Scanning probe methods are being used to study an increasing range of surface phenomena. These surface phenomena include surface topography, dielectric and magnetic properties, molecular manipulation and many other effects that occur on the micrometer to the subnanometer scale. Scanning probe microscopes include scanning tunneling microscopes (STMs) and atomic force microscopes (AFMs). Both STMs and AFMs utilize a sharp probe tip to interact with the target surface to generate a topographic surface scan. The parameters used for image formation are different for STMs and AFMs. In an STM system, the modulations of the tunneling current are used to determine surface topography while in an AFM system, the varying force between the probe tip and the target surface is used to determine the surface topography.
STM systems may also be employed to monitor the tunneling current versus the bias voltage at fixed locations on the target surface. The STM probe tip is positioned at a particular location above the target surface. A spectrum is produced by sweeping the bias voltage and measuring the tunneling current. This provides information regarding the density of surface states in the target surface proximate to the STM probe tip.
The use of STM and AFM systems has enabled relatively routine imaging of surfaces at nanoscale resolutions, but more information may be desired about the surface. A desirable measurement modality is one that allows surface imaging of unaltered samples while at the same time providing a means for characterizing a physical property associated with the localized target surface.
In accordance with the invention, resonant quantum tunneling microscopy allows surface imaging while allowing characterization of physical properties associated with the surface.
a-b show embodiments in accordance with the invention.
a-c show the affect of various applied voltages on the Fermi level in accordance with the invention.
a shows the tunneling current as a function of tunneling voltage in accordance with the invention.
b shows the differential conductance as a function of tunneling voltage in accordance with the invention.
a-c show the effect of various ramp voltages on the Fermi level in accordance with the invention.
An embodiment in accordance with the invention is shown in
Tip support structure 150 functions as the base to which tunneling electrodes 120 and 130 are attached and can be made in a manner similar to the way that SPM support structures are manufactured. SPM support structures may be microfabricated from silicon in a manner similar to computer chips and produced to have a pyramidal shape. In order for tip support structure 150 to be insulating, the silicon can be oxidized prior to the attachment or growth of tunneling electrodes 120 and 130. Other typical methods of fabricating appropriately dimensioned and insulating tip support structure 150 may also be used. Tunneling electrodes 120 and 130 may be attached to tip support structure 150 in a number of ways in accordance with the invention. For example, small metal precursor regions may be lithographically patterned and deposited on tip support structure 150. The carbon nanotubes are then grown (e.g., M. Jose-Yacaman, M. Miki-Yoshida, L. Rendon, T. Santiesteban, Appl. Phys. Lett. 62, 202 (1993) incorporated herein by reference) on the small metal precursor regions to form tunneling electrodes 120 and 130. Any other available technique to attach carbon nanotubes to tip support structure 150 may also be used to create tunneling electrodes 120 and 130. Other structures may be manufactured and attached to tip support structure 150 to form tunneling electrodes 120 and 130 in accordance with the invention and includes structures typically used for STM tips. For example, a typical STM tip is made from tungsten polycrystalline wire, platinum wire or iridium wire which is electrochemically etched to provide a suitably sharp tip. Tip support structure 150 typically has a lateral dimension in the submicron range. Tunneling electrodes 120 and 130 typically have a separation on the order of nanometers or less at their furthest extent from tip support structure 150 to create adequate tunneling current, IT.
Tunneling electrodes 120 and 130 are connected to variable voltage source 140 and current measuring device 145. The variable voltage source 140 and current measuring device 145 are similar to those typically used in STMs. Electrical connections to variable voltage source 140 and to current measuring device 145 from tunneling electrodes 120 and 130 may be accomplished in a number of ways in accordance with the invention. For example, metal lines may be created lithographically from the attachment sites of tunneling electrodes 120 and 130 to larger metal bonding pads (not shown). This may be accomplished during the microfabrication of tip support structure 150.
