Cotunneling theory and multiplet excitations: emergence of asymmetric line shape in inelastic scanning tunneling spectroscopy of correlated molecules on surfaces

This paper extends cotunneling theory to general multireference systems, demonstrating that the combination of multireference character and asymmetric tip/substrate couplings generates pronounced asymmetric line shapes in inelastic scanning tunneling spectroscopy, offering a new microscopic explanation for features frequently observed in experiments on correlated molecules.

The Gist

The Big Picture: Listening to Molecules Sing

Imagine you have a tiny, single molecule sitting on a metal table. You want to know what it "sounds" like when you poke it with electricity. Scientists use a tool called a Scanning Tunneling Microscope (STM). Think of the STM tip as a super-sharp needle that hovers just above the molecule. When you push a little voltage through the needle, electrons tunnel across the gap, creating a tiny electric current.

Usually, when you measure this current, you see a smooth line. But sometimes, right at a specific voltage, you see a sudden "step" or a "dip" in the line. This is the molecule saying, "Hey! I just used some energy to jump to a higher excited state!"

For a long time, scientists thought these steps were always symmetrical (looking the same on the left and right side of the zero point). They believed this was caused by simple magnetic spins flipping, like a tiny compass needle turning.

This paper says: "Not always!"

The authors discovered that sometimes these steps are wildly asymmetrical. They look like a steep cliff on one side and a gentle slope on the other. They figured out why this happens, and it has to do with how complex the molecule's internal structure is and how the needle touches it.

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The Key Concepts (The "Cast of Characters")

1. The "Chameleon" Molecule (Multireference Systems)

Most simple molecules are like a choir singing in perfect unison; everyone is doing the same thing. But some complex molecules (like the ones studied here) are like a jazz band where everyone is improvising.

• The Analogy: Imagine a molecule that doesn't have just one "personality" (one way the electrons are arranged). Instead, it's a superposition of many different personalities at once. It's like a chameleon that is simultaneously red, blue, and green.
• The Science: In physics terms, these are called multireference systems. The ground state isn't just one simple arrangement of electrons; it's a messy mix of many different arrangements.

2. The "One-Sided" Touch (Asymmetric Coupling)

Now, imagine you are trying to tap this jazz band with a drumstick (the STM tip).

• The Analogy: If the band members are standing in a circle, and you tap only the person on the far left, you hear a different sound than if you tap the person on the far right.
• The Science: The electrons in these complex molecules live in different "orbits" (orbitals). Some orbits stick out toward the tip, while others hide near the surface. If the tip touches the "sticking out" orbit strongly but the "hiding" orbit weakly, the connection is asymmetric.

3. The "Ghost" Jump (Cotunneling)

To make the molecule jump to a higher energy state, an electron has to borrow energy.

• The Analogy: Imagine you want to jump over a wall, but you can't do it alone. You need a friend to give you a boost. In this quantum world, the electron briefly "borrows" a charge state (becoming a charged ion for a split second) before settling back down. This is called cotunneling.
• The Twist: Because the molecule is a "chameleon" (mix of personalities) and the tip is touching it "one-sidedly," the rules for borrowing that charge change depending on which way you push the voltage.

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The "Aha!" Moment: Why the Shape is Weird

The authors ran a simulation to see what happens when you combine these two things: a complex, mixed-up molecule and a one-sided touch.

The Result:

• Symmetric Case: If the molecule is simple (like a perfect pair of electrons) or the tip touches it perfectly evenly, the signal is a nice, symmetrical step.
• Asymmetric Case: If the molecule is complex (a mix of states) and the tip touches it unevenly, the signal becomes a lopsided cliff.
  • Why? Think of it like a door.
    • If you push the door from the "easy" side (where the tip touches strongly), the door swings open easily (a big signal).
    • If you push from the "hard" side (where the tip touches weakly), the door is jammed and won't open (no signal).
  • Because the molecule is a mix of different states, the "door" only opens for one direction of voltage, creating that weird, asymmetric shape.

