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The search for residual resistivity dipoles by scanning tunneling potentiometry
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
The search for residual resistivity dipoles by scanning tunneling potentiometry R. M. Feenstra† Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
B. G. Briner‡ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
(Received 22 August 1997) A review is given of the application of scanning tunneling potentiometry (STP) to the study of potential variations due to scattering from localized centers. Landauer has theoretically investigated such scattering in the ballistic regime where the electronic mean free path exceeds the size of the scattering centers. Landauer’s analysis predicts the formation of a residual resistivity dipole (RRD) in the electrochemical potential. The dipole moment of the RRD is larger than that expected from conventional diffusive scattering theory. We have applied STP to search for RRDs in the vicinity of surface voids in thin bismuth films. Although these experiments were carried out at room temperature ballistic, transport effect became apparent, but diffusive scattering (phonon scattering) still accounted for a significant part of the observed dipoles. This review also includes a discussion of scattering from other types of defects such as grain boundaries and misfit dislocations. c 1998 Academic Press Limited
Key words: residual resistivity, dipole, scanning tunneling potentiometry, ballistic transport, bismuth.
1. Introduction In a classic paper in 1957 Landauer presented a novel way to describe the influence of scattering centers in metallic films on the spatial variation of currents and fields [1]. Up until that time, charge transport in a film had been viewed according to Ohm’s law which says that a current flows in response to an externally applied electric field. The response function of the metal, its conductivity σ , was usually assumed to be constant on a microscopic scale (that is on the scale of the mean free path for phonon scattering). Landauer’s considerations dealt with scattering centers which are small compared with the mean free path, so that the conventional treatment would not be applicable. Scattering from such centers is responsible for the residual resistivity of a metal at low temperatures. Landauer investigated local current and field variations near point scatterers and created the concept of the residual resistivity dipole (RRD) as the elementary form of potential drop due to scattering. His approach treats the electric field not as the source for the charge transport but .
† E-mail address: [email protected] ‡ E-mail address: [email protected]
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as the response of the metal caused by scattering of the injected current. Pile-up of charge at the scatterer leads to the formation of a dipole in the electrochemical potential (quasi-Fermi level). These dipoles have a magnitude which differs from that given by the conventional treatment using Ohm’s law [1–3]. A quantum mechanical analysis of defect scattering in the ballistic regime has indicated that the RRD is accompanied by an oscillatory perturbation of the charge density [4]. This perturbation derives from the Friedel oscillations which appear when an ionic impurity is screened by a free-electron gas in the limit of zero current flow. In addition to point scatterers Landauer’s 1957 paper also considered thin planar barriers. As discussed there, it is not surprising that highly localized charge density and field variations appear near the barrier interfaces (indeed, it is difficult to view a tunneling barrier in any other way). When a certain current density is injected, scattering at the barrier interfaces leads to a pile-up of charge on one side of the barrier and to a deficit on the other. This charge separation produces the potential drop across the barrier from which the resistance is computed. Again, a quantum mechanical treatment of the problem shows that interference phenomena between incoming and scattered electrons give rise to charge density oscillations in the vicinity of a planar obstacle. As indicated above, carrier transport at small length scales compared with a mean free path can produce many novel and unexpected physical phenomena. So far this ballistic transport regime has been studied mainly by means of magnetotransport experiments. An alternative method of investigation which could offer new insights is the direct imaging of scattering-induced potential variations using scanning tunneling potentiometry (STP). This experimental technique which expands the scanning tunneling microscope (STM) [5] to probe the local electrochemical surface potential in metal films was introduced by Muralt and Pohl [6]. Several researchers, including the present authors, have used STP for probing potential variations at grain boundaries, steps, and localized defects in metal films [7–12]. Initial studies were hampered by tip–sample convolution artifacts in the STM data [7–9], but more recent experiments have succeeded in unambiguously detecting potential variations due to defect scattering [11, 12]. Steps in the electrochemical potential have been observed near film discontinuities and grain boundaries. For the case of RRD, observations have indeed revealed dipoleshaped variations in the electrochemical potential around small scattering centers. The magnitude of these dipoles was, however, not much greater than that expected for diffusive scattering indicating that the defects were not much smaller than the mean free path. In all cases, the experiments have proven to be substantially more difficult than originally conceived. Nevertheless, substantial progress has been made, and it seems quite possible that the next generation of experiments will succeed in visualizing a truly point-like resistivity dipole. This paper provides a short review of the progress to date in the use of STP for the observation of scatteringinduced potential variations. Section 2 presents a theoretical discussion of the RRD, focusing on the differences between the RRD and the classical dipole which forms around a macroscopic cavity in a current-carrying metal film. Section 3 briefly summarizes previous STP studies [7–10] followed by a description of our own work [11, 12]. Finally, in Section 4 we consider another aspect of scattering from planar barriers which is not related to any potentiometry measurements but involves scattering of a two-dimensional gas from an array of dislocations [13]. This problem was encountered by one of the authors (RMF) during the course of our STP studies. .
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2. Theoretical considerations In this section we consider the magnitude of potential dipoles which are induced by scattering from spherically shaped defects. The discussion focuses on the difference between results for obstacles which are larger, or smaller than the mean free path of the conduction electrons. The latter case corresponds to the RRD (ballistic scattering). The result for the former case will be called a diffusive dipole, because in this case the electrons are diverted around the defect by means of diffusive scattering (i.e. scattering with phonons). To obtain the magnitude of a diffusive dipole we determine the potential distribution around a macroscopic circular cavity using Ohm’s law J = σ E and current continuity (∇ · J = 0). The electrostatic potential V is
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Potential
RRD Cavity Diffusive
Distance Fig. 1. Potential distribution resulting from current flowing past a cavity of radius a. The solid line gives the result for purely diffusive scattering, using Ohm’s law. The dashed line schematically represents a RRD resulting from ballistic scattering. Possible oscillations in the potential variation for the RRD are not shown. .
obtained by solving the Laplace equation with the boundary condition of no current flow into the cavity [14]. For an electric field, E 0 , directed along the z-axis far from the cavity the potential has the form .