In operation, xyz scanner 160 positions tunneling electrodes 120 and 130 proximate to target surface 110 and over scanned spot 135. The tunneling voltage, Vt, between tunneling electrodes 120 and 130 is ramped over a prescribed range, and the tunneling current, IT, is measured as a function of time using current measuring device 145. The predominant physical mechanism for generating the tunneling current is resonant quantum tunneling (RQT). The tunneling differential conductance, σ=dIT/dVt, for RQT peaks sharply when two physical conditions are satisfied. First, the integrated magnitudes of the tunneling barriers between tunneling electrodes 120, 130 and scanned spot 135, which functions as the intermediate well region, must be equal. Second, the Fermi energy in negatively biased tunnel electrode 120 must precisely match an energy level of scanned spot 135 which functions as the intermediate well region. The tunneling differential conductance, σ, may be enhanced by orders of magnitude when these two conditions are met.
During operation, as tunneling electrodes 120 and 130 move to each scanned spot 135 on target surface 110, the tunneling voltage, Vt, is ramped over the prescribed range. The precise position and extent of scanned spot 135 is self-selected by the requirement that the integrated magnitudes of the tunneling barriers between tunneling electrodes 120, 130 and scanned spot 135 are equal. Typically, this implies that scanned spot 135 is approximately equidistant from tunneling electrodes 120 and 130, although there may be some deviation, for example, if the electron emissivity of scanned spot 135 has inherent asymmetries and the effective barrier potential varies accordingly. Target surfaces with significant roughness, where the ratio of surface roughness to the spacing between tunneling electrodes 120 and 130 is less than unity will have a bias in the position of scanned spot 135; from being centered between electrodes 120 and 130 to being off-center. The spatial extent of scanned spot 135 is determined by how rapidly the tunneling probability decreases as the tunneling barriers with respect to tunneling electrodes 120 and 130 become asymmetric. Hence, even if tunneling electrodes 120 and 130 are spatially separated by a distance on the order of several nanometers, the spatial extent of scanned spot 135 is determined by the symmetry requirement for the tunneling barriers, i.e. only that limited spatial region with a symmetric barrier to tunneling electrodes 120 and 130 is interrogated. This typically results in sizes for scanned spot 135 on the order of several Angstroms. Tunneling current contributions from surface regions proximate to tunneling electrodes 120 and 130 that have asymmetric tunneling barriers contribute to background noise but this noise contribution is typically many orders of magnitude smaller than RQT currents. In directions perpendicular to the tunneling current flow, the size of scanned spot 135 is determined by tunneling suppression due to increased barrier width and not by the condition of tunneling barrier symmetry.
As the tunneling voltage, Vt, is ramped over the prescribed range during a measurement, the peaks in the tunneling differential conductance, σ, occur at those voltages where the Fermi energy of negatively biased tunnel electrode 120 matches an energy level of scanned spot 135. The measurement provides detailed information about the electron energy spectrum of scanned spot 135 which can be used to determine local elemental or molecular composition and deviations from expected spectra measured from control regions. The magnitude of the tunneling current, IT, off-resonance can be used as a standard STM signal to obtain a measure of the separation of tunneling electrodes 120 and 130 from target surface 110 to obtain surface topography. The measured energy spectra are referenced to the potentials of tunneling electrodes 120 and 130 and surface charges or other sources of potential offsets at scanned spot 135 may produce measurable voltage shifts that may need to be considered in spectral identification.
