1. Theory

Definition: A scanning tunnelling microscope brings a sharp metal tip within a few angstroms of a conducting surface and measures the quantum tunnelling current under a small bias while scanning (Sinica slides; Ervasti et al.).
Exponential sensitivity: The current decays as $e^{- 2 κ z}$ , about an order of magnitude per angstrom, giving sub-angstrom vertical sensitivity (Sinica slides; Ervasti et al.).
Electronic contrast: In the Tersoff-Hamann picture the current follows the sample local density of states at the tip position, so images mix geometry and electronic structure (Ervasti et al.).
Spectroscopy: Scanning tunnelling spectroscopy (STS) measures dI/dV, proportional to the LDOS at energy eV (Ervasti et al.; Oldenkotte et al.).

2. Working Principle

Feedback: In constant-current mode a piezo scanner adjusts tip height to keep the current at a setpoint, typically 0.01 to 50 $nA$ (Sinica slides).
Lock-in spectroscopy: A small bias modulation, about 1 $kHz$ , is added and the first-harmonic current gives dI/dV; height modulation gives the barrier height (Sinica slides; Oldenkotte et al.).
Isolation: Spring, eddy-current and pneumatic stages suppress building vibration (Sinica slides).
Environment: Ultrahigh vacuum and low temperature, such as 77 $K$ , give clean surfaces and better energy resolution (Oldenkotte et al.; background knowledge).

3. Applications

Atomic-scale surface structure: Reconstructions such as Si(111)-(7x7) and the Au(111) herringbone (Sinica slides; Oldenkotte et al.).
Semiconductor spectroscopy: Band edges, surface states and dopant states, with care for tip-induced band bending (background knowledge, Feenstra et al.).
2-D materials and molecules: LDOS of graphene, molecular orbitals and many-body states (Ervasti et al.; background knowledge).
Work-function mapping: Local barrier-height maps alongside dI/dV (Oldenkotte et al.).

Tunnelling Physics

Name	Formula	Related Formulae	Variables/Constants
Tunnelling current and decay constant	$I \propto \frac{V}{d} exp (- A ϕ^{1/2} d), A = 1.025 e V^{- 1/2} \overset{˚}{A}^{- 1}$	$I \propto e^{- 2 κ z}, κ = \frac{2 m ϕ}{ℏ}$	$I : t u nn e l l in g c u r r e n t$ $V : bia s, d or z : g a p$ $ϕ : ba r r i er h e i g h t$ $κ : in v er se d ec a y l e n g t h; S ini c a s l i d es : f i r s t f or m, ϕ ab o u t 4 t o 5 eV, ab o u t t e n f o l d c u r r e n t p er 1 \overset{˚}{A}; E r v a s t i e t a l . : e x p o n e n t ia l f or m (p r in t e d a s 2 m ϕ /ℏ), w or k f u n c t i o n s a r o u n d 5 eV, o n e or d er o f ma g ni t u d e p er 1 \overset{˚}{A}; O l d e nk o tt e e t a l . : ρ (ε, V, z) \sim ρ (ε, V, 0) exp (- 2 κ (V) z) (see t h e Q / A)$ $(co m p u t e d : ϕ = 4.5 eV g i v es κ = 1.09 \overset{˚}{A}^{- 1} an d a f a c t or o f 8.8 p er \overset{˚}{A}; 5 eV g i v es 1.15 \overset{˚}{A}^{- 1} an d 9.9)$
Bardeen and Tersoff-Hamann current	$I = \frac{2 π e}{ℏ} \sum_{μ, ν} f (E_{μ}) [1 - f (E_{ν} + e V)] ∣ M_{μν} ∣^{2} δ (E_{μ} - E_{ν})$	$M_{μν} \propto ψ_{ν} (r_{0}) \Rightarrow I \propto \int_{0}^{e V} ρ_{s} (r_{0}, E_{F} - e V + ϵ) ρ_{t} T d ϵ$	$f : F er mi f u n c t i o n$ $M : t u nn e l l in g ma t r i x e l e m e n t, ψ_{ν} : s am pl e w a v e f u n c t i o n$ $r_{0} : t i p ce n t r e$ $ρ_{s}, ρ_{t} : s am pl e an d t i p D O S$ $T : t r an s mi ss i o n; E r v a s t i e t a l . : B a r d ee n e x p r ess i o n (p r in t e d w i t h ℏ^{2}, h er e ℏ f or d im e n s i o na l co n s i s t e n cy, m y r e a d in g) an d s - w a v e T er so f f - H amann m o d e l; S ini c a s l i d es : f e a t u r e l ess t i p D O S a ss u m e d$ $(m y r e a d in g : co n s t an t - c u r r e n t t o p o g r a p h y i s a co n t o u r o f co n s t an t in t e g r a t e d L D O S, n o t o f a t o m p os i t i o n s)$
Apparent barrier height	$\frac{d I / d z}{I} \approx - 2 κ$	$ϕ_{a pp} = \frac{ℏ ^{2}}{8 m} (\frac{d l n I}{d z})^{2}$	$ϕ_{a pp} : a pp a r e n t ba r r i er h e i g h t; O l d e nk o tt e e t a l . : d I / d z \approx - 2 κ I, z m o d u l a t i o n o f 0.27 \overset{˚}{A}, ba r r i er hi g h er o n h c p t han f cc an d hi g h es t o n b r i d g e r e g i o n s, κ n e a r l y in d e p e n d e n t o f bia s o n A u (111); ba r r i er - h e i g h t e x p r ess i o n (ba c k g r o u n d k n o w l e d g e, C h e n)$ $(co m p u t e d : 0.27 \overset{˚}{A} m o d u l a t i o n w i t h κ = 1.09 \overset{˚}{A}^{- 1} g i v es 59 p er ce n t c u r r e n t m o d u l a t i o n)$

