the LSs. The number of features in the Z-V curve does not necessarily imply the number of LSs in the chain. For three LSs, there are three steps, and for four LSs, the analysis is rather complicated and there are six or more steps in the Z-V curve [ 141. The steps are not necessarily equally spaced due to the fact that UCdepends on r and Vhas to be increased to overcome UC. The conditions under which the chains of LS appear are discussed in Ref. [ 15 1. The number of LSs in a chain increases with temperature. The incoherent tunneling theory predicts only step like structures in the Z-V curve. However, we do observe peaks as well in the experiment. Peaks are usually associated with some kinds of resonances. However, when dealing with localized impurities, resonances with a localized state manifest themselves as steps rather than peaks in the I- V curves [ 15,16 1, Z(V)=
+qv-v,))
(5)
241
where O(x) is the unit step function, Vti is the threshold voltage of the resonances, and r is the width of the localized state. At room temperature, kgT will smear out the width of the state and hence resonance wiIl be destroyed. Since the experiment is performed at room temperature, resonant tunneling is not likely to occur. In fact part of the peak is a decrease in current just after the onset of a step. Therefore the peaks can be regarded as some small corrections to the basic steplike structure. We tentatively attribute the mechanism that produces the peaks to the interaction between the LS and the conduction electrons in the metallic electrode as described by Matveev and Larkin [ 18 1. If the LS has a positive charge, its net charge will be zero when an electron tunneling from the semiconductor electrode occupies the SL. As soon as the electron leaves the SL to tunnel into the metallic electrode, a positive change appears at the SL and an attractive potential is created for the conduction electron which has just tunnelled to the metallic electrode. The electron can possibly be scattered in momentum space in the metallic electrode by the attractive potential. Thus the total tunneling event from the LS to the metallic electrode consists of the direct (unscattered) component and the scattered component. By considering only the first-order correction of the scattering process to the total tunneling amplitude, it can be shown that a logarithmic dependence of the tunneling amplitude on the biasing voltage appears. Therefore the total tunneling current acquires a logarithmic dependence on the biasing voltage when the biasing voltage is increased beyond Vth. This logarithmic dependence is general in origin, and should show up in the beginning of the steps. In conclusion, we made Z-V measurements on passivated Si samples using a STM at room temperature. Steps and peaks appear in the Z-V curves. From the step spacing and the result of changing the junction capacitance we conclude that the features are not due to the conventional mechanism of single electron charging. Also since the experiment was performed at room temperature, resonant tunneling can be precluded. We interpret the measurements in the frame of the theory of incoherent sequential charging of localized impurity states in the passivated layer of the Si sample. This kind of single electron transport takes place from zero temperature up to considerably elevated temperatures, including room temperature. The
242
S.-T. Yau /Physics
Letters A 189 (1994) 237-242
anoearance of neaks in the I- I/ curves mieht be due __ to scattering of the conduction electrons in the electrode by the potential of the localized impurities. The author would like to express his gratitude to M. Raikh for advice, help, and information.
References [ 1] K.K. Likharevand T. Claeson, Sci. Am. June ( 1992) 50; M.H. Devon%, D. Esteve and C. Urbina, Nature 360 ( 1992) 547. [2] U. Meirav, M.A. Kastner and S.J. Wind, Phys. Rev. Lett. 65 (1990) 771. [ 31 V. Chandrasekhar, Z. Ovadyahu and R. Webb, Phys. Rev. Lett. 67 ( 199 1) 2862. [4] K. MuBen, E. Ben-Jacob, RC. Jaklevic and Z. Schuss, Phys. Rev. B 37 (1988) 98. 15lK.K. Likharev, IBM J. Res. Dev. 32 (1988) 144. i6 j S.-T. Yau, X. Zheng and M.H. Nayfeh, Appl. Phys. Len. 59 (1991) 2457.
