Scanning tunneling microscope - STM Fundamentals

Scanning Tunneling Microscope (STM): Principles and Practice Introduction The scanning tunneling microscope is a powerful tool for imaging surfaces at the atomic scale. Unlike optical microscopes that use light, STM exploits a quantum mechanical phenomenon called tunneling to "see" individual atoms and map their electronic properties. This makes it uniquely capable of achieving sub-angstrom resolution—allowing visualization of features smaller than a single atom. To understand how STM works and how to interpret its measurements, you need to grasp the quantum mechanical basis of electron tunneling and how the tunneling current relates to the physical and electronic structure of surfaces. Quantum Tunneling: The Physical Foundation Why Electrons Can Cross a Vacuum Gap In classical physics, if an electron doesn't have enough energy to overcome a potential barrier (like the vacuum gap between a microscope tip and a sample), it simply cannot cross. But quantum mechanics tells us electrons don't have definite trajectories—they are described by wavefunctions that can extend into classically forbidden regions. This means there is a non-zero probability that an electron can "tunnel" through the barrier, even though it lacks the energy to climb over it. The probability of tunneling depends exponentially on the distance the electron must tunnel. Specifically, as the gap width increases by just 1 Ångstrom (≈ 0.1 nanometers), the tunneling probability drops by roughly an order of magnitude. This extreme sensitivity to distance is what makes STM so powerful—small changes in tip height produce large changes in current, allowing measurement of tiny surface features. Rectangular Barrier Model To calculate tunneling quantitatively, physicists approximate the vacuum gap as a rectangular potential barrier of height $U$ and width $w$. The height $U$ is approximately equal to the work function $W$ of the sample material (typically 4–6 eV). The key result is the transmission probability: $$T \approx e^{-2\kappa w}$$ where the decay constant is: $$\kappa = \frac{\sqrt{2me(U - E)}}{\hbar}$$ Here $me$ is the electron mass, $E$ is the electron energy, and $\hbar$ is the reduced Planck constant. The exponential dependence on $w$ is crucial: this is why tunneling current drops so dramatically with distance, giving STM its exceptional distance sensitivity. Connecting Tunneling Current to Surface Properties Tersoff-Hamann Approximation While the rectangular barrier model explains why tunneling decays with distance, it doesn't tell us what property of the sample the tunneling current actually measures. This is where the Tersoff-Hamann approximation becomes essential. In this model, developed by Tersoff and Hamann in the mid-1980s, the microscope tip is treated as a simple spherical point source (an $s$-wave source in quantum language). This simplification leads to a powerful result: $$I \propto \rho{\text{sample}}(EF, \mathbf{r}0)$$ The tunneling current is proportional to the local density of states (LDOS) of the sample at the Fermi level, evaluated at the tip position $\mathbf{r}0$. This relationship is fundamental to understanding STM images. It tells us that what we're actually measuring is not just the topography (physical height) of the surface, but the electronic structure of the surface. Regions with more available electronic states near the Fermi level appear brighter in STM images—they conduct more current. Why This Matters The Tersoff-Hamann result explains a crucial fact about STM: the images show a convolution of topography and electronic structure. An atom that is physically higher might appear dimmer than a lower atom if it has fewer electronic states available for tunneling. Conversely, adsorbates or impurities that locally change the density of states will show up as bright or dark features even if they're only slightly raised or lowered. Scanning Modes: How STM Images Are Made Constant-Current Mode In constant-current mode, the microscope uses feedback control to maintain a constant tunneling current as it scans. Here's how it works: As the tip scans laterally across the surface, it encounters bumps and valleys A feedback circuit continuously adjusts the vertical position (z-piezo voltage) to keep the current constant The vertical adjustments are recorded—this creates the image What do the recorded z-values represent? The height values encode information about both the surface topography and local electronic density. When the current would drop (either because the surface dips down or because electronic states decrease), the feedback loop pushes the tip closer to maintain constant current. Conversely, when the current would increase, the tip retracts. Advantage: This mode directly shows features that are electronically relevant. Impurities or adsorbates that matter electronically will appear as prominent features. Disadvantage: Constant-current mode is slower because the feedback loop must settle at every pixel of the image, taking time to adjust the z-piezo. Constant-Height Mode In constant-height mode, the z-piezo voltage is simply held fixed. As the tip scans, the tunneling current fluctuates naturally—and these current variations are recorded directly to form the image. Advantages: This mode is much faster since there's no feedback loop settling time. Disadvantages: This mode risks crashing the tip on rough surfaces or tall adsorbates. If the tip dips too close, the current spikes and can damage the tip or sample. This mode requires