Spectroscopy

Name	Formula	Related Formulae	Variables/Constants
Differential conductance and LDOS	$\frac{d I}{d V} \propto ρ_{s} (E_{F} + e V)$	$\frac{d I / d V}{I / V} (normalized conductance)$	$E r v a s t i e t a l . : d I / d V p r o p or t i o na l t o s am pl e D O S a t V; O l d e nk o tt e e t a l . : d I / d V p r o p or t i o na l t o L D O S; S ini c a s l i d es : d i f f er e n t ia t i o n y i e l d s D O S, ab o u t 1 kHz bia s d i t h er w i t h l oc k - in; n or ma l i z e d co n d u c t an ce r e m o v in g t h e e x p o n e n t ia l g a p d e p e n d e n ce (ba c k g r o u n d k n o w l e d g e, F ee n s t r a e t a l .)$ $(m y r e a d in g : a t co n s t an t c u r r e n t t h e e x p o n e n t ia l f a c t or mak es r a w d I / d V ma p s mi x L D O S w i t h se tp o in t e f f ec t s)$
Dual modulation lock-in	$V = V_{0} + \overset{v}{^} sin (ω_{V} t + φ_{V}), z = z_{0} + \overset{z}{^} sin (ω_{z} t + φ_{z})$	$f_{c} = \frac{1}{2 π τ}$	$\overset{v}{^}, \overset{z}{^} : m o d u l a t i o n am pl i t u d es$ $ω_{V}, ω_{z} : m o d u l a t i o n f r e q u e n c i es$ $τ : l oc k - in t im e co n s t an t; O l d e nk o tt e e t a l . : 20 mV an d 0.27 \overset{˚}{A}, 2247 an d 2929 Hz (F i g . 1), 2117 an d 2643 Hz (F i g s . 2 an d 3), τ = 1 m/s an d 24 d B p er oc t a v e, se tp o in t s - 50 an d - 100 mV a t 400 pA, 77 K, s u bnan o m e t r e r eso l u t i o n in b o t h ma p s$ $(co m p u t e d : τ = 1 m/s g i v es 159 Hz, b e l o w t h e 526 t o 682 Hz se p a r a t i o n o f t h e tw o c a r r i er s)$
Energy resolution	$Δ E_{t h} \approx 3.5 k_{B} T$	$Δ E \approx Δ E_{t h}^{2} + Δ E_{m o d}^{2}, Δ E_{m o d} \sim e \overset{v}{^}$	$Δ E_{t h} : t h er ma l b r o a d e nin g o f t h e F er mi e d g e$ $Δ E_{m o d} : m o d u l a t i o n b r o a d e nin g; t h er ma l an d q u a d r a t u r e es t ima t es (ba c k g r o u n d k n o w l e d g e, C h e n); E r v a s t i e t a l . : f ini t e r eso l u t i o n f r o m t e m p er a t u r e, e l ec t r i c a l n o i se an d o t h er b r o a d e nin g; O l d e nk o tt e e t a l . : A u (111) s u r f a ce - s t a t e k ink n e a r - 0.45 V a t 77 K (see t h e Q / A)$ $(co m p u t e d : 3.5 k_{B} T = 23 meV a t 77 K; w i t h 20 mV m o d u l a t i o n, ab o u t 31 meV co mbin e d)$