The Real-World Example: The Metal Porphyin

To prove this wasn't just math, they looked at a real molecule called Cobalt Porphyrin (a ring-shaped molecule with a cobalt atom in the middle).

• This molecule has electrons in the ring (the "ligand") and electrons in the cobalt atom (the "metal center").
• These two groups of electrons are far apart but connected.
• When they simulated the STM tip hovering over this molecule, they found that because the electrons in the cobalt atom stick out differently than the electrons in the ring, the tip touches them with different strengths.
• The Outcome: The simulation produced exactly the kind of asymmetric, lopsided peaks that experimentalists see in real life but couldn't explain before.

The Takeaway

Before this paper: If you saw a weird, lopsided bump in your data, you might have blamed it on a specific magnetic effect or a "Kondo" effect (a complex interaction involving many electrons).

After this paper: You now know that if you see a lopsided bump, it might just be because:

1. The molecule is a complex mix of different electron states (a "multireference" system).
2. The STM tip is touching the molecule unevenly.

The Metaphor Summary:
Imagine trying to push a swing. If the swing is simple and you push from the center, it goes back and forth evenly. But if the swing is made of two different materials (one heavy, one light) and you push it from the heavy side, it will swing wildly in one direction and barely move in the other. This paper explains that the "weird swings" we see in molecular experiments are often just the result of pushing a complex, uneven swing from an uneven angle.

This discovery gives scientists a new tool: by looking at the shape of the signal (symmetric vs. asymmetric), they can now tell if a molecule is a simple spin or a complex, multi-faceted quantum object.

Technical Summary

1. Problem Statement

Scanning Tunneling Spectroscopy (STS) is a primary tool for investigating low-energy magnetic and vibrational excitations in individual molecules on surfaces. Typically, spin excitations mediated by magnetic anisotropy appear as symmetric step-like features in the differential conductance ($dI/dV$), while zero-bias peaks are often attributed to Kondo screening or Fano resonances.

However, experimental STS spectra frequently reveal pronounced asymmetric line shapes (peaks or dips appearing only on one side of the bias voltage) near the Fermi level. The physical origin of these asymmetries in correlated molecular systems remains debated. Existing theories often rely on single-reference models (single Slater determinants) or assume symmetric coupling between the STM tip, the molecule, and the substrate. This paper addresses the gap in understanding how multireference electronic character (where the ground state is a superposition of multiple configurations) combined with asymmetric tip/molecular orbital coupling generates these asymmetric features.

2. Methodology

The authors develop a theoretical framework extending the cotunneling formalism to general multireference systems.

• Hamiltonian Model: The system is modeled as a quantum dot (the molecule) coupled to metallic leads (Tip and Substrate). The molecular Hamiltonian ($H_{dot}$) includes hopping terms and Coulomb interactions, solved via Exact Diagonalization (ED) to obtain many-body eigenstates (multiplets) represented as expansions of Slater determinants.
• Cotunneling Formalism: The inelastic current is calculated using second-order perturbation theory. The current expression accounts for virtual charging processes ($N \to N \pm 1$) that mediate transitions between neutral many-body states ($J \to K$).
  • The transition rates depend on the matrix elements $\Gamma$, which involve the overlap between the initial/final many-body states and the intermediate charged states.
  • Crucially, the model explicitly includes the tunneling amplitudes ($V_{\alpha i}$) between the leads and specific molecular orbitals ($i$).
• Models Investigated:
  1. Extended Hubbard Dimer: A simplified two-orbital model (HOMO/LUMO) tuned to vary the "diradical character" (from a perfect diradical to a closed-shell system) by adjusting orbital energy differences ($\epsilon_1, \epsilon_2$).
  2. Metal Porphyrin (Co-Por): A realistic, strongly correlated molecule (Cobalt cyano-porphyrin) adsorbed on Au(111). The authors used ab initio CASCI calculations to identify the active space, reducing it to a tetramer model (four $d$-orbitals of the Co center) to make the many-body problem computationally tractable while preserving the essential multireference physics.