Vin = − 32 E 0r cos θ
(2.1)
Vout = −E 0r cos θ − 12 E 0
3
a cos θ r2
(2.2)
inside and outside the cavity, respectively. The electric field, E = −∇V , inside the cavity is constant. Outside the cavity there is a constant field from the first term of eqn (2.2) plus a dipole field from the second term. The source of the diffusive dipole is a sheet of surface charge which is accumulated on the wall of the cavity. This dipole has a moment given by p = 12 a 3 E 0 =
1 J0 a 2 σ
(2.3)
3
(2.4)
where J0 is the current density far from the cavity. Equation (2.3) indicates that the dipole moment is determined entirely by the applied field and the radius a, so that for a constant applied field E 0 the dipole is independent of the current. Equivalently, it can be seen from eqn (2.4) that for a constant applied current density J0 the diffusive dipole is inversely proportional to the conductivity. Figure 1 illustrates the shape of a diffusive dipole around a cavity of radius a. A characteristic feature of this dipole is that the normal component of the field approaches zero at the boundary of the cavity. Within this simple model the charge density is constant everywhere in the conductor except for the surface charge on the cavity walls (in reality this excess charge is distributed over a screening length). In contrast to the above results for a macroscopic cavity let us now go to the opposite extreme and consider a point-like defect (or more realistically a spherical scattering center which is small compared with the mean free path). Then, as discussed by Landauer [1], the charge density in the vicinity of the scatterer will no longer be constant. On the side where electrons impinge on the defect, the superposition of incoming and scattered carriers gives rise to an enhanced electron concentration, whereas on the other side of the scatterer the charge density becomes depleted. This variation of the electron concentration over a length scale of the mean free path λ is, of course, screened by the remaining conduction electrons. To describe the spatial charge density variation in the vicinity of a point scatterer one must use the electrochemical potential, which is the quasi-Fermi level needed to achieve a given electron occupation [15]. (In the presence of additional diffusive scattering, the electrochemical potential is the sum of the electrostatic potential and the quasi-Fermi level.) In three dimensions, Landauer’s theory gives the following expression for the dipole moment of a small spherical .
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scatterer with the ‘geometric’ scattering cross section πa 2 : p=
3π 2 h¯ J0 a 2 4k F 2 e2
(2.5)
where k F denotes the Fermi wavevector and J0 denotes the current density far from the defect. An analogous formula has been derived for the two-dimensional case [16]. Comparing this result to that of eqns (2.3) and (2.4), we see that the moment for the RRD differs from that for the diffusive dipole in two important ways. First, the RRD is independent of the conductivity and the scattering time. The moment of an RRD is entirely determined by the source current J0 , the Fermi wavevector k F and the scattering cross section. Secondly, the moment of the RRD is proportional to a 2 whereas the diffusive dipole depends on a 3 . Thus, the RRD result will dominate for small radii and the diffusive result is applicable for large radii, as expected. The crossover point for this behavior can be determined by equating eqns (2.4) and (2.5), from which we find a = λ/2 for a mean free path of λ = v F τ , where v F is the Fermi velocity and τ is the scattering time. Besides the different magnitude there are other features which may be used to distinguish between the RRD and a diffusive dipole. The RRD should be accompanied by Doppler-shifted charge-density oscillations [4]. In principle, such waves could be a convenient fingerprint to locate an RRD, but we presume that they can only be observed in a low-temperature experiment. If the current density is so high that the drift velocity exceeds the Fermi velocity, the RRD is expected to become asymmetric, with a more extended lobe on the downstream side of the scatterer and a more pronounced charge-density overshoot on the upstream side [4]. We return to these features which affect the appearance of the dipoles following our discussion of the experimental results. .
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3. Experimental results The technique of STP was developed to image potential distributions with high resolution [6]. For this purpose, a probe tip is placed within tunneling distance of a sample which typically consists of a thin conducting film on an insulating substrate. Two electrodes are attached to the film to allow for the injection of a lateral current. This current induces a potential distribution which can be probed by adjusting the bias voltage V of the STM tip such that no tunnel current flows between tip and sample. In operation, one means of accomplishing this procedure is to measure, at each pixel in the image, a current vs. voltage curve with fixed tip–sample separation. Later this I (V ) data is analyzed to extract a map of the voltages at which the tunnel current equals zero (i.e. a map of the electrochemical potential). More sophisticated methods of data acquisition have been developed to minimize the noise level in the potential measurements [7, 8]. Given the high spatial resolution of the scanning tunneling microscope and the ability to perform potentiometric measurements by STP, an ideal application for the method would be to study spatial variations in the electrochemical potential due to scattering by localized centers [6, 7]. The first work in this regard following that of Muralt and Pohl [6] was that of Kirtley et al. [7], who presented data apparently showing potential steps near grain boundaries in AuPd films. However, it was later pointed out by Pelz et al. [9] that such potential steps could arise from artifacts of the STP apparatus due to convolution between the tip shape and the sample topography. The origin of these types of artifacts is as follows. If the tip apex is substantially more blunt than the sample topography, then the point of closest approach between tip and sample will switch discontinuously as the tip is scanned across a topographic discontinuity on the sample. This type of convolution or ‘tip-switching’ effect was quite well known in conventional topographic STM images of rough films. Pelz et al. [9] identified its particularly pernicious influence for potentiometric images, showing that a potential scan will contain a series of abrupt steps at the locations where the tunneling position switches. Thus, what may be an apparently discontinuous potential contour can, in reality, arise simply from this tip convolution effect. More recent STP measurements also appear to be affected by such tip-related artifacts [10]. Careful examination of simultaneously recorded topographic and potentiometric images can be used, in certain cases, to unambiguously identify real potential variations on the sample [8]. However, as noted by .
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Vx 1000 Å
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In0.53Ga0.47As, n+ × 1019 cm–3 InP wafer,
×
n+
1018
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IT
Ix
= ,d
30
Å
film
Bi-
Vbias
[110]
Fig. 2. An illustration of sample geometry for cross-sectional STP experiments. Bismuth is deposited onto the cleaved edge of a n − i − n heterostructure. The potential drop on the portion of the Bi film bridging the i layer is probed with the STM (reprinted from [12]). .
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Pelz et al. [9], it is preferable to perform STP experiments on samples with very flat topography. Ideally one would like to use atomically flat samples, allowing for some inevitable topography variations associated with whatever scattering centers are being investigated. With this requirement in mind we have proposed a new sample geometry which allows for the deposition of flat films in ultra-high-vacuum while simultaneously permitting a convenient means of forming the lateral contacts to the films [11, 12]. This geometry is illustrated in Fig. 2. A cross-sectional plane of a multilayer semiconductor structure provides the substrate surface onto which a thin conducting film is deposited. An atomically flat and clean surface is obtained by cleaving the multilayer substrate in an ultra-high vacuum. The conducting layers of the substrate are used as contacts to the film. The applied voltage is dropped across a 500 nm thick insulating layer between the contacts. For the metal film itself one must use a material which maintains a flat morphology upon deposition and which also forms relatively good ohmic contacts to the conducting layers of the underlying semiconductor heterostructure. The semimetal bismuth was chosen, because prior studies indicated at least some possibility for achieving both of these requirements [17, 18]. Deposition of Bi at a temperature of 140 K was found to result in a flat morphology, and careful choice of the semiconducting contact layers resulted in good ohmic contacts to the film [11]. The use of a semimetal with a very-low-conduction electron density (n ≈ 3 × 1017 cm−3 for bulk Bi) has the additional advantage that the small Fermi wavevector leads to a large dipole moment of the RRD (eqn (2.5)). The STP experiments with the configuration depicted in Fig. 2 were successful. Using this sample geometry it was possible to study charge transport with current densities up to 8 × 106 A cm−2 , two orders of magnitude higher than that achieved in prior studies [11, 19]† . Results are presented in Fig. 3 [12]. Figure 3A shows the ˚ thick Bi film. The film surface is atomically flat apart from some irregularly shaped voids topography of a 30 A ˚ which form naturally during the deposition process. The simultaneously or holes with a depth of about 12 A recorded potential distribution in this film is shown in Fig. 3B. For this image which covers a potential range .