RQTSM 100 can be used to simultaneously obtain both topographical and spectral information about target surface 110. The spatial resolution is typically substantially better than the separation distance between tunneling electrodes 120 and 130, on the order of Angstroms. Operation of RQTSM 100 may in some cases be affected by thermal noise issues which may result in signal to noise issues. A typical solution to thermal noise issues is to cool a system to low temperature where thermal broadening is less than the energy spacing between the targeted differential conductance, σ, peaks (see
b shows an embodiment in accordance with the invention. Here, the tunneling voltage, Vt, applied between tunneling electrodes 120 and 130 is kept constant and the ramped voltage, VR, is applied by ramped voltage source 190 between metal layer 175 beneath target surface 110 and voltage source 185. The tunneling current, IT, and the differential conductance, σ, as a function of voltage provide the same information as in the embodiment shown in
In accordance with the invention as shown in
The effects of a non-zero temperature for tunneling electrodes 120 and 130 are potentially significant. At zero temperature, the Fermi level denotes the energy below which all electron levels in the Fermi sea of electrodes 120 and 130 are occupied, and above which all electron levels in the Fermi sea of electrodes 120 and 130 are unoccupied. Thus, as the tunneling voltage, Vt, is increased and the Fermi level passes through a narrow well bound state energy, a very sharp turn-on of the tunneling current, IT, occurs, as shown in
In an embodiment in accordance with the invention as shown in
An examination of the one-dimensional quantum mechanical double-barrier transmission problem shows the conditions for resonant quantum tunneling (RQT). The Appendix shows the exact solution for the transmission problem for the one-dimensional quantum mechanical double-barrier and shows that the transmission probability only becomes unity when the energy, Ee, of an incident particle matches a bound state energy level and the integrated magnitude of potential barriers 215 and 225 are equal as noted above. For the double-barrier structure shown in
Ttot˜TL·TR. (1)
where TL and TR are the quantum mechanical tunneling probabilities through left potential barrier 215 and right potential barrier 225, respectively. The quantum mechanical tunneling probabilities, TL,R have an exponential dependence on the integrated magnitude of the potential barriers:
where l is the barrier width, μ is the charge carrier mass and (VL−Ee) is the energy difference between the top of the left potential barrier 215, VL, and the incident particle energy, Ee and (VR−Ee) is the energy difference between the top of the right potential barrier 225, VR, and the incident particle energy, Ee. For an order of magnitude estimate, it can be assumed that a single potential barrier has a typical height of about one eV and the single potential barrier width is typically about one nanometer for scanned spot 135 so that a single potential barrier tunneling probability is TL,R≈3×10−5. These small probabilities are the limiting factors for the expected magnitude of the tunneling current, IT.
A property of the double-barrier structure shown in
where A, B, C, D are factors of order unity that depend weakly on the incident particle energy, Ee. For non-resonant tunneling, TL and TR are much less than unity as discussed above so that Eq. (4) for Ttot is dominated by the first term in the denominator giving Eq. (1)
Ttot˜TL·TR (1)
When the resonant quantum tunneling conditions are met, the coefficient A(Ee) in Eq. (4) goes to zero as shown in the Appendix. Eq. (4) is then dominated by a non-leading term. If potential barriers 215 and 225 are of equal integrated magnitudes, TL=TR, the total transmission probability, Ttot, is of order unity. The Appendix shows that for equal potential barriers at resonance
Ttot=1 (5)
However, if TL≢TR, the transmission probability remains negligible. For example, if TL<<TR, the dominant term in Eq. (4) becomes
which is much less than unity. Similarly, if TL>>TR, the dominant term in Eq. (4) becomes
as expected from symmetry considerations.
The enhanced spatial resolution provided in accordance with the invention using RQTSM can then be easily understood. For those regions on target surface 110 separated from scanned spot 135, tunneling barriers 215 and 225 are unequal and the left and right tunneling probabilities TL and TR, respectively, are typically of significantly different magnitudes due to the exponential dependence of TL and TR on the integrated barrier magnitudes as shown by Eq. (2). In this case, as shown by Eqs. (6) and (7), the gain in transmission probability, Ttot and tunneling current, IT, is small. Only for scanned spot region 135 is the large tunneling current enhancement due to RQT operative.
