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Band offset
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Band offset describes the relative alignment of the energy bands at a semiconductor heterojunction.
Introduction
[edit]At semiconductor heterojunctions, energy bands of two different materials come together, leading to an interaction. Both band structures are positioned discontinuously from each other, causing them to align close to the interface. This is done to ensure that the Fermi energy level stays continuous throughout the two semiconductors. This alignment is caused by the discontinuous band structures of the semiconductors when compared to each other and the interaction of the two surfaces at the interface. This relative alignment of the energy bands at such semiconductor heterojunctions is called the Band offset.
The band offsets can be determined by both intrinsic properties, that is, determined by properties of the bulk materials, as well as non-intrinsic properties, namely, specific properties of the interface. Depending on the type of the interface, the offsets can be very accurately considered intrinsic, or be able to be modified by manipulating the interfacial structure.[1] Isovalent heterojunctions are generally insensitive to manipulation of the interfacial structure, whilst heterovalent heterojunctions can be influenced in their band offsets by the geometry, the orientation, and the bonds of the interface and the charge transfer between the heterovalent bonds.[2] The band offsets, especially those at heterovalent heterojunctions depend significantly on the distribution of interface charge.
The band offsets are determined by two kinds of factors for the interface, the band discontinuities and the built-in potential. These discontinuities are caused by the difference in band gaps of the semiconductors and are distributed between two band discontinuities, the valence-band discontinuity, and the conduction-band discontinuity. The built-in potential is caused by the bands which bend close at the interface due to a charge imbalance between the two semiconductors, and can be described by Poisson's equation.
Semiconductor types
[edit]
The behaviour of semiconductor heterojunctions depend on the alignment of the energy bands at the interface and thus on the band offsets. The interfaces of such heterojunctions can be categorized in three types: straddling gap (referred to as type I), staggered gap (type II), and broken gap (type III).
These representations do not take into account the band bending, which is a reasonable assumption if you only look at the interface itself, as band bending exerts its influence on a length scale of generally hundreds of angström. For a more accurate picture of the situation at hand, the inclusion of band bending is important.
Experimental methods
[edit]Two kinds of experimental techniques are used to describe band offsets. The first is an older technique, the first technique to probe the heterojunction built-in potential and band discontinuities. This methods are generally called transport methods. These methods consist of two classes, either capacitance-voltage (C-V) or current-voltage (I-V) techniques. These older techniques were used to extract the built-in potential by assuming a square-root dependence for the capacitance C on bi - qV, with bi the built-in potential, q the electron charge, and V the applied voltage. If band extrema away from the interface, as well as the distance between the Fermi level, are known parameters, known a priori from bulk doping, it becomes possible to obtain the conduction band offset and the valence band offset. This square root dependence corresponds to an ideally abrupt transition at the interface and it may or may not be a good approximation of the real junction behaviour.[1]
The second kind of technique consists of optical methods. Photon absorption is used effectively as the conduction band and valence band discontinuities define quantum wells for the electrons and the holes. Optical techniques can be used to probe the direct transitions between sub-bands within the quantum wells, and with a few parameters known, such as the geometry of the structure and the effective mass, the transition energy measured experimentally can be used to probe the well depth. Band offset values are usually estimated using the optical response as a function of certain geometrical parameters or the intensity of an applied magnetic field. Light scattering could also be used to determine the size of the well depth.
Alignment
[edit]Prediction of the band alignment is at face value dependent on the heterojunction type, as well as whether or not the heterojunction in question is heterovalent or isovalent. However, quantifying this alignment proved a difficult task for a long time. Anderson's rule is used to construct energy band diagrams at heterojunctions between two semiconductors. It states that during the construction of an energy band diagram, the vacuum levels of the semiconductors on either side of the heterojunction should be equal.[1]
Anderson's rule states that when we construct the heterojunction, we need to have both semiconductors on an equal vacuum energy level. This ensures that the energy bands of both the semiconductors are being held to the same reference point, from which ΔEc and ΔEv, the conduction band offset and valence band offset can be calculated. By having the same reference point for both semiconductors, ΔEc becomes equal to the built-in potential, Vbi = Φ1 - Φ2, and the behaviour of the bands at the interface can be predicted as can be seen at the picture above.