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† The current density in [7], p.1548 should correctly read as j = 4×104 A cm−2 [19]. (J. Kirtley, private communication). .
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Topography image
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100 Å B
Potential image
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Potential (mV)
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C B–B
10 5
30 Å
0 –5
A–A
–10 –15
50 Å
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˚ thick Bi film. The potential discontinuity at grain Fig. 3. A, STM-topograph, B, simultaneously recorded STP-image on a 30 A boundary is highlighted. C, Cuts across the potential image as marked in B, and expected potential slopes if steps were entirely caused by tip convolution (reprinted from [12]). .
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of 34 mV the Bi film has been subject to a lateral current density of about 3 × 106 A cm−2 . Figure 3B displays two distinct features which reflect two different scattering mechanisms contributing to the resistivity of the film. First, phonon scattering is clearly evident as a smooth ramp in the grey scale. Secondly, there are potential steps of 2–4 mV coinciding with the position of the surface voids. These small steps can be seen more clearly in the line cuts of Fig. 3C. An upper bound to the possible influence of tip-convolution artifacts on the shape of the line cuts can be determined from the topographic information on the defect size and the measured potential ramp. The two voids which cause the depicted potential steps have a horizontal extension ˚ Assuming that no voltage drop occurs at the voids, the distortion of the uniform ramp a of 30 and 50 A. by tip convolution could at most lead to voltage steps of 1V = E x a. As indicated in Fig. 3C, these steps are significantly smaller than the observed discontinuities. In contrast a tip convolution, which only locally
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distorts the potential ramp, the steps in the experimental data lead to an offset between the flat parts of the potential curve, i.e. they represent a real resistance increase. Tip convolution effects may still contribute to the observed potential variations within a void, but on the atomically flat portions of the film the experimental data can be confidently assigned to a true variation of the electrochemical potential. A network of faint lines connecting the surface defects is visible in Fig. 3A. These lines are grain boundaries which separate differently oriented crystallites of the Bi film. The potential distribution near a vertical grain boundary has been highlighted in Fig. 3B by locally changing the gray-scale repartition. A small step of <1 mV is visible, providing evidence for localized scattering at the grain boundary. Such steps were close to the detection limit of our STP experiments arising from external noise sources. For thinner Bi films, the grain boundaries gave rise to more pronounced discontinuities [12], although even in those cases the features were somewhat obscured due to overlapping potential variations from other defects. Therefore, it was not possible to quantitatively determine the resistance of a grain boundary. During prolonged current injection, the Bi films undergo a strain relaxation process [11]. As a consequence, ˚ deep voids disappear, and the surface now mainly consists of large terraces separated most of the 12 A ˚ by 4 and 8 A high steps [12]. However, a few voids even increase in depth and become very dominant scatterers, which permit investigation of the transport-induced potential variations in more detail. The images ˚ thick Bi film carrying a current density of 4 × 106 A cm−2 . A shown in Fig. 4 were obtained on a 30 A ˚ 24 A deep void located in the center of the topography image (Fig. 4A) gives rise to the surface potential presented in Fig. 4B. For the gray-scale image of the potential distribution we have subtracted a linear ramp ˚ −1 ) to highlight the part of the potential distribution which is induced by defect scattering. In (E x = 44 µV A addition, the superimposed contours (1V = 1 mV between adjacent lines) indicate the potential distribution before background subtraction. Figure 4B reveals a very large dipole with lobes extending far beyond the topographical size of the defect. A horizontal line cut through the center of the dipole (Fig. 4C) illustrates the ˚ extension of the potential tails over several hundred A. As discussed in Section 2, the RRD arising from ballistic scattering and the diffusive dipole can be distinguished on the basis of their magnitude. For a given current density J0 the moment of the diffusive dipole is determined by the conductivity σ of the surrounding medium (eqn (2.4)). Therefore, the strength of a diffusive dipole can be determined from the slope of the observed potential ramp. If ballistic scattering is present in addition to phonon scattering, the observed dipole moment should be larger than that predicted by the diffusive model. The RRD does not depend on σ and its moment exceeds the diffusive dipole if the defect size 2a is smaller than the mean free path λ = v F τ . This situation occurs either for small defects or at low temperature. To determine in our experiment whether the observed dipole reflects ballistic scattering we compare the line cut in Fig. 4C with the calculated shape of a diffusive dipole (dashed line). This calculated dipole represents the potential around a rectangular void with dimensions equal to that seen in Fig. 4A passing through a two-dimensional film. There are no free parameters which enter the computation other than the dimensions of the void and the magnitude of the field far from the void, both of which are accurately known from the experiment. As seen in Fig. 4C, the observed potential dipole is substantially larger than the computed diffusive results, thus demonstrating that ballistic scattering is definitely present in the experiment. On the other hand, the magnitude of the observed dipole is not so large as to permit its exclusive assignment to ballistic scattering. Oscillations in the potential which have been theoretically predicted [4] are not observed in our experiments. We believe that this can be explained with the argument that, at room temperature, the Fermi energy of Bi is comparable to k B T , so that the absence of a sharp cut-off in the density of states would inhibit these oscillations. An asymmetry of the dipole lobes which has been predicted for high drift velocities [4] is indeed found in the experiment. The contours on Fig. 4B show a small region on the left side of the void where the local field is inverted with respect to the background field. This overshoot on the upstream side of the defect is in qualitative accordance with theory. We conclude that the dipole shown in Fig. 4B is not a pure RRD but rather a superposition of ballistic and diffusive dipoles. This result indicates that the defect .
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Topography image
A
Potential image
B
15
C
Potential (mV)
10 5 0 –5
–10 –15 –300 –200 –100 0 100 200 300 400 Scan distance (Å) ˚ 2 area of 30 A ˚ thick Bi film with a 24 A ˚ deep void in the center; pixel resolution Fig. 4. A, The STM-topograph of 640 × 800 A ˚ B, The reduced STP potential after subtraction of a linear background. Contours indicate STP potential before background = 3.5 A. subtraction. C, Cross-sectional cut along the line marked by arrows in B (solid line), and computed curve for purely diffusive transport around the deep void (dashed line). All three figures share the same x-axis calibration. .
dimension is still comparable to the mean free path. Obviously, transport through surface states significantly ˚ [11]. To obtain reduces the mean free path in thin Bi films in comparison to the bulk value (λ ≈ 1000 A) evidence for purely ballistic transport it will probably be necessary to perform experiments at low temperature and maybe to choose a metal with a longer mean free path. .