[ 71 E.H. Rhode&k and R.H. Williams, Metal-semiconductor
contacts,2nd Ed. (Clarendon, Oxford, 1988). [S] H. Nejoh, Nature 353 (1991) 640. [ 91M. Reeve, 0. Symko and R. Li, Mod. Phys. Lett. B 6 (1992) 273. [lo] C. Schonenberger, H. van Houten and H.C. Donkersloot, Europhys. Lett. 20 ( 1992) 249. [ 111 R. Wilkins, M. Amman, E. Ben-Jacob and R.C. Jaklevic, Phys. Rev. B 42 (1990) 8698. [ 121 M.E. Raikh and A. Asenov, Superlatt. Microstruct. 11 (1992) 325. [ 131 A.L. Efros and B.L. Shklovskii, in: Electron%lectron interactions in disordered systems, eds. A.L. Efros and M. Pollak (North-Holland, Amsterdam, 1985). [ 141 M. Raikh, private communication. [ 151 L.I. Glazman and K.A. Matveev, Zh. Eksp. Teor. Fiz. 94 (1988) 332 [Sov. Phys. JETP 67 (1988) 12761. [ 161 A.V. Chaplii and M.V. Entin, Zh. Eksp. Teor. Fiz. 67 ( 1974) 208 [Sov. Phys. JETP 40 (1974) 1061. [ 171 LM. Lifshitz and V.Ya. Kirpichenkov, Zh. Eksp. Teor. Fiz. 77 (1979) 989 [Sov. Phys. JETP 50 (1979) 4991. [ 181 K.A. Matveev and A.I. Larkin, Phys. Rev. B 46 (1992) 15337.
LETTERS
A
Physics Letters A 189 ( 1994) 237-242
ELSEVIER
Incohekent single electron effect in a disordered tunneling barrier S.-T. Yau School cefElectrica1and ElectronicEngineering, Nanyang TechnologicalUniversity,NanyangAvenue, 2263, Singapore
Received 24 January 1994; revised manuscript received 16 March 1994;accepted for publication 5 April 1994 Communicated by J. Plouquet
Insight on electron transport through a disordered tunneling barrier is obtained using a scanning tunneling microscope operated at room temperature. We interpret the experimental results within the frame of incoherent sequential charging of localized states in the tunneling barrier by single electrons. This kind of single electron transport mechanism is somewhat different from the one described by the conventional theory of single electron tunneling. We explain the difference and present experimental evidence.
The understanding of single electron transport in metals and semiconductors is of fundamental importance in the electron transport mechanism at the mesoscopic level, Single electron charging, an effect associated with the phenomenon of Coulomb blockade, has been theorized and experimentally observed recently in various systems at temperatures ranging from millikelvins to above room temperature (for a general review, see Ref. [ 1 ] ) . The most common system which allows single electron charging to be observed, is a voltage-driven array of ultra-small capao itors with capacitance C in the range of 10T8 F connected in series. The small capacitance is required in order that the charging energy of an electron e*/2Cis greater than the thermal noise kBT. Steps in the current-voltage (Z-V) curves or equally spaced peaks in the conductance curves manifest the discrete nature of the charging mechanism. The small capacitor can be made artificially with the present fabrication techniques. However, one can also use “unintentionally made” capacitors to observe single-electron charging. In a one-dimensional Si device, Meirav et al. [ 2 ] observed periodic oscilRlsevier Science B.V. SS’DIO375-9601(94)00267-S
lations in the conductance curve, which are attributed to the presence of impurities in the semiconductor. Recently Chandrasekhar et al. [ 31 observed the single electron charging effect in disordered indium oxide wires. It is suggested that series of localized states (LS) in the wire behave as arrays of serially connected capacitors. In these works, the concept of capacitance is used in the interpretation of the experimental results. In this Letter, we present the observation of features in the Z-V curves of a nanometer size metalinsulator-semiconductor tunneling junction at room temperature. The junction is formed between the tip of a scanning tunneling microscope (STM) and a Si substrate, whose surface is chemically passivated. The features in the Z-V curves reflect correlated electrostatic interactions between occupied localized states in the tunneling barrier. We interpret our results in terms of incoherent sequential tunneling via chains of localized states. In our formalism, the localized states are in some way treated as capacitors, since they can accommodate electrons. However, the concept of capacitance is not utilized. In the conventional treat-