Anderson's rule fails to predict real band offsets. This is primarily due to the fact that Anderson's model implies that the materials are assumed to behave the same as if they were separated by a large vacuum distance, however at these heterojunctions consisting of solids filling the space, there is no vacuum, and the use of the electron affinities at vacuum leads to wrong results. Anderson's rule ignores actual chemical bonding effects that occur on small vacuum separation or non-existent vacuum separation, which leads to wrong predictions about the band offsets.
A better theory for predicting band offsets has been linear-response theory. In this theory, interface dipoles have a significant impact on the lining up of the bands of the semiconductors. These interface dipoles however are not ions, rather they are mathematical constructs based upon the difference of charge density between the bulk and the interface. Linear-response theory is based on first-principles calculations, which are calculations aimed at solving the quantum-mechanical equations, without input from experiment. In this theory, the band offset is the sum of two terms, the first term is intrinsic and depends solely on the bulk properties, the second term, which vanishes for isovalent and abrupt non-polar heterojunctions, depends on the interface geometry, and can easily be calculated once the geometry is known, as well as certain quantities (such as the lattice parameters).
The goal of the model is to attempt to model the difference between the two semiconductors, that is, the difference with respect to a chosen optimal average (whose contribution to the band offset should vanish). An example would be GaAs-AlAs, constructing it from a virtual crystal of Al0.5Ga0.5As, then introducing an interface. After this a perturbation is added to turn the crystal into pure GaAs, whilst on the other side, the perturbation transforms the crystal in pure AlAs. These perturbations are sufficiently small so that they can be handled by linear-response theory and the electrostatic potential lineup across the interface can then be obtained up to the first order from the charge density response to those localized perturbations. Linear response theory works well for semiconductors with similar potentials (such as GaAs-AlAs) as well as dissimilar potentials (such as GaAs-Ge), which was doubted at first. However predictions made by linear response theory coincide exactly with those of self-consistent first principle calculations. If interfaces are polar however, or nonabrupt nonpolar oriented, additional effects must be taken into account. These are additional terms which require simple electrostatics, which is within the linear response approach.
References
[edit]- ^ a b c Franciosi, Alfonse; Van de Walle, Chris G. (1996). "Heterojunction band offset engineering". Surface Science Reports. 25 (1): 1–140. Bibcode:1996SurSR..25....1F. doi:10.1016/0167-5729(95)00008-9.
- ^ Tung, Raymond T.; Kronik, Leeor (2018). "Charge Density and Band Offsets at Heterovalent Semiconductor Interfaces". Advanced Theory and Simulations. 1 1700001. doi:10.1002/adts.201700001. S2CID 103991603.
See also
[edit]Band offset
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Basic Concepts
Definition
In semiconductor physics, band offset refers to the discontinuity in the energy levels of the valence band maximum (ΔE_v) and conduction band minimum (ΔE_c) at the interface between two different semiconductors, forming a heterojunction.[6][7] These discontinuities arise due to differences in the electronic structures of the materials and play a crucial role in determining charge carrier confinement and transport properties in heterostructure devices.[6] A heterojunction is an interface between two dissimilar semiconductors, in contrast to a homojunction, which occurs within a single semiconductor material where the band structure is continuous except for doping-induced changes.[8][9] The valence band offset (VBO), denoted as ΔE_v, represents the difference in valence band maximum positions, while the conduction band offset (CBO), denoted as ΔE_c, represents the difference in conduction band minimum positions across the interface.[6] These offsets are related to the bandgaps of the two semiconductors, E_{g1} and E_{g2}, through the equation \Delta E_c + \Delta E_v = E_{g1} - E_{g2} (assuming the wider-bandgap material is labeled as 1, with E_{g1} > E_{g2}, and offsets taken as positive magnitudes), which follows from the alignment of band edges relative to a common reference such as the vacuum level.[9] The valence band offset can be expressed as \Delta E_v = E_{v2} - E_{v1}, where E_v denotes the energy of the valence band edge relative to the vacuum level for each semiconductor.[7] This formulation assumes no interface dipole effects and aligns the materials based on their intrinsic properties. The concept of band offset was first introduced by R. L. Anderson in 1960 as a fundamental parameter for understanding carrier behavior in heterojunctions, particularly for enabling confinement in devices like transistors and optoelectronics.[10]Band Diagrams at Interfaces
Band diagrams at semiconductor heterojunction interfaces are constructed by plotting the conduction band edge and valence band edge as functions of position along a spatial coordinate perpendicular to the interface. In isolated semiconductors, these bands are flat, but upon forming the heterojunction, discontinuities appear at the interface: the valence band offset and conduction band offset