4. Scattering from an array of dislocations During the course of our STP studies of Bi films, one of us (RMF) working in collaboration with F. Stern of IBM Research encountered a separate problem involving application of Landauer’s formula for the resistance
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ζ (z)
x
707
100 Å
Screened potential (meV)
15 10 5
EF
0 –5 –10 –15 –500
–250
0 250 Distance (Å)
500
Fig. 5. Screened potential resulting from the strain field of a misfit dislocation located at the lower interface of a quantum channel ˚ The channel layer and misfit dislocation are shown schematically in the upper part of the figure, along layer, with thickness of 100 A. with the subband wavefunction ζ (z). The solid line for the potential gives the result for the effective potential, integrated over z with ˚ from the bottom of the channel, near the peak position weighting factor ζ 2 (z). The dashed line gives the potential at a distance of 80 A of ζ (z) (reprinted from [13]). .
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of a planar barrier. We briefly discuss here the background behind this problem and the computations which were performed to resolve it [13]. The problem originated in measurements by Ismail and coworkers of the lowtemperature mobility of a two-dimensional electron gas at a SiGe/Si heterojunction [20]. The heterostructure consists of a Si channel, grown on top of a strain-relaxed Si0.7 Ge0.3 buffer layer. The mobility of the electron gas was found to be a very strong function of the thickness of the channel layer, displaying a very sharp drop ˚ Plan-view electronmicroscopy measurements when the channel thickness exceeded a value of about 100 A. revealed the presence of a two-dimensional array of misfit dislocations formed at the SiGe/Si interface [20]. Theses dislocations form to relieve strain in the Si channel, and the channel thickness for this system (see [21] for a similar example of misfit dislocation formation, in another materials system). The dislocation does, however, produce strain inhomogeneities in the channel layer, which act to scatter the carriers. Initial estimates of the influence of the strain on mobility yielded good qualitative agreement with the observed trends [20], although a quantitative computation of this effect was lacking. The influence of the dislocations on the mobility of the electron gas can be evaluated by considering their strain field perturbing the conduction band edge through a known deformation potential constant. The resulting perturbation acts as a scattering potential, and reflection and transmission coefficients for this potential can be numerically evaluated in the usual manner [13]. The geometry is illustrated in Fig. 5, showing in the upper ˚ thick Si channel layer with misfit dislocation at the lower interface. The electron gas is part the Si 100 A represented by its envelope function ζ (z). The scattering potential is shown, including the effects of screening using the random-phase approximation. From this potential, the reflection coefficient R(φ) is obtained, as a function of the scattering angle φ. Given the reflection coefficient for scattering from a single barrier, one is able to compute the conductance for incoherent scattering from an array of dislocations from the formula of Landauer [1, 22] Z 2π 1 − R(φ) 2e2 k F dφ (4.1) | cos φ| σ = 4π 2 h¯ N 0 R(φ) .
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where N is the linear density of misfit dislocations (known in this case from experiment [20]), and a factor of 2 .
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R (φ)
Electron mobility (103 cm2/Vs)
T = 20 K
0.8 0
60
φ
π
40
20
0 50
100
150
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Channel thickness (Å) Fig. 6. Computed mobilities (lines) and measured mobilities (symbols, from [20] for Ge fraction of 0.34) as a function of channel thickness. Measurements are for a temperature of 20 K and theory is for 0 K. Theory considers scattering from strain variations induced by misfit dislocations only. The solid line gives an exact computation of localized scattering, and the dashed line gives results using the Born approximation. The inset shows the reflection coefficient (for the exact computation) as a function of the angle φ relative to the ˚ (reprinted from [13]). dislocation normal, for a channel thickness of 100 A .
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has been included in the prefactor to account for the two conduction-band minima in strained Si. This formula was originally derived for current flowing at normal incidence to a linear array of dislocations, but it should also apply to a two-dimensional array of orthogonal dislocations since the component of momentum along the normal to one set of dislocations is not affected by scattering from the perpendicular dislocations. From the conductance computed from eqn (4.1), the mobility is obtained by dividing by en s where n s = 5 × 1011 cm−2 is the electron density in the channel. The result for the above scattering computation is shown by the solid line in Fig. 6, and is compared with experimental results for the mobility. The dominant behavior there comes from the onset of the dislocation ˚ In addition, the mobility (and conductance) increases with formation at a channel thickness of about 100 A. channel thickness since the electrons see a smaller (albeit wider) scattering potential. The theoretical results are in quite good agreement with the experimental results of Ismail et al. [20], thus providing some support for their model of deformation potential scattering to account for the observed sharp drop in mobility with increasing channel thickness. Also shown by the dashed line in Fig. 6 is a more approximate computation of the mobility, making use of a Born approximation for the conductance [13, 22]. Substantial deviations occur between the approximate and exact computations, arising from the fact that the dislocations are moderately strong scattering centers. .
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5. Conclusions We have presented a brief review of the application of STP to the study of potential variations due to scattering from localized centers. Although STP provides an apparently ideal method for spatially resolving such phenomena [6, 7], there are numerous practical problems which must be overcome before such experiments come to fruition. Tip-convolution effects were identified as an important aspect of the measurement which leads to artifacts in the results [9]. The best method to avoid such artifacts is to perform the experiments on atomically flat films, as was achieved in measurements of Bi films deposited on cleaved InP surfaces .
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[11, 12]. True potential variations due to localized scattering were observed there, although the contribution of phonon scattering to the measurements was still substantial. Further progress in this area will probably rely on low-temperature experiments with different types of metal films. .
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Acknowledgements—We gratefully acknowledge the contributions of our collaborators, T. P. Chin and J. M. Woodall, for growth of the semiconductors heterostructures used in our potentiometry studies. We also thank J. R. Kirtley, R. Landauer, J. P. Pelz, and F. Stern for helpful discussions during the course of this work. This work was supported by the U.S. National Science Foundation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
}R. Landauer, IBM J. Res. Develop. 1, 223 (1957). }R. Landauer, Z. Phys. B21, 247 (1975). }R. Landauer, J. Phys. C8, 761 (1975). }W. Zwerger, L. B¨onig, and K. Sch¨onhammer, Phys. Rev. B43, 6434 (1991). }G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Phys. Rev. Lett. 49, 57 (1982). }P. Muralt and D. Pohl, Appl. Phys. Lett. 48, 514 (1986); P. Muralt, D. W. Pohl, and W. Denk, IBM J. Res. Develop. 30, 443 (1986). }J. R. Kirtley, S. Washburn, and M. J. Brady, Phys. Rev. Lett. 60, 1546 (1988). }J. P. Pelz and R. H. Koch, Rev. Sci. Instrum. 60, 301 (1989). }J. P. Pelz and R. H. Koch, Phys. Rev. B41, 1212 (1990). }J. Besold, G. Reiss, and H. Hoffmann, Appl. Surf. Sci. 65/66, 23 (1993). }B. G. Briner, R. M. Feenstra, T. P. Chin, and J. M. Woodall, Phys. Rev. B54, R5283 (1996). }B. G. Briner, R. M. Feenstra, T. P. Chin, and J. M. Woodall, Semicond. Sci. Technol. 11, 1575 (1996). }R. M. Feenstra and M. A. Lutz, J. Appl. Phys. 78, 6091 (1995). }J. D. Jackson, Classical Electrodynamics (New York, Wiley, 1975), 2nd edn, p. 151. }R. Landauer, IBM J. Res. Develop. 32, 306 (1988). }R. S. Sorbello and C. S. Chu, IBM J. Res. Develop. 32, 58 (1988). }J. C. Patrin, Y. Z. Li, M. Chander, and J. H. Weaver, J. Vac. Sci. Technol. A11, 2073 (1993). }R. H. Williams, D. R. T. Zahn, N. Esser and W. Richter, J. Vac. Sci. Technol. B7, 997 (1989). }J. Kirtley, private communication. }K. Ismail, F. K. LeGoues, K. L. Saenger, M. Arafa, J. O. Chu, P. M. Mooney, and B. S. Meyerson, Phys. Rev. Lett. 73, 3447 (1994). }N. Frank, G. Springholz, and G. Bauer, Phys. Rev. Lett. 73, 2236 (1994). }R. Landauer, Phys. Rev. B52, 11225 (1995). .