3. Key Contributions

• Extension of Cotunneling Theory: The authors generalize the standard cotunneling theory to handle multireference states, where the wavefunction is a linear combination of multiple Slater determinants with different weights.
• Mechanism for Asymmetry: They identify a new microscopic mechanism for asymmetric STS line shapes: the interplay between multireference character and orbital-dependent coupling.
  • In multireference systems, the ground and excited states are superpositions of configurations.
  • If the STM tip couples asymmetrically to the specific orbitals involved in these configurations (e.g., coupling strongly to HOMO but weakly to LUMO), the accessibility of virtual charged states ($N+1$ or $N-1$) becomes biased.
  • This bias suppresses either the electron addition or removal channel, leading to a signal that appears only at positive or negative bias.
• Distinction from Anisotropy: The paper demonstrates that while magnetic anisotropy (spin-orbit coupling) typically produces symmetric features, the presence of low-lying multiplet excitations in multireference systems is the primary driver of strong asymmetry.

4. Key Results

• Hubbard Dimer Simulations:
  • For a perfect diradical (equal weight of configurations), the $dI/dV$ signal remains symmetric regardless of tip coupling ratios.
  • As radicality is quenched (approaching a closed-shell system), the signal becomes increasingly asymmetric.
  • In the limit of a closed-shell system with highly asymmetric coupling (e.g., tip couples only to HOMO), the signal vanishes on one side of the bias (e.g., only positive bias shows a peak) because the virtual transition to the $N+1$ state (requiring LUMO occupation) is blocked, while the $N-1$ state remains accessible.
• Triplet Ground States:
  • Similar asymmetry is observed for triplet ground states when transitions occur between multiplets of different total spin ($S$).
  • However, if magnetic anisotropy splits the triplet states (transitions within the same $S$ multiplet), the features revert to being symmetric, regardless of coupling asymmetry. This confirms that the asymmetry is a signature of multiplet transitions involving different spin configurations.
• Cobalt Porphyrin (Co-Por) Application:
  • The Co-Por molecule has a high-spin ground state with five unpaired electrons spatially separated between the Co center ($d$-orbitals) and the ligand ($\pi$-orbitals).
  • The first excitation involves a transfer of an electron between specific $d$-orbitals ($d_{xy} \to d_{yz}$).
  • Due to the different spatial extension of these $d$-orbitals along the $z$-axis, the tip coupling is naturally asymmetric.
  • Simulations show that this realistic setup produces a strongly asymmetric $dI/dV$ line shape (a peak at negative bias and a vanishing signal at positive bias), matching experimental observations of asymmetric features in similar systems.

5. Significance

• New Interpretation Framework: This work provides an alternative explanation for asymmetric STS features that does not rely on Kondo physics or Fano resonances. It suggests that such asymmetries are a direct fingerprint of multireference character and spatially separated radical sites.
• Quantification of Radicality: The degree of asymmetry in the STS spectrum can be used as a probe to quantify the "diradical character" of a molecule. By scanning the tip position (changing coupling ratios), one can map the radical nature of the system.
• Design of Molecular Spintronics: Understanding how orbital-specific coupling influences inelastic spectra is crucial for designing molecular spintronic devices where specific spin states need to be addressed or read out.
• Theoretical Advancement: The methodology bridges the gap between high-level quantum chemistry (multireference wavefunctions) and transport spectroscopy, offering a more accurate tool for interpreting complex STS data of correlated molecules.

In summary, the paper establishes that asymmetric line shapes in inelastic STS are not merely artifacts but are fundamental signatures of multireference electronic structures interacting with spatially non-uniform tip couplings. This insight refines the theoretical toolkit for analyzing quantum materials on surfaces.