238
S.-T. Yau /Physics
LettersA 189 (1994) 237-242
ment of single electron tunneling [ 4,5], in which electrons tunnel coherently through a central electrode of a continuous energy spectrum, the steps in the I-Vcurve are evenly spaced due to the constant values of the junction capacitances. In our formalism, the steps are not necessarily evenly spaced. The spacing depends on the distance between neighboring localized states. We show that this effect occurs at zero temperature and elevated temperatures. In this experiment, an STM was used at room temperature. When taking images, the STM was operated in the constant-current mode. I- I’ curves were generated by disabling the feedback loop and ramping the biasing voltage between the tip and the sample while recording the tunneling current. Each point of the curves shown in this Letter is averaged over ten data points. The n-type Si sample has an orientation of ( 111) and a resistivity of 1 Q cm. It was degreased in a solution of HzS04 and HzOz (4 : 1 by volume), rinsed in de-ionized water, etched in 49% HF, and finally rinsed with de-ionized water. Mechanically cut gold and iridium-platinum wires were used as tunneling tips. The polarities in the I- T/curves indicate the tip bias polarities. As shown in the STM image in Fig. 1, the surface layer of the passivated Si consists of clusters. These clusters are organic (hydrocarbon) species as determined by X-ray photoelectron spectroscopy [ 61. Therefore the insulating tunneling barrier consists of the air-gap and the organic layer. Z-V curves are generated with the tip positioned at random locations on the surface. Since the tunneling barrier is a Schottky barrier with an inter-facial layer [ 7 1, its I- V curves as the one shown in Fig. 2, show the typical rectifying characteristics: in the reverse-bias region, the current remains almost constant at a low level; in the forward-bias region, the current rises exponentially. This suggests that in the forward-bias region the dominant mechanism of electron transfer across the barrier is thermionic emission of electrons from the semiconductor to the metallic tip. The curve in Fig. 2 is typical of the many curves, which show the rectifying characteristics. However, sometimes we obtained Z-V curves which have small features superimposed upon them in the forward-bias region. The features appear in the forms of steps and peaks. Fig. 3a shows eleven steps in the Z-V curve, while in Fig. 3b there is also a peak in ad-
Fig. 1. STM image ( 1000 A x 1000 8, ) of the passivated Si surface. The image was obtained with a tunneling current of 0.86 nA and a tip biasing voltage of 1.2 V. Organic species appear as clusters with a dimension of 100 A x 100 A.The span of the grey scale is 17A.
n
0.9 I
0.8
I
v 0.1
Ow --_______-___
-0.1 r -1.2
-0.5
I I /I
-_____----_______
1
i
0 VOLTAGE
0.5
1.2
(V)
Fig. 2. Current-voltage characteristic of the passivated Si surface. The shape of the curve suggests that the electron transfer mechanism is thermionic emission of electrons.
dition to the ten steps. In all of the curves that show features, we observe a threshold of 0.7 V, below which no features appears. The features are not equally spaced along the biasing voltage axis. This phenomenon can readily be discerned by observing the spacing separating the peaks in the numerically derived conductance curve (dI/dV versus V) in Fig. 3a. In general, the spacings of a curve range between 30 and 60 mV. The range of the biasing voltage is N 1 V. Within this range, the number of peaks in the con-
S.-T. Yau /Physics LettersA 189 (1994) 237-242
641
-0.11 -1.1
I -0.5
0
0.5
1.1
VOLTAGE (V)
Fig. 3. Current-vpltage characteristics of the passivated Si surface. (a) Curve shpws several steps, which are not equally spaced. Inset: conductance curve numerically derived from the I-Vcurve. (b) Curve shows several steps as well as a peak.