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The search for residual resistivity dipoles by scanning tunneling potentiometry R. M. Feenstra† Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
B. G. Briner‡ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
(Received 22 August 1997) A review is given of the application of scanning tunneling potentiometry (STP) to the study of potential variations due to scattering from localized centers. Landauer has theoretically investigated such scattering in the ballistic regime where the electronic mean free path exceeds the size of the scattering centers. Landauer’s analysis predicts the formation of a residual resistivity dipole (RRD) in the electrochemical potential. The dipole moment of the RRD is larger than that expected from conventional diffusive scattering theory. We have applied STP to search for RRDs in the vicinity of surface voids in thin bismuth films. Although these experiments were carried out at room temperature ballistic, transport effect became apparent, but diffusive scattering (phonon scattering) still accounted for a significant part of the observed dipoles. This review also includes a discussion of scattering from other types of defects such as grain boundaries and misfit dislocations. c 1998 Academic Press Limited
Key words: residual resistivity, dipole, scanning tunneling potentiometry, ballistic transport, bismuth.
1. Introduction In a classic paper in 1957 Landauer presented a novel way to describe the influence of scattering centers in metallic films on the spatial variation of currents and fields [1]. Up until that time, charge transport in a film had been viewed according to Ohm’s law which says that a current flows in response to an externally applied electric field. The response function of the metal, its conductivity σ , was usually assumed to be constant on a microscopic scale (that is on the scale of the mean free path for phonon scattering). Landauer’s considerations dealt with scattering centers which are small compared with the mean free path, so that the conventional treatment would not be applicable. Scattering from such centers is responsible for the residual resistivity of a metal at low temperatures. Landauer investigated local current and field variations near point scatterers and created the concept of the residual resistivity dipole (RRD) as the elementary form of potential drop due to scattering. His approach treats the electric field not as the source for the charge transport but .
† E-mail address: [email protected] ‡ E-mail address: [email protected]
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as the response of the metal caused by scattering of the injected current. Pile-up of charge at the scatterer leads to the formation of a dipole in the electrochemical potential (quasi-Fermi level). These dipoles have a magnitude which differs from that given by the conventional treatment using Ohm’s law [1–3]. A quantum mechanical analysis of defect scattering in the ballistic regime has indicated that the RRD is accompanied by an oscillatory perturbation of the charge density [4]. This perturbation derives from the Friedel oscillations which appear when an ionic impurity is screened by a free-electron gas in the limit of zero current flow. In addition to point scatterers Landauer’s 1957 paper also considered thin planar barriers. As discussed there, it is not surprising that highly localized charge density and field variations appear near the barrier interfaces (indeed, it is difficult to view a tunneling barrier in any other way). When a certain current density is injected, scattering at the barrier interfaces leads to a pile-up of charge on one side of the barrier and to a deficit on the other. This charge separation produces the potential drop across the barrier from which the resistance is computed. Again, a quantum mechanical treatment of the problem shows that interference phenomena between incoming and scattered electrons give rise to charge density oscillations in the vicinity of a planar obstacle. As indicated above, carrier transport at small length scales compared with a mean free path can produce many novel and unexpected physical phenomena. So far this ballistic transport regime has been studied mainly by means of magnetotransport experiments. An alternative method of investigation which could offer new insights is the direct imaging of scattering-induced potential variations using scanning tunneling potentiometry (STP). This experimental technique which expands the scanning tunneling microscope (STM) [5] to probe the local electrochemical surface potential in metal films was introduced by Muralt and Pohl [6]. Several researchers, including the present authors, have used STP for probing potential variations at grain boundaries, steps, and localized defects in metal films [7–12]. Initial studies were hampered by tip–sample convolution artifacts in the STM data [7–9], but more recent experiments have succeeded in unambiguously detecting potential variations due to defect scattering [11, 12]. Steps in the electrochemical potential have been observed near film discontinuities and grain boundaries. For the case of RRD, observations have indeed revealed dipoleshaped variations in the electrochemical potential around small scattering centers. The magnitude of these dipoles was, however, not much greater than that expected for diffusive scattering indicating that the defects were not much smaller than the mean free path. In all cases, the experiments have proven to be substantially more difficult than originally conceived. Nevertheless, substantial progress has been made, and it seems quite possible that the next generation of experiments will succeed in visualizing a truly point-like resistivity dipole. This paper provides a short review of the progress to date in the use of STP for the observation of scatteringinduced potential variations. Section 2 presents a theoretical discussion of the RRD, focusing on the differences between the RRD and the classical dipole which forms around a macroscopic cavity in a current-carrying metal film. Section 3 briefly summarizes previous STP studies [7–10] followed by a description of our own work [11, 12]. Finally, in Section 4 we consider another aspect of scattering from planar barriers which is not related to any potentiometry measurements but involves scattering of a two-dimensional gas from an array of dislocations [13]. This problem was encountered by one of the authors (RMF) during the course of our STP studies. .
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2. Theoretical considerations In this section we consider the magnitude of potential dipoles which are induced by scattering from spherically shaped defects. The discussion focuses on the difference between results for obstacles which are larger, or smaller than the mean free path of the conduction electrons. The latter case corresponds to the RRD (ballistic scattering). The result for the former case will be called a diffusive dipole, because in this case the electrons are diverted around the defect by means of diffusive scattering (i.e. scattering with phonons). To obtain the magnitude of a diffusive dipole we determine the potential distribution around a macroscopic circular cavity using Ohm’s law J = σ E and current continuity (∇ · J = 0). The electrostatic potential V is
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Potential
RRD Cavity Diffusive
Distance Fig. 1. Potential distribution resulting from current flowing past a cavity of radius a. The solid line gives the result for purely diffusive scattering, using Ohm’s law. The dashed line schematically represents a RRD resulting from ballistic scattering. Possible oscillations in the potential variation for the RRD are not shown. .
obtained by solving the Laplace equation with the boundary condition of no current flow into the cavity [14]. For an electric field, E 0 , directed along the z-axis far from the cavity the potential has the form .