extremely flat surfaces. The choice between modes depends on the sample: flat, well-prepared surfaces allow fast constant-height imaging, while rougher surfaces require the slower but safer constant-current approach. Scanning Tunneling Spectroscopy: Measuring Electronic Structure What Is STS? While standard STM imaging maps spatial variations in tunneling current, scanning tunneling spectroscopy (STS) measures how the tunneling current changes with energy. This gives direct information about the electronic density of states—a fundamental property of any material. How STS Measurements Work The procedure is straightforward: The tip is brought to a specific position on the sample and held fixed The sample bias voltage is swept over a range (typically −1 to +1 V, but variable) At each voltage step, the tunneling current is measured The result is a plot of current $I$ versus bias voltage $V$ To increase measurement precision, a small AC modulation is typically applied to the voltage, and the derivative $dI/dV$ (differential conductance) is measured directly using a lock-in amplifier. This derivative contains the most information about electronic states. Interpreting STS Data: Relationship to Density of States The fundamental relationship is: $$I(V) \propto \int{0}^{eV} \rho{\text{sample}}(E)\,\rho{\text{tip}}(EF)\,e^{-2\kappa w}\,dE$$ At low temperature and small bias, this simplifies: the differential conductance $dI/dV$ is approximately proportional to the sample's density of states at energy $E = eV$. This is profound: by varying the bias voltage, you can "tune" which electronic energies contribute to the tunneling, and measure the density of states as a function of energy. What do peaks mean? Peaks in a $dI/dV$ vs. $V$ plot indicate electronic states or band edges—places where there are many available electronic states at that energy. Valleys (low $dI/dV$) indicate energy gaps where few states are available. By comparing spectra taken on different parts of the surface (e.g., on and off an impurity), you can see how local defects or adsorbates modify the electronic structure. <extrainfo> Temperature Effects STS experiments are often performed at liquid-helium temperature (≈ 4 K) because this sharpens the Fermi-Dirac distribution $f(E)$—the function describing how many electrons occupy each energy state. At room temperature, thermal energy broadens the distribution, smearing out fine features in $dI/dV$ spectra. Low temperature gives better energy resolution. </extrainfo> Key Practical Considerations Tip Materials and Sharpness STM tips are typically made from tungsten, platinum-iridium alloy, or gold. The choice depends on the application and sample reactivity. What matters most is tip sharpness—the radius of curvature of the tip apex. A sharp tip (small radius) provides better spatial resolution because the current is more sensitive to the local density of states immediately below the tip. A common problem is double-tip artifacts: when the tip has two sharp apices rather than one, both contribute to tunneling, and the image shows broadened or duplicated features. This is why tip conditioning is important. Sample Requirements For STM to work effectively, samples must be: Atomically clean: Contamination ruins the ability to resolve atomic features. Surfaces typically require ultra-high-vacuum preparation, including ion sputtering and thermal annealing to remove adsorbates. Electrically conductive: There must be a conducting path for electrons to tunnel. Insulating samples cannot be imaged directly (unless the tip itself is coated with a conductive layer). Mechanically stable: Surface atoms must not move or vibrate significantly during imaging. This is another reason vacuum is crucial—it eliminates adsorbate-driven diffusion. Image Artifacts and Limitations <extrainfo> Several artifacts can distort STM images: Tip convolution: The finite size of the tip means spatial features appear broader and can be inverted (what should appear as a depression appears as a protrusion, or vice versa). Thermal drift: Temperature fluctuations cause the piezos and scanner to expand or contract slowly, distorting the image. This is why STM systems often include thermal stabilization and drift-compensation electronics. Piezo creep: Piezoelectric actuators can exhibit time-dependent deformation under constant voltage, causing image distortion. Regular recalibration helps mitigate this. </extrainfo> Summary of Core Concepts The STM's power comes from combining two ideas: Exponential tunneling decay ($T \propto e^{-2\kappa w}$) gives sub-angstrom distance sensitivity—small changes in tip height produce huge current changes. Tersoff-Hamann theory tells us that tunneling current measures the local density of electronic states, not just physical height. This makes STM an electronic microscope as much as a topographic one. In constant-current mode, the feedback-controlled z-position maps both topography and LDOS. In constant-height mode, current fluctuations directly reveal electronic structure variations. By using spectroscopy (varying bias voltage), one can measure how LDOS varies with energy, accessing the electronic band structure of individual atoms and molecules. Understanding these principles is essential for interpreting real STM data: features in images can arise from topographic steps, from regions of different electronic density, or from a combination of both. Spectroscopy then provides the missing piece—the energy-resolved electronic information needed to fully characterize a surface.