ductance curve is approximately the same and the peaks are approximately located at the same positions. Sometimes curves with identical peak characteristics were obtained. The tip-sample separation was varied by varying the set-point tunneling current to obtain I- Vctuves. The set-point current and hence the junction resistance was varied over two orders of magnitude. This change corresponds to a change of N 2 A in the di$tance between the tip and the sample. However, we did not observe changes in the manner the spacings appear. The fact that ismall features appear in the I- Vcurves suggest that there are some kind of central electrodes
239
existing between the tip and the bulk Si. We believe that these electrodes are in fact impurity species imbedded in the passivated Si layer. The impurities are very likely some transition metals, which are contaminants in the chemicals used to prepare the Si surface. These impurity species induce localized states in the tunneling barrier. There are several publications on single electron charging at room temperature [ 8-101. These works all show equally spaced steps in the I-Vcurve, which result from the constant capacitances between the central electrode and the two outer electrodes. In our curves, although the spacings are on the same order, the steps are not equally spaced. Therefore our room temperature measurements cannot be explained by the conventional theory of single electron charging [ 431. Also according to the conventional theory, changing the junction capacitance will change the step spacing and if the change in capacitance is large enough, steps will disappear [ 111. We changed the tip-sample distance and hence the junction capacitance as described above, but we did not observe these phenomena. We believe that our measurements are the results of the incoherent sequential tunneling via chains of localized states in the tunneling barrier as recently predicted by Raikh and Asenov [ 12 1. We first consider the effect of two localized states. Consider the energy diagram of a tunneling junction in Fig. 4a. Two LSs, LSl and LS2, are present in the barrier between the semiconductor and the metallic electrodes. The bias-dependent energies of the LSs are denoted by El ( V) and E2( V). When the temperature is high enough so that thermionic emission is the dominant electron transport mechanism, the thermally emitted electrons tunnel via the top of the barrier Es from the semiconductor to LSl (S+LSl). The S4LSl and LS2+M transitions are elastic processes. The LSl +LS2 transition is an inelastic process, in which the electron localized at LSl hops into LS2 accompanied by the emission of a phonon. Each LS can be occupied by only one electron since the strong repulsive Hubbard energy forbids two electrons to be on the same impurity site simultaneously. Therefore each LS can accommodate only one electron. When a LS is occupied by an electron, the energy of neighboring LSs is shifted due to Coulomb interaction as shown
240
S.-T. Yau /Physics Letters A 189 (I 994) 237-242
=EF-E2(V)-@z,
_____
%m
Semiconductor
Met&II
Semiconductor
Fig. 4. Energy diagrams of a metal-insulator-semiconductor tunneling junction. There are two localized states, LSl and IS2, in a chain in the insulating layer. (a) The two localized states are not occupied by electrons. (b) LS2 is occupied by an electron and the energy level of L!31 is raised by U,.
in Fig. 4b. The Hamiltonian of the pair of LSs can be expressed as [ 13 ] H(n19 n,)=Jff,(V)n,
+&(vn2+Qn,n2,
(1)
where n, and n, are the occupation numbers of the LSs, which takes on values of either 1 or 0. Uc is the Coulomb shift; it is equal to e’/C,,E,r where r is the distance between the two LSs. In general the S+LSl transition is allowed if As-LSI =H( 1,
~2)
-H(O,
n2)
=E,(V)+Ucn2-Es
-Es
(2)
The corresponding conditions for the LS 1+LS2 and LS2+ M transitions are hs1432
=H(O,
=E2(V)-&(V
and
1)
-H(
1,O)
(3)
(4)
When LS2 is not occupied ( n2= 0 ) and if V is increased such that Eq. (2) gives Es 2 El ( V), then the S+ LS 1 transition will occur. If LS2 is occupied by an electron ( n2= 1 ), then at the previous value of V the S-+LSl transition cannot occur since Eq. (2) does not hold due to the presence of the Coulomb interaction term Uc. Here we see that the energy of LSl depends on the occupation status of LS2. Thus the S+LSI transition is blocked. The blockade is lifted when the electron occupying LS2 tunnels into the metal electrode. However, if V is further increased such that Eq. (2) holds, the S+LSl transition becomes allowed independently of the occupation of LS2. Thus, a time correlation of tunneling processes between S-+LSl and LS2+M is established and it produces a step in the I- T/curve. For appropriate values of V, El ( V) and E2( V), the steps can also be caused by the blockade of the LSZ-+M transition usingEq. (4). At lower temperatures, when thermionic emission via the top of the barrier Es is suppressed, thermionic-field emission [ 71 of electrons below Es becomes the dominant transport mechanism for moderately doped semiconductors. Thus the voltage position of a given step in the I- V curve may shift as temperature varies. The shift may also occur at temperatures high enough such that thermionic emission above Es is possible. In this case the inequality sign in Eq. (2) is reversed. Fig. 5 shows the results of a Monte Carlo simulation of a metal-insulator-metal junction at zero and finite temperatures [ 121. If we assume that the thickness of the surface layer is 30 8, and that there are five LSs aligned spatially in a chain, the average distance (r) between two nearest LSs is 6 A. For a relative dielectric constant E, of 5, the formula Uc= e2/4neoq( r) yields a value of 7.7x 10e20 J. The actual numerical value of UC should be about three times less because of the screening provided by the presence of the metallic tip. Therefore UC is approximately 3x 10m20J and the ratio kT/U, is 0.14 at room temperature. According to the estimation, steps can readily be observed at room temperature as shown in Fig. 5. In the above, we considered the situation in