Vin = − 32 E 0r cos θ
(2.1)
Vout = −E 0r cos θ − 12 E 0
3
a cos θ r2
(2.2)
inside and outside the cavity, respectively. The electric field, E = −∇V , inside the cavity is constant. Outside the cavity there is a constant field from the first term of eqn (2.2) plus a dipole field from the second term. The source of the diffusive dipole is a sheet of surface charge which is accumulated on the wall of the cavity. This dipole has a moment given by p = 12 a 3 E 0 =
1 J0 a 2 σ
(2.3)
3
(2.4)
where J0 is the current density far from the cavity. Equation (2.3) indicates that the dipole moment is determined entirely by the applied field and the radius a, so that for a constant applied field E 0 the dipole is independent of the current. Equivalently, it can be seen from eqn (2.4) that for a constant applied current density J0 the diffusive dipole is inversely proportional to the conductivity. Figure 1 illustrates the shape of a diffusive dipole around a cavity of radius a. A characteristic feature of this dipole is that the normal component of the field approaches zero at the boundary of the cavity. Within this simple model the charge density is constant everywhere in the conductor except for the surface charge on the cavity walls (in reality this excess charge is distributed over a screening length). In contrast to the above results for a macroscopic cavity let us now go to the opposite extreme and consider a point-like defect (or more realistically a spherical scattering center which is small compared with the mean free path). Then, as discussed by Landauer [1], the charge density in the vicinity of the scatterer will no longer be constant. On the side where electrons impinge on the defect, the superposition of incoming and scattered carriers gives rise to an enhanced electron concentration, whereas on the other side of the scatterer the charge density becomes depleted. This variation of the electron concentration over a length scale of the mean free path λ is, of course, screened by the remaining conduction electrons. To describe the spatial charge density variation in the vicinity of a point scatterer one must use the electrochemical potential, which is the quasi-Fermi level needed to achieve a given electron occupation [15]. (In the presence of additional diffusive scattering, the electrochemical potential is the sum of the electrostatic potential and the quasi-Fermi level.) In three dimensions, Landauer’s theory gives the following expression for the dipole moment of a small spherical .
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scatterer with the ‘geometric’ scattering cross section πa 2 : p=
3π 2 h¯ J0 a 2 4k F 2 e2
(2.5)
where k F denotes the Fermi wavevector and J0 denotes the current density far from the defect. An analogous formula has been derived for the two-dimensional case [16]. Comparing this result to that of eqns (2.3) and (2.4), we see that the moment for the RRD differs from that for the diffusive dipole in two important ways. First, the RRD is independent of the conductivity and the scattering time. The moment of an RRD is entirely determined by the source current J0 , the Fermi wavevector k F and the scattering cross section. Secondly, the moment of the RRD is proportional to a 2 whereas the diffusive dipole depends on a 3 . Thus, the RRD result will dominate for small radii and the diffusive result is applicable for large radii, as expected. The crossover point for this behavior can be determined by equating eqns (2.4) and (2.5), from which we find a = λ/2 for a mean free path of λ = v F τ , where v F is the Fermi velocity and τ is the scattering time. Besides the different magnitude there are other features which may be used to distinguish between the RRD and a diffusive dipole. The RRD should be accompanied by Doppler-shifted charge-density oscillations [4]. In principle, such waves could be a convenient fingerprint to locate an RRD, but we presume that they can only be observed in a low-temperature experiment. If the current density is so high that the drift velocity exceeds the Fermi velocity, the RRD is expected to become asymmetric, with a more extended lobe on the downstream side of the scatterer and a more pronounced charge-density overshoot on the upstream side [4]. We return to these features which affect the appearance of the dipoles following our discussion of the experimental results. .
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3. Experimental results The technique of STP was developed to image potential distributions with high resolution [6]. For this purpose, a probe tip is placed within tunneling distance of a sample which typically consists of a thin conducting film on an insulating substrate. Two electrodes are attached to the film to allow for the injection of a lateral current. This current induces a potential distribution which can be probed by adjusting the bias voltage V of the STM tip such that no tunnel current flows between tip and sample. In operation, one means of accomplishing this procedure is to measure, at each pixel in the image, a current vs. voltage curve with fixed tip–sample separation. Later this I (V ) data is analyzed to extract a map of the voltages at which the tunnel current equals zero (i.e. a map of the electrochemical potential). More sophisticated methods of data acquisition have been developed to minimize the noise level in the potential measurements [7, 8]. Given the high spatial resolution of the scanning tunneling microscope and the ability to perform potentiometric measurements by STP, an ideal application for the method would be to study spatial variations in the electrochemical potential due to scattering by localized centers [6, 7]. The first work in this regard following that of Muralt and Pohl [6] was that of Kirtley et al. [7], who presented data apparently showing potential steps near grain boundaries in AuPd films. However, it was later pointed out by Pelz et al. [9] that such potential steps could arise from artifacts of the STP apparatus due to convolution between the tip shape and the sample topography. The origin of these types of artifacts is as follows. If the tip apex is substantially more blunt than the sample topography, then the point of closest approach between tip and sample will switch discontinuously as the tip is scanned across a topographic discontinuity on the sample. This type of convolution or ‘tip-switching’ effect was quite well known in conventional topographic STM images of rough films. Pelz et al. [9] identified its particularly pernicious influence for potentiometric images, showing that a potential scan will contain a series of abrupt steps at the locations where the tunneling position switches. Thus, what may be an apparently discontinuous potential contour can, in reality, arise simply from this tip convolution effect. More recent STP measurements also appear to be affected by such tip-related artifacts [10]. Careful examination of simultaneously recorded topographic and potentiometric images can be used, in certain cases, to unambiguously identify real potential variations on the sample [8]. However, as noted by .
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Vx 1000 Å
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In0.53Ga0.47As, n+ × 1019 cm–3
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×
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IT
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30
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film
Bi-
Vbias
[110]
Fig. 2. An illustration of sample geometry for cross-sectional STP experiments. Bismuth is deposited onto the cleaved edge of a n − i − n heterostructure. The potential drop on the portion of the Bi film bridging the i layer is probed with the STM (reprinted from [12]). .