Instrument and Semiconductor Surfaces

Name	Formula	Related Formulae	Variables/Constants
Piezo scanner and stability	$Δ z = d_{33} V_{p}, Δ x = d_{31} V_{p}$	$\frac{Δ I}{I} = 2 κ δ z$	$d_{33}, d_{31} : p i ez o coe f f i c i e n t s$ $V_{p} : p i ez o v o l t a g e$ $δ z : g a p n o i se; S ini c a s l i d es : d 31 ab o u t - 1 \overset{˚}{A} /V, d 33 ab o u t 3 \overset{˚}{A} /V, g a p h e l d w i t hin 0.01 \overset{˚}{A}, c u r r e n t 0.01 t o 50 nA, t u b e an d t r i p o d sc ann er s$ $(co m p u t e d : 100 V g i v es 300 \overset{˚}{A} o f z t r a v e l; 0.01 \overset{˚}{A} n o i se i s 2.2 p er ce n t c u r r e n t n o i se; 0.01 t o 50 nA s p an s a g a p r an g e o f 3.9 \overset{˚}{A})$
Vibration isolation	$ω_{0} = \frac{k}{m}$	$∣ H (ω) ∣ \approx (\frac{ω _{0}}{ω})^{2} (ω ≫ ω_{0}, light damping)$	$ω_{0} : i so l a t or r eso nan ce$ $H : t r an s mi ss ibi l i t y; S ini c a s l i d es : ω_{0} r e l a t i o n, s p r in g s w i t h e dd y - c u r r e n t d am p in g, s t a c k e d pl a t es, p n e u ma t i c t ab l es, b u i l d in g v ib r a t i o n a t 15 t o 20 Hz an d p eo pl e a t 2 t o 4 Hz; t r an s mi ss ibi l i t y (ba c k g r o u n d k n o w l e d g e, C h e n)$ $(a ss u m e d : 2 Hz i so l a t or; co m p u t e d : 20 Hz b u i l d in g v ib r a t i o n a tt e n u a t e d 100 t im es; 2 t o 4 Hz f oo t s t e p s a r e ba r e l y a tt e n u a t e d)$
Semiconductor surfaces and band gaps	$E_{g} \approx e (V_{C B} - V_{V B}) (onsets in d I / d V)$	$tip-induced band bending shifts apparent onsets$	$V_{C B}, V_{V B} : co n d u c t i o n - an d v a l e n ce - ban d o n se t bia ses; ban d - g a p S T S an d t i p - in d u ce d ban d b e n d in g (ba c k g r o u n d k n o w l e d g e, F ee n s t r a e t a l .); S ini c a s l i d es : S i (111) - (7 x 7) a d a t o m s, r es t a t o m s w i t h d an g l in g b o n d s, f a u l t e d an d u n f a u l t e d ha l v es, S T M an d S T S ima g es; O l d e nk o tt e e t a l . : A u (111) (22 b y r oo t 3) h er r in g b o n e w i t h 23 a t o m s o v er 22 an d b r i d g es ab o u t 10 pm hi g h er$ $(co m p u t e d : 23 a t o m s o v er 22 i s a 4.5 p er ce n t co m p r ess i o n)$

Important Questions

Ques: Why is the tunnelling current so sensitive to tip height?

