Electron–Hole Puddles

Updated 11 November 2025

Electron–hole puddles are regions of localized n-type or p-type carriers arising from long-range disorder and imperfect screening in low-density electronic systems.
Their characteristic size, amplitude, and scaling properties in materials like graphene and topological insulators are set by impurity distributions and non-Gaussian potential fluctuations.
Experimental methods such as STM/STS, Hall measurements, and optical spectroscopy reveal that puddles crucially influence transport phenomena, quantum Hall effects, and device performance.

Electron–hole puddles are spatially inhomogeneous regions of locally excess electron-like (n-type) or hole-like (p-type) carriers that appear in low-dimensional electron systems due to long-range potential fluctuations. These inhomogeneities emerge prominently near the charge neutrality point in materials such as graphene, graphite multilayers, two-dimensional semimetals, and fully or nearly compensated bulk semiconductors and topological insulators. Electron–hole puddles fundamentally influence low-density transport, screening, localization, and collective quantum phenomena across a wide range of platforms.

1. Fundamental Origin and Physical Mechanisms

Electron–hole puddles originate from the interplay between spatial disorder and imperfect screening, which create local fluctuations in the electrostatic potential landscape. Key mechanisms include:

Random charged impurities: Trapped charges in substrates or in the host material act as long-range Coulomb scatterers (Li et al., 2011, Adam et al., 2011). In graphene on SiO₂, correlated disorder engenders potential fluctuations with rms amplitude $s \sim 20$ meV and spatial correlation length set by impurity density and depth.
Remote donor/acceptor statistics: In compensated semiconductors or topological insulators, the random distribution of amphoteric donors and acceptors produces net charge fluctuations within volume $R^3$ of amplitude $\delta Q \sim e\sqrt{N_{\mathrm{def}} R^3}$ , and corresponding potential fluctuations $\delta V(R) \sim e^2 N_{\mathrm{def}}^{1/2} R^{1/2}/\varepsilon$ (Borgwardt et al., 2015). Once these reach $\sim \Delta/2$ (half the gap), the local band edge crosses the Fermi level, resulting in n- or p-type puddles.
Surface/corrugation effects: Inhomogeneous local curvature, strain, or intercalated molecular species can generate smooth scalar and vector potentials (Martin et al., 2013, Gibertini et al., 2011). For instance, graphene nanoripples, evidenced by STM/STS, correlate with density variations at nanometer scales.
Potential asymmetry and local statistical properties: Nonzero third cumulants (skewness) of the disorder potential, such as in graphene with particle–hole–asymmetric potential, shift the spatial balance of n- and p-type regions and the position of minimal conductivity (Hering et al., 2015).
Compensation and band-edge bending: In three-dimensional topological insulators, even full donor/acceptor compensation cannot globally neutralize local Coulomb potential excursions; metallic puddles are thus inevitable (Bömerich et al., 2017).

These mechanisms operate in both two-dimensional and three-dimensional systems, differing quantitatively due to the nature of screening and the electronic density of states.

2. Statistical Characterization, Scaling, and Geometric Properties

The characteristic features of electron–hole puddles—size, amplitude, spatial correlations, and statistical exponents—are determined by the competition between disorder, electronic interactions, and screening:

Length scale and amplitude: In graphene on SiO₂, puddles typically have diameters $\xi \sim 3$ –$40$ nm; monolayer graphene exhibits gate- and impurity-density-dependent scaling, while bilayer graphene maintains $\xi \approx 3.5$ nm nearly independent of doping, due to the finite DOS at the charge neutrality point (Adam et al., 2011).
RMS amplitude: $s \approx 20$ meV for monolayer graphene on SiO₂; in BLG or on hBN, typical fluctuation strengths are an order of magnitude lower (Tuan et al., 2016, Adam et al., 2011).
Statistical distribution: The distribution $P(n)$ of carrier density is generically non-Gaussian, exhibiting exponential tails in $\sqrt{|n|}$ and strong skewness, particularly near charge neutrality (Najafi et al., 2016, Najafi et al., 2016).
Fractal and scaling properties: At zero chemical potential, the EHP landscape realizes a self-similar random surface, with local roughness exponent $\alpha_l \approx 0.35$ for $n(\mathbf{r})$ , fractal dimension for contours $D_f \approx 1.39$ , and loop correlation exponent $x_l \approx 0.5$ (Najafi et al., 2016, Najafi et al., 2016). These exponents satisfy Kondev–Henley hyperscaling relations, even for non-Gaussian underlying fields.
Conformal invariance: Analysis of zero-density contours via Schramm–Loewner evolution establishes that EHP boundaries in ungated graphene are conformally invariant (SLE $_\kappa$ with $\kappa=1.8\pm0.2$ ), defining a novel universality class (Najafi et al., 2016).

Table 1: Puddle Correlation Lengths in Common Systems

System	Correlation Length $\xi$	RMS Amplitude $s/\Delta$
Monolayer graphene / SiO₂	$3$–$40$ nm	$\sim 20$ meV
Bilayer graphene / SiO₂	$\sim 3.5$ nm	$\sim 20$ –$35$ meV
Graphene / hBN	$\sim 3.5$ nm	$\sim 11$ meV
Compensated TI, $N\sim 10^{19}$ cm $^{-3}$	$20$–$30$ nm	$E_c\sim 3$ meV

The typical length and amplitude sets the scale for transport inhomogeneity and thus the mesoscopic behavior of the system.

3. Experimental Detection and Quantitative Imaging

Direct and indirect manifestations of electron–hole puddles have been attained via a range of experimental techniques:

STM/STS mapping: Nanoscale imaging of density fluctuations, using the position of the Dirac point ( $E_D(\mathbf{r})$ ) as a proxy for local carrier density. Electron–hole puddles with $L_p \sim 5$ –$20$ nm have been imaged on various substrates, with clear correlation to topographic ripples or impurity locations (Martin et al., 2013, Adam et al., 2011).
Single-electron transistor (SET) and Coulomb blockade: Spatially resolved charge density measurements establish puddle amplitudes $\sim 10^{11}$ – $10^{12}$ cm $^{-2}$ (Adam et al., 2011).
Magnetotransport and Hall effect: Analysis of temperature-dependent Hall resistivity, minimum conductivity, and plateau transitions quantifies the potential fluctuation amplitude $s$ and residual puddle carrier density $n_0$ (Kurganova et al., 2013).
Optical spectroscopy: Drude-like absorption at low T in compensated TIs reveals puddle-induced metallic regions even when dc transport is insulating. Collapse of this spectral weight at a temperature scale set by $E_c$ confirms non-linear screening physics (Borgwardt et al., 2015).
Conductance fingerprinting: In p–n graphene devices under quantizing fields, the disruption of snake-state transport allows the extraction of puddle size and position via mesoscopic oscillations (Milovanovic et al., 2016).

Measurement of the gate-voltage range over which both carrier types coexist can be linked to characteristic puddle density and spatial inhomogeneity (Poumirol et al., 2010).

4. Puddles and Electronic Transport Phenomena

Electron–hole puddles dramatically affect transport properties at low carrier density in both 2D and 3D systems:

Two-component transport: Conductivity at/near neutrality is governed by the parallel sum of metallic (diffusive) and activated (hopping across potential hills) channels (Li et al., 2011). This model explains insulating, nonmonotonic, and minimum-conductivity phenomena.
Quantum Hall regime: The coexistence of localized electrons and holes near the charge neutrality point results in the absence of a diverging resistivity (even at 57 T in graphene), persistence of $R_{xy}=0$ , and enhanced resistance fluctuations due to finite-size puddles (Poumirol et al., 2010). Thermal smearing and localization produce distinctive Hall overshoots (Kurganova et al., 2013).
Hopping and percolation: In compensated semiconductors and topological insulators, charge transport occurs via variable-range (Efros–Shklovskii) hopping between puddles, yielding activated and nondiverging resistivities at low T (Rischau et al., 2016, Bömerich et al., 2017).
Spin dynamics: In bilayer graphene, substrate-induced puddles alter the energy dependence of the spin lifetime through modulation of momentum relaxation times. At low energy, puddle-induced scattering strengthens Dyakonov–Perel spin relaxation, inverting the canonical M-shaped $\tau_s(E)$ profile (Tuan et al., 2016).
Recombination: Under optical pumping, interband tunneling between adjacent puddles creates a fast, nonmonotonic recombination channel leading to potential hysteresis in optoelectronic response (Ryzhii et al., 2011).

Residual conductivity, minimum quantum capacitance, and broadened Dirac peaks in graphene field-effect transistors are all traceable to the inhomogeneous density of puddles, with precise scaling to interface trap capacitance and impurity disorder strength (Zebrev et al., 2010).

5. Theoretical Modeling and Universality

Multiple theoretical frameworks have been developed to capture puddle formation and their influence on observable properties:

Thomas–Fermi–Dirac (TFD) theory: Semi-classical minimization of an energy functional incorporating local kinetic, Hartree, exchange-correlation, and disorder terms yields a highly inhomogeneous, scale-invariant carrier landscape at charge neutrality (Najafi et al., 2016, Najafi et al., 2016).
Random potential statistics: Mapping to Gaussian or non-Gaussian disorder distributions, with specified second (variance) and third (skewness) moments, determines the residual and imbalance densities (Hering et al., 2015).
Quantum transport and effective-medium theory: Two-component and resistor-network models analytically and numerically reproduce low-T resistivity upturns, non-divergence at zero temperature, and the percolative nature of EHP transport for both weak and strong disorder (Knap et al., 2014).
Scaling laws and asymptotics: In 3D compensated systems, puddle separation $l_p$ scales as $(\Delta/E_c)^2$ or, more accurately, $l_p \sim (\Delta/E_c)^2/\ln(\Delta/E_c)$ for large band gaps (Bömerich et al., 2017).

In all these approaches, the dimensionality of the system, nature of the band structure, and screening properties play crucial roles in determining the size, density, and transport relevance of puddles.

6. Broader Significance, Material Dependencies, and Universal Features

Electron–hole puddles are a generic consequence of potential fluctuations in low-density electron systems, manifesting regardless of the details of disorder source:

Materials dependence: Monolayer and bilayer graphene differ quantitatively in screening and puddle size persistence; trilayer graphene and TIs show related phenomena but with modified length and density scales.
Universality: The percolation, scaling, and transport phenomena arising from puddle inhomogeneity—a two-component fluid, activated plus diffusive transport, fractal and conformal statistical properties—appear consistently across carbon allotropes, compensated semiconductors, and topological insulators (Borgwardt et al., 2015, Bömerich et al., 2017, Rischau et al., 2016).
Interpretation caution: The presence of background conduction from bulk puddles in TIs challenges the identification of surface-state transport unless bulk insulating behavior can be assured via defect minimization or dimensional suppression (film thickness below characteristic puddle separation) (Bömerich et al., 2017).
Noncharged impurity mechanisms: Puddles may arise even in the absence of charged impurities, for instance through corrugation-induced strain fields, which produce density inhomogeneity at scales exceeding those attributable to atomic-scale disorder (Gibertini et al., 2011).
Measurement and device engineering: Extracting puddle parameters via transport, STM/STS, and optical spectroscopy provides a quantitative avenue for engineering cleaner electronic devices, maximizing mobility, and controlling spin and optoelectronic function.

Electron–hole puddles thus represent a central paradigm for understanding and engineering the mesoscopic regime of low-density electronic transport in both two- and three-dimensional semimetals and semiconductors.