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Pelz et al. [9], it is preferable to perform STP experiments on samples with very flat topography. Ideally one would like to use atomically flat samples, allowing for some inevitable topography variations associated with whatever scattering centers are being investigated. With this requirement in mind we have proposed a new sample geometry which allows for the deposition of flat films in ultra-high-vacuum while simultaneously permitting a convenient means of forming the lateral contacts to the films [11, 12]. This geometry is illustrated in Fig. 2. A cross-sectional plane of a multilayer semiconductor structure provides the substrate surface onto which a thin conducting film is deposited. An atomically flat and clean surface is obtained by cleaving the multilayer substrate in an ultra-high vacuum. The conducting layers of the substrate are used as contacts to the film. The applied voltage is dropped across a 500 nm thick insulating layer between the contacts. For the metal film itself one must use a material which maintains a flat morphology upon deposition and which also forms relatively good ohmic contacts to the conducting layers of the underlying semiconductor heterostructure. The semimetal bismuth was chosen, because prior studies indicated at least some possibility for achieving both of these requirements [17, 18]. Deposition of Bi at a temperature of 140 K was found to result in a flat morphology, and careful choice of the semiconducting contact layers resulted in good ohmic contacts to the film [11]. The use of a semimetal with a very-low-conduction electron density (n ≈ 3 × 1017 cm−3 for bulk Bi) has the additional advantage that the small Fermi wavevector leads to a large dipole moment of the RRD (eqn (2.5)). The STP experiments with the configuration depicted in Fig. 2 were successful. Using this sample geometry it was possible to study charge transport with current densities up to 8 × 106 A cm−2 , two orders of magnitude higher than that achieved in prior studies [11, 19]† . Results are presented in Fig. 3 [12]. Figure 3A shows the ˚ thick Bi film. The film surface is atomically flat apart from some irregularly shaped voids topography of a 30 A ˚ which form naturally during the deposition process. The simultaneously or holes with a depth of about 12 A recorded potential distribution in this film is shown in Fig. 3B. For this image which covers a potential range .
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† The current density in [7], p.1548 should correctly read as j = 4×104 A cm−2 [19]. (J. Kirtley, private communication). .
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Topography image
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Potential image
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10 5
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0 –5
A–A
–10 –15
50 Å
0
200
400 600 Scan distance (Å)
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˚ thick Bi film. The potential discontinuity at grain Fig. 3. A, STM-topograph, B, simultaneously recorded STP-image on a 30 A boundary is highlighted. C, Cuts across the potential image as marked in B, and expected potential slopes if steps were entirely caused by tip convolution (reprinted from [12]). .
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of 34 mV the Bi film has been subject to a lateral current density of about 3 × 106 A cm−2 . Figure 3B displays two distinct features which reflect two different scattering mechanisms contributing to the resistivity of the film. First, phonon scattering is clearly evident as a smooth ramp in the grey scale. Secondly, there are potential steps of 2–4 mV coinciding with the position of the surface voids. These small steps can be seen more clearly in the line cuts of Fig. 3C. An upper bound to the possible influence of tip-convolution artifacts on the shape of the line cuts can be determined from the topographic information on the defect size and the measured potential ramp. The two voids which cause the depicted potential steps have a horizontal extension ˚ Assuming that no voltage drop occurs at the voids, the distortion of the uniform ramp a of 30 and 50 A. by tip convolution could at most lead to voltage steps of 1V = E x a. As indicated in Fig. 3C, these steps are significantly smaller than the observed discontinuities. In contrast a tip convolution, which only locally
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distorts the potential ramp, the steps in the experimental data lead to an offset between the flat parts of the potential curve, i.e. they represent a real resistance increase. Tip convolution effects may still contribute to the observed potential variations within a void, but on the atomically flat portions of the film the experimental data can be confidently assigned to a true variation of the electrochemical potential. A network of faint lines connecting the surface defects is visible in Fig. 3A. These lines are grain boundaries which separate differently oriented crystallites of the Bi film. The potential distribution near a vertical grain boundary has been highlighted in Fig. 3B by locally changing the gray-scale repartition. A small step of <1 mV is visible, providing evidence for localized scattering at the grain boundary. Such steps were close to the detection limit of our STP experiments arising from external noise sources. For thinner Bi films, the grain boundaries gave rise to more pronounced discontinuities [12], although even in those cases the features were somewhat obscured due to overlapping potential variations from other defects. Therefore, it was not possible to quantitatively determine the resistance of a grain boundary. During prolonged current injection, the Bi films undergo a strain relaxation process [11]. As a consequence, ˚ deep voids disappear, and the surface now mainly consists of large terraces separated most of the 12 A ˚ by 4 and 8 A high steps [12]. However, a few voids even increase in depth and become very dominant scatterers, which permit investigation of the transport-induced potential variations in more detail. The images ˚ thick Bi film carrying a current density of 4 × 106 A cm−2 . A shown in Fig. 4 were obtained on a 30 A ˚ 24 A deep void located in the center of the topography image (Fig. 4A) gives rise to the surface potential presented in Fig. 4B. For the gray-scale image of the potential distribution we have subtracted a linear ramp ˚ −1 ) to highlight the part of the potential distribution which is induced by defect scattering. In (E x = 44 µV A addition, the superimposed contours (1V = 1 mV between adjacent lines) indicate the potential distribution before background subtraction. Figure 4B reveals a very large dipole with lobes extending far beyond the topographical size of the defect. A horizontal line cut through the center of the dipole (Fig. 4C) illustrates the ˚ extension of the potential tails over several hundred A. As discussed in Section 2, the RRD arising from ballistic scattering and the diffusive dipole can be distinguished on the basis of their magnitude. For a given current density J0 the moment of the diffusive dipole is determined by the conductivity σ of the surrounding medium (eqn (2.4)). Therefore, the strength of a diffusive dipole can be determined from the slope of the observed potential ramp. If ballistic scattering is present in addition to phonon scattering, the observed dipole moment should be larger than that predicted by the diffusive model. The RRD does not depend on σ and its moment exceeds the diffusive dipole if the defect size 2a is smaller than the mean free path λ = v F τ . This situation occurs either for small defects or at low temperature. To determine in our experiment whether the observed dipole reflects ballistic scattering we compare the line cut in Fig. 4C with the calculated shape of a diffusive dipole (dashed line). This calculated dipole represents the potential around a rectangular void with dimensions equal to that seen in Fig. 4A passing through a two-dimensional film. There are no free parameters which enter the computation other than the dimensions of the void and the magnitude of the field far from the void, both of which are accurately known from the experiment. As seen in Fig. 4C, the observed potential dipole is substantially larger than the computed diffusive results, thus demonstrating that ballistic scattering is definitely present in the experiment. On the other hand, the magnitude of the observed dipole is not so large as to permit its exclusive assignment to ballistic scattering. Oscillations in the potential which have been theoretically predicted [4] are not observed in our experiments. We believe that this can be explained with the argument that, at room temperature, the Fermi energy of Bi is comparable to k B T , so that the absence of a sharp cut-off in the density of states would inhibit these oscillations. An asymmetry of the dipole lobes which has been predicted for high drift velocities [4] is indeed found in the experiment. The contours on Fig. 4B show a small region on the left side of the void where the local field is inverted with respect to the background field. This overshoot on the upstream side of the defect is in qualitative accordance with theory. We conclude that the dipole shown in Fig. 4B is not a pure RRD but rather a superposition of ballistic and diffusive dipoles. This result indicates that the defect .
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Topography image
A
Potential image
B
15
C
Potential (mV)
10 5 0 –5
–10 –15 –300 –200 –100 0 100 200 300 400 Scan distance (Å) ˚ 2 area of 30 A ˚ thick Bi film with a 24 A ˚ deep void in the center; pixel resolution Fig. 4. A, The STM-topograph of 640 × 800 A ˚ B, The reduced STP potential after subtraction of a linear background. Contours indicate STP potential before background = 3.5 A. subtraction. C, Cross-sectional cut along the line marked by arrows in B (solid line), and computed curve for purely diffusive transport around the deep void (dashed line). All three figures share the same x-axis calibration. .
dimension is still comparable to the mean free path. Obviously, transport through surface states significantly ˚ [11]. To obtain reduces the mean free path in thin Bi films in comparison to the bulk value (λ ≈ 1000 A) evidence for purely ballistic transport it will probably be necessary to perform experiments at low temperature and maybe to choose a metal with a longer mean free path. .
4. Scattering from an array of dislocations During the course of our STP studies of Bi films, one of us (RMF) working in collaboration with F. Stern of IBM Research encountered a separate problem involving application of Landauer’s formula for the resistance
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EF
0 –5 –10 –15 –500
–250
0 250 Distance (Å)
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Fig. 5. Screened potential resulting from the strain field of a misfit dislocation located at the lower interface of a quantum channel ˚ The channel layer and misfit dislocation are shown schematically in the upper part of the figure, along layer, with thickness of 100 A. with the subband wavefunction ζ (z). The solid line for the potential gives the result for the effective potential, integrated over z with ˚ from the bottom of the channel, near the peak position weighting factor ζ 2 (z). The dashed line gives the potential at a distance of 80 A of ζ (z) (reprinted from [13]). .
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of a planar barrier. We briefly discuss here the background behind this problem and the computations which were performed to resolve it [13]. The problem originated in measurements by Ismail and coworkers of the lowtemperature mobility of a two-dimensional electron gas at a SiGe/Si heterojunction [20]. The heterostructure consists of a Si channel, grown on top of a strain-relaxed Si0.7 Ge0.3 buffer layer. The mobility of the electron gas was found to be a very strong function of the thickness of the channel layer, displaying a very sharp drop ˚ Plan-view electronmicroscopy measurements when the channel thickness exceeded a value of about 100 A. revealed the presence of a two-dimensional array of misfit dislocations formed at the SiGe/Si interface [20]. Theses dislocations form to relieve strain in the Si channel, and the channel thickness for this system (see [21] for a similar example of misfit dislocation formation, in another materials system). The dislocation does, however, produce strain inhomogeneities in the channel layer, which act to scatter the carriers. Initial estimates of the influence of the strain on mobility yielded good qualitative agreement with the observed trends [20], although a quantitative computation of this effect was lacking. The influence of the dislocations on the mobility of the electron gas can be evaluated by considering their strain field perturbing the conduction band edge through a known deformation potential constant. The resulting perturbation acts as a scattering potential, and reflection and transmission coefficients for this potential can be numerically evaluated in the usual manner [13]. The geometry is illustrated in Fig. 5, showing in the upper ˚ thick Si channel layer with misfit dislocation at the lower interface. The electron gas is part the Si 100 A represented by its envelope function ζ (z). The scattering potential is shown, including the effects of screening using the random-phase approximation. From this potential, the reflection coefficient R(φ) is obtained, as a function of the scattering angle φ. Given the reflection coefficient for scattering from a single barrier, one is able to compute the conductance for incoherent scattering from an array of dislocations from the formula of Landauer [1, 22] Z 2π 1 − R(φ) 2e2 k F dφ (4.1) | cos φ| σ = 4π 2 h¯ N 0 R(φ) .
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where N is the linear density of misfit dislocations (known in this case from experiment [20]), and a factor of 2 .
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R (φ)
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φ
π
40
20
0 50
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Channel thickness (Å) Fig. 6. Computed mobilities (lines) and measured mobilities (symbols, from [20] for Ge fraction of 0.34) as a function of channel thickness. Measurements are for a temperature of 20 K and theory is for 0 K. Theory considers scattering from strain variations induced by misfit dislocations only. The solid line gives an exact computation of localized scattering, and the dashed line gives results using the Born approximation. The inset shows the reflection coefficient (for the exact computation) as a function of the angle φ relative to the ˚ (reprinted from [13]). dislocation normal, for a channel thickness of 100 A .
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has been included in the prefactor to account for the two conduction-band minima in strained Si. This formula was originally derived for current flowing at normal incidence to a linear array of dislocations, but it should also apply to a two-dimensional array of orthogonal dislocations since the component of momentum along the normal to one set of dislocations is not affected by scattering from the perpendicular dislocations. From the conductance computed from eqn (4.1), the mobility is obtained by dividing by en s where n s = 5 × 1011 cm−2 is the electron density in the channel. The result for the above scattering computation is shown by the solid line in Fig. 6, and is compared with experimental results for the mobility. The dominant behavior there comes from the onset of the dislocation ˚ In addition, the mobility (and conductance) increases with formation at a channel thickness of about 100 A. channel thickness since the electrons see a smaller (albeit wider) scattering potential. The theoretical results are in quite good agreement with the experimental results of Ismail et al. [20], thus providing some support for their model of deformation potential scattering to account for the observed sharp drop in mobility with increasing channel thickness. Also shown by the dashed line in Fig. 6 is a more approximate computation of the mobility, making use of a Born approximation for the conductance [13, 22]. Substantial deviations occur between the approximate and exact computations, arising from the fact that the dislocations are moderately strong scattering centers. .
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5. Conclusions We have presented a brief review of the application of STP to the study of potential variations due to scattering from localized centers. Although STP provides an apparently ideal method for spatially resolving such phenomena [6, 7], there are numerous practical problems which must be overcome before such experiments come to fruition. Tip-convolution effects were identified as an important aspect of the measurement which leads to artifacts in the results [9]. The best method to avoid such artifacts is to perform the experiments on atomically flat films, as was achieved in measurements of Bi films deposited on cleaved InP surfaces .
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[11, 12]. True potential variations due to localized scattering were observed there, although the contribution of phonon scattering to the measurements was still substantial. Further progress in this area will probably rely on low-temperature experiments with different types of metal films. .
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Acknowledgements—We gratefully acknowledge the contributions of our collaborators, T. P. Chin and J. M. Woodall, for growth of the semiconductors heterostructures used in our potentiometry studies. We also thank J. R. Kirtley, R. Landauer, J. P. Pelz, and F. Stern for helpful discussions during the course of this work. This work was supported by the U.S. National Science Foundation.
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