Because it decays exponentially with the gap. The Sinica slides give $I \propto (V / d) exp (- A ϕ^{1/2} d)$ with A = 1.025 per root electronvolt per angstrom and barrier heights of 4 to 5 $eV$ , and Ervasti et al. write $I \propto e^{- 2 κ z}$ . For 4.5 $eV$ , $κ = 2 m ϕ /ℏ$ = 1.09 $\overset{˚}{A}^{- 1}$ , so the current changes by a factor of 8.8 per angstrom; at 5 $eV$ it is 9.9 (computed). Holding the gap within 0.01 $\overset{˚}{A}$ , as the slides require, keeps current noise near 2.2 per cent (computed). The setpoint range of 0.01 to 50 $nA$ corresponds to only 3.9 $\overset{˚}{A}$ of gap change (computed). This is why the feedback loop resolves height changes far smaller than an atom. Reference: TIGP Nano Science and Technology Program, Academia Sinica, ‘Scanning Tunneling Microscopy’ lecture slides (2007) (fetch-tool summary; partly garbled); M. M. Ervasti, F. Schulz, P. Liljeroth and A. Harju, ‘Single- and many-particle description of scanning tunneling spectroscopy’, arXiv:1607.00240 (fetch-tool summary).

Ques: Does a 1 angstrom change really change the current tenfold?

Approximately, and only for barriers near 5 $eV$ . Both the Sinica slides and Ervasti et al. state about one order of magnitude per angstrom. From $e^{2 κ Δ z}$ the factor is 9.9 for 5 $eV$ , 8.8 for 4.5 $eV$ and 7.8 for 4 $eV$ (computed), so tenfold is a rounded upper value. Measured apparent barriers are often lower than work functions because of image-potential rounding and tip-sample contact effects, which reduces the factor further (background knowledge, Chen). Oldenkotte et al. find barrier variations even within one surface, higher on hcp than fcc and highest on bridge regions of Au(111). Note also that Ervasti’s printed $κ = 2 m ϕ /ℏ$ is a typesetting slip for $2 m ϕ /ℏ$ (my reading). Reference: TIGP Nano Science and Technology Program, Academia Sinica, ‘Scanning Tunneling Microscopy’ lecture slides (2007) (fetch-tool summary; partly garbled); M. M. Ervasti, F. Schulz, P. Liljeroth and A. Harju, ‘Single- and many-particle description of scanning tunneling spectroscopy’, arXiv:1607.00240 (fetch-tool summary); R. M. Feenstra, J. A. Stroscio and A. P. Fein, ‘Tunneling spectroscopy of the Si(111)2x1 surface’, Surf. Sci. 181, 295 (1987), and C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press 1993 (background knowledge; not opened); V. J. S. Oldenkotte, F. J. Witmans, M. H. Siekman, P. L. de Boeij, K. Sotthewes, C. Castenmiller, M. D. Ackermann, J. M. Sturm and H. J. W. Zandvliet, ‘Dual Modulation STM: simultaneous high-resolution mapping of the differential conductivity and local tunnel barrier height demonstrated on Au(111)’, J. Appl. Phys. 129, 225301 (2021) (fetch-tool summary).

Ques: What does an STM image actually show?

A contour of constant integrated local density of states, not atomic positions directly. In the Bardeen formalism the current sums tunnelling matrix elements between tip and sample states weighted by Fermi factors, and Tersoff and Hamann’s s-wave tip makes $M \propto ψ_{ν} (r_{0})$ , so the current follows the sample LDOS at the tip centre integrated from $E_{F}$ to $E_{F} - e V$ (Ervasti et al.). The Sinica slides likewise assume a featureless tip DOS. Changing bias therefore changes which states are imaged, so filled and empty states of Si(111)-(7x7) look different, as the slides show (background knowledge, Chen). Bridge regions of the Au(111) herringbone appear about 10 $pm$ higher in Oldenkotte’s topography. Reference: M. M. Ervasti, F. Schulz, P. Liljeroth and A. Harju, ‘Single- and many-particle description of scanning tunneling spectroscopy’, arXiv:1607.00240 (fetch-tool summary); TIGP Nano Science and Technology Program, Academia Sinica, ‘Scanning Tunneling Microscopy’ lecture slides (2007) (fetch-tool summary; partly garbled); R. M. Feenstra, J. A. Stroscio and A. P. Fein, ‘Tunneling spectroscopy of the Si(111)2x1 surface’, Surf. Sci. 181, 295 (1987), and C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press 1993 (background knowledge; not opened); V. J. S. Oldenkotte, F. J. Witmans, M. H. Siekman, P. L. de Boeij, K. Sotthewes, C. Castenmiller, M. D. Ackermann, J. M. Sturm and H. J. W. Zandvliet, ‘Dual Modulation STM: simultaneous high-resolution mapping of the differential conductivity and local tunnel barrier height demonstrated on Au(111)’, J. Appl. Phys. 129, 225301 (2021) (fetch-tool summary).