A1. Double Barrier Solution
It is assumed that the energy of the incident particle, Ee , is greater than the potential energy in all regions except for barrier regions 2 and 4. Under this condition, the general solutions, Ψi, to the one-dimensional, time independent Schroedinger equation in each of the five regions shown in
Ψ1=A1eik
Ψ2=A2eik
Ψ3=A3eik
Ψ4=A4eik
Ψ5=A5eik
wherek1,3,5=√{square root over (2μ(Ee−V1,3,5))} (A6)
andk2,4=√{square root over (2μ(V2,4−Ee).)} (A7)
The solution specific solution is determined by matching Ψ and dΨ/dx at the region interfaces. This procedure may be accomplished as a pair of subproblems. Matching the boundary conditions across the first barrier allows the wavefunction coefficients in region 1, A1, B1 to be written in terms of the wavefunction coefficients in region 3, A3, B3
Similarly, matching the boundary conditions across the second barrier allows the wavefunction coefficients in region 3, A3, B3 to be written in terms of the coefficients in region 5, A5, B5
The full expression connecting the wavefunction coefficients of region 1 with those of region 5 is then determined by the linear transformation using Eqs. (A8) and A(15)
The full transmission coefficient is determined by applying the boundary condition
which corresponds to an incident wave of unit amplitude from the left (A1=1) in
Explicit evaluation of Eq. (A24) and collecting and grouping terms gives
and the terms of F are listed in descending powers of the large “barrier suppression factors” and where
Φ1≡k3L (A27)
γ2≡2k2a (A28)
γ4≡2k4b. (A29 )
A2. Resonance Condition
Assuming the potential baffiers are strong impediments to particle transmission, i.e. e2γ
Ttot˜e−2γ
However, for the situation when
φ1−φ3−φ4=nπ (A31)
the coefficients of the first three terms in Eq. (A26) are zero. If the two potential baffiers are of equal integrated magnitudes, i.e. γ2=γ4, then the leading term in Eq. (A26) is of the order unity, and the total transmission coefficient can be shown to approach unity. This is the condition for resonant quantum tunneling and exhibits the property of total transmission through a double potential barrier structure, regardless of the strength of the individual bafflers as long as the potential barriers are of equal integrated magnitudes.
It is important to understand the physical significance of the resonance condition of Eq. (A31). To simplify the discussion, the completely symmetric case will be considered where
φ3=a tan(k2/k3)=a tan(k4/k3)=φ4 (A32)
which gives
sin(φ1−2φ3)=0. (A33)
Using trigonometric identities and substituting the definitions for φ1 and φ3, allows Eq. (A33) to be rewritten as
If the arbitrary baseline for the potential energy is chosen to be V3≡0 and V2 is renamed V0, Eq. (A34) becomes
Eq. (A35) is the eigenvalue equation for the energy levels of a quantum square well potential with the parameters as defined above. This shows why the phenomenon of total transmission through a double-well structure is called resonant quantum tunneling. The condition for resonant quantum tunneling is that the energy, Ee, of the incident particle matches any of the bound state energy levels of the quantum square well. Whenever the incident energy, Ee, matches any of the bound state energy levels, the total transmission probability increases dramatically, as long as the double potential barriers are symmetric.
A3. Tunneling Current on Resonance
As discussed above, for a symmetric potential barrier structure, the transmission probability becomes unity when the incident energy, Ee, passes through a bound state energy level of the quantum square well. However, the situation is different for a double potential barrier structure that has asymmetric potential barriers. For the general asymmetric potential barrier structure on resonance, it follows from Eq. (A26) that the dominant terms have the form
F˜e2γ
This implies that for the situation where the left potential baffler in
and for the reverse situation (γ4>>γ2)
This shows the different resonant tunneling behavior for the asymmetric double potential barrier structure. If the potential baffler structure is highly asymmetric, there is very little gain in the tunneling probability as the resonance condition is approached. Only under the condition of double potential baffler symmetry is resonant quantum tunneling in effect.
Number | Name | Date | Kind |
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7410564 | Flory | Aug 2008 | B2 |
Number | Date | Country | |
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20080179516 A1 | Jul 2008 | US |