S.-T. Yau /Physics Letters A 189 (1994) 237-242
‘\
I 0.4 0.2 1 0
---a 2
d) TIUp6.7 r
I
I
,
1
1
2
3
4
I 6
V/UC Fig. 5. Monte Carlo simulations at zero and finite temperatures (reprinted from Ref. [ 121). The tunneling junction consists of two metallic electrodes denoted by P and r. There are two localized states locatgd in the insulating region between the eleo trodes. The biasi@ voltage is normalized to the Coulomb shift Uc and the tunneiling current is normalized to a maximum current I-. (SeeRef. [ 121 for the definition of I-. )
which two LSs are in a chain connecting the two electrodes. Using the same arguments, it can be shown that the Z-V curve of chains consisting of more than two LSs exhibits more than one step. Each step occurs at the voltage at which it is energetically favorable for an additional electron to occupy one of the LSs. The number of features in the Z-V curve does not necessarily imply the number of LSs in the chain. For three LSs, there are three steps, and for four LSs, the analysis is rather complicated and there are six or more steps in the Z-V curve [ 141. The steps are not necessarily equally spaced due to the fact that UCdepends on r and Vhas to be increased to overcome UC. The conditions under which the chains of LS appear are discussed in Ref. [ 15 1. The number of LSs in a chain increases with temperature. The incoherent tunneling theory predicts only step like structures in the Z-V curve. However, we do observe peaks as well in the experiment. Peaks are usually associated with some kinds of resonances. However, when dealing with localized impurities, resonances with a localized state manifest themselves as steps rather than peaks in the I- V curves [ 15,16 1, Z(V)=
+qv-v,))
(5)
241
where O(x) is the unit step function, Vti is the threshold voltage of the resonances, and r is the width of the localized state. At room temperature, kgT will smear out the width of the state and hence resonance wiIl be destroyed. Since the experiment is performed at room temperature, resonant tunneling is not likely to occur. In fact part of the peak is a decrease in current just after the onset of a step. Therefore the peaks can be regarded as some small corrections to the basic steplike structure. We tentatively attribute the mechanism that produces the peaks to the interaction between the LS and the conduction electrons in the metallic electrode as described by Matveev and Larkin [ 18 1. If the LS has a positive charge, its net charge will be zero when an electron tunneling from the semiconductor electrode occupies the SL. As soon as the electron leaves the SL to tunnel into the metallic electrode, a positive change appears at the SL and an attractive potential is created for the conduction electron which has just tunnelled to the metallic electrode. The electron can possibly be scattered in momentum space in the metallic electrode by the attractive potential. Thus the total tunneling event from the LS to the metallic electrode consists of the direct (unscattered) component and the scattered component. By considering only the first-order correction of the scattering process to the total tunneling amplitude, it can be shown that a logarithmic dependence of the tunneling amplitude on the biasing voltage appears. Therefore the total tunneling current acquires a logarithmic dependence on the biasing voltage when the biasing voltage is increased beyond Vth. This logarithmic dependence is general in origin, and should show up in the beginning of the steps. In conclusion, we made Z-V measurements on passivated Si samples using a STM at room temperature. Steps and peaks appear in the Z-V curves. From the step spacing and the result of changing the junction capacitance we conclude that the features are not due to the conventional mechanism of single electron charging. Also since the experiment was performed at room temperature, resonant tunneling can be precluded. We interpret the measurements in the frame of the theory of incoherent sequential charging of localized impurity states in the passivated layer of the Si sample. This kind of single electron transport takes place from zero temperature up to considerably elevated temperatures, including room temperature. The
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Letters A 189 (1994) 237-242
anoearance of neaks in the I- I/ curves mieht be due __ to scattering of the conduction electrons in the electrode by the potential of the localized impurities. The author would like to express his gratitude to M. Raikh for advice, help, and information.
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