Ques: How is dI/dV measured and what limits its energy resolution?

With a lock-in amplifier and small bias modulation; resolution is set by temperature and modulation. Ervasti et al. state $d I / d V \propto ρ_{s} (E_{F} + e V)$ , and the Sinica slides describe a bias dither of about 1 $kHz$ with lock-in detection. Oldenkotte et al. used 20 $mV$ modulation at about 2.2 $kHz$ with a 1 $m/s$ time constant, a 159 $Hz$ cut-off (computed). Thermal broadening is about $3.5 k_{B} T$ , 23 $meV$ at 77 $K$ , and combined with a broadening of order the 20 $mV$ modulation it gives about 31 $meV$ (background knowledge, Chen; computed). Ervasti et al. add electrical noise to the broadening sources. Normalizing as $(d I / d V) / (I / V)$ reduces the exponential gap dependence (background knowledge, Feenstra et al.). Reference: M. M. Ervasti, F. Schulz, P. Liljeroth and A. Harju, ‘Single- and many-particle description of scanning tunneling spectroscopy’, arXiv:1607.00240 (fetch-tool summary); TIGP Nano Science and Technology Program, Academia Sinica, ‘Scanning Tunneling Microscopy’ lecture slides (2007) (fetch-tool summary; partly garbled); V. J. S. Oldenkotte, F. J. Witmans, M. H. Siekman, P. L. de Boeij, K. Sotthewes, C. Castenmiller, M. D. Ackermann, J. M. Sturm and H. J. W. Zandvliet, ‘Dual Modulation STM: simultaneous high-resolution mapping of the differential conductivity and local tunnel barrier height demonstrated on Au(111)’, J. Appl. Phys. 129, 225301 (2021) (fetch-tool summary); R. M. Feenstra, J. A. Stroscio and A. P. Fein, ‘Tunneling spectroscopy of the Si(111)2x1 surface’, Surf. Sci. 181, 295 (1987), and C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press 1993 (background knowledge; not opened).

Ques: Is the Au(111) surface-state onset at -0.45 V or -0.5 V?

Oldenkotte et al. place the kink in dI/dV attributed to the Au(111) two-dimensional surface state at about -0.45 V at 77 $K$ , whereas values near -0.48 to -0.5 $eV$ are widely quoted from low-temperature work (background knowledge, Chen). The 30 to 50 $meV$ difference is comparable to their energy resolution of about 31 $meV$ from 23 $meV$ thermal broadening and 20 $mV$ modulation (computed), and a kink at the band bottom is read where the conductance rises, which sits above the true onset once broadened (my reading). The lower value near -0.48 $eV$ is therefore more likely the true band bottom, and -0.45 V the broadened kink position (my reading). Tip state and herringbone stacking can also shift it. Reference: V. J. S. Oldenkotte, F. J. Witmans, M. H. Siekman, P. L. de Boeij, K. Sotthewes, C. Castenmiller, M. D. Ackermann, J. M. Sturm and H. J. W. Zandvliet, ‘Dual Modulation STM: simultaneous high-resolution mapping of the differential conductivity and local tunnel barrier height demonstrated on Au(111)’, J. Appl. Phys. 129, 225301 (2021) (fetch-tool summary); R. M. Feenstra, J. A. Stroscio and A. P. Fein, ‘Tunneling spectroscopy of the Si(111)2x1 surface’, Surf. Sci. 181, 295 (1987), and C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press 1993 (background knowledge; not opened).

Ques: What does dual-modulation STM add?

Simultaneous maps of dI/dV and local barrier height. Oldenkotte et al. modulate bias and height at different frequencies, $V = V_{0} + \overset{v}{^} sin (ω_{V} t)$ and $z = z_{0} + \overset{z}{^} sin (ω_{z} t)$ , with 20 $mV$ at 2247 $Hz$ and 0.27 $\overset{˚}{A}$ at 2929 $Hz$ , and read each with a lock-in. Since $(d I / d z) / I \approx - 2 κ$ , the height channel gives $κ$ and the barrier height. A 0.27 $\overset{˚}{A}$ modulation produces about 59 per cent current modulation for $κ$ = 1.09 $\overset{˚}{A}^{- 1}$ (computed). The 159 $Hz$ lock-in bandwidth is well below the 682 $Hz$ separation of the carriers, so the channels do not mix (computed). They found $κ$ nearly bias independent and resolved hcp, fcc and bridge regions with subnanometre resolution at 400 $pA$ and 77 $K$ . Reference: V. J. S. Oldenkotte, F. J. Witmans, M. H. Siekman, P. L. de Boeij, K. Sotthewes, C. Castenmiller, M. D. Ackermann, J. M. Sturm and H. J. W. Zandvliet, ‘Dual Modulation STM: simultaneous high-resolution mapping of the differential conductivity and local tunnel barrier height demonstrated on Au(111)’, J. Appl. Phys. 129, 225301 (2021) (fetch-tool summary).

Ques: What complicates STS on semiconductor surfaces?

Tip-induced band bending and low carrier density near the gap. On a semiconductor, part of the bias drops inside the sample because screening is weak, so band onsets in dI/dV appear at larger biases than the true edges and the apparent gap exceeds the bulk value (background knowledge, Feenstra et al.). The gap is read from valence- and conduction-band onsets, $E_{g} \approx e (V_{C B} - V_{V B})$ , after correcting for this shift. Normalized conductance $(d I / d V) / (I / V)$ helps compare data across the gap where current is small (Feenstra et al.). Surface states can also pin the Fermi level, as on Si(111)-(7x7) with its adatom and rest-atom dangling bonds shown in the Sinica slides. Constant-height or variable-gap spectroscopy extends dynamic range (background knowledge, Chen). Reference: R. M. Feenstra, J. A. Stroscio and A. P. Fein, ‘Tunneling spectroscopy of the Si(111)2x1 surface’, Surf. Sci. 181, 295 (1987), and C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press 1993 (background knowledge; not opened); TIGP Nano Science and Technology Program, Academia Sinica, ‘Scanning Tunneling Microscopy’ lecture slides (2007) (fetch-tool summary; partly garbled).

References

TIGP Nano Science and Technology Program, Academia Sinica, “Scanning Tunneling Microscopy,” lecture slides, 2007. Available: https://tigpnano.rcas.sinica.edu.tw/Course/2007_Spring/Class%20Notes/CMNS_STM_20070313.pdf
M. M. Ervasti, F. Schulz, P. Liljeroth and A. Harju, “Single- and many-particle description of scanning tunneling spectroscopy,” preprint arXiv:1607.00240. Available: https://arxiv.org/pdf/1607.00240
R. M. Feenstra, J. A. Stroscio and A. P. Fein, “Tunneling spectroscopy of the Si(111)2x1 surface,” Surface Science, vol. 181, pp. 295-306, 1987; C. J. Chen, Introduction to Scanning Tunneling Microscopy, Oxford University Press, 1993 (cited from background knowledge).
V. J. S. Oldenkotte, F. J. Witmans, M. H. Siekman, P. L. de Boeij, K. Sotthewes, C. Castenmiller, M. D. Ackermann, J. M. Sturm and H. J. W. Zandvliet, “Dual Modulation STM: simultaneous high-resolution mapping of the differential conductivity and local tunnel barrier height demonstrated on Au(111),” Journal of Applied Physics, vol. 129, 225301, 2021. Available: https://arxiv.org/pdf/2107.14612

Quartz 5

Explorer

Scanning Tunneling Microscope (STM) and Spectroscopy (STS)