Scanning tunneling microscope Study Guide

📖 Core Concepts Quantum tunneling – electrons cross the vacuum gap between tip and sample despite a classically forbidden barrier; the probability drops exponentially with distance. Decay constant $\kappa = \sqrt{2me (U-E)}/\hbar$ (units m⁻¹) determines how fast the tunneling probability falls. Transmission probability $T \approx e^{-2\kappa w}$ → a ≈1 Å increase in gap cuts the current by 10×. Local density of states (LDOS) – the number of electronic states per energy at a specific location; in STM the current is directly proportional to the sample LDOS at the tip position (Tersoff‑Hamann). Feedback control – in constant‑current mode a closed‑loop adjusts the $z$‑piezo voltage to keep $I$ at a setpoint, mapping surface height. Scanning tunneling spectroscopy (STS) – measures $dI/dV$ vs. bias to probe the energy‑resolved LDOS. --- 📌 Must Remember Exponential distance dependence: $I \propto e^{-2\kappa w}$ → 1 Å ↑ → ≈10× ↓ in current. Tersoff‑Hamann result: $I \propto \rho{\text{sample}}(EF,\mathbf r0)$. Bardeen current formula: $I \propto \sum{\mu,\nu} |M{\mu\nu}|^2 [f(E\mu)-f(E\nu)]$. Constant‑current vs. constant‑height: Current mode → height image, slower, safer on rough surfaces. Height mode → current image, faster, risk of tip crash. STS relation (low T, small bias): $dI/dV \approx \rho{\text{sample}}(eV) \,\rho{\text{tip}}(EF)\,e^{-2\kappa w}$. Tip materials: W, Pt‑Ir, Au; tip radius dictates ultimate resolution; double‑tip artefacts distort images. --- 🔄 Key Processes Approach & Engage: Coarse approach brings tip within 1 nm. Detect tunneling current → engage fine piezo scanner. Feedback Loop (constant‑current): Measure $I$. Compare to setpoint $I0$. Error → adjust $z$‑piezo voltage via PID controller. Raster Scan: Step tip laterally (x‑y) in a grid. At each pixel, let feedback settle → record $z$ (height mode) or $I$ (height mode). STS Measurement: Hold tip at fixed $(x,y)$. Sweep bias $V$ while adding small AC modulation. Lock‑in detect $dI/dV$ → plot vs. $V$. --- 🔍 Key Comparisons Constant‑Current vs. Constant‑Height Current: maps topography + LDOS; slower; safe on uneven surfaces. Height: maps LDOS directly via current variations; faster; risk of tip‑sample contact. Rectangular Barrier Model vs. Bardeen Formalism Rectangular barrier: simple exponential $T$; good for intuition of distance sensitivity. Bardeen: full quantum‑mechanical overlap integral; basis for quantitative current calculations. Tip Materials (W vs. Pt‑Ir vs. Au) W: high stiffness, easy to sharpen → best spatial resolution. Pt‑Ir: chemically inert, good for ambient/air operation. Au: excellent conductivity, useful for spin‑polarized STM when coated. --- ⚠️ Common Misunderstandings “STM measures topography only.” – In constant‑current mode the recorded height also includes variations in LDOS; apparent corrugations may be electronic, not geometric. “Higher bias gives better resolution.” – Large bias broadens the energy window, mixing states and reducing spectroscopic fidelity; low bias (and low T) sharpens the Fermi edge. “Any conductive sample works.” – Surface must be atomically clean and stable; contamination or adsorbates create artefacts and tip crashes. --- 🧠 Mental Models / Intuition “Tunneling as a leak.” Imagine the vacuum gap as a thin wall; the thinner the wall, the more water (electrons) leaks through exponentially. “Feedback as a cruise‑control for height.” The controller constantly adjusts the “gas pedal” (piezo voltage) to keep the “speed” (current) at the set value, thereby “steering” the tip height. “LDOS as a spotlight.” The tip’s s‑wave acts like a tiny spotlight that shines on the sample’s electronic states; brighter spots in the image mean higher LDOS at $EF$. --- 🚩 Exceptions & Edge Cases Double‑tip artefact: Two apexes simultaneously tunneling produce doubled features or inverted contrast. Thermal drift & piezo creep: At long scan times the image can stretch or shift; require drift compensation or frequent recalibration. Spin‑polarized STM: Requires a ferromagnetic tip; normal STM equations hold but the measured LDOS is spin‑dependent. --- 📍 When to Use Which Choose constant‑current when scanning rough or unknown surfaces, or when you need reliable height information. Choose constant‑height for high‑speed imaging of atomically flat regions or when mapping rapid LDOS variations. Use STS (dI/dV) to identify electronic states, band edges, or impurity resonances; perform at cryogenic temperature for best energy resolution. Select tip material based on environment: W for UHV/high‑resolution; Pt‑Ir for ambient; Au for spin‑polarized measurements. --- 👀 Patterns to Recognize Exponential current decay: A linear plot of $\ln I$ vs. $z$ indicates proper tunneling regime. Symmetric vs. asymmetric $dI/dV$ spectra: Symmetry suggests identical tip and sample LDOS; asymmetry points to tip states or band bending. Repeated corrugation period matching lattice constant → true atomic resolution; irregular spacing often signals tip artefacts. --- 🗂️ Exam Traps “Current decreases linearly with distance.” – The correct relationship is exponential; linear answers are distractors. Confusing work function $W$ with barrier height $U$ – $U \approx W$, but $U$ appears in $\kappa$; exams may swap symbols to test understanding. Selecting constant‑height for rough surfaces – This leads to tip crash; the correct choice is constant‑current. Assuming $dI/dV$ equals total DOS – It actually reflects the sample LDOS at the energy $eV$, modulated by tip DOS and the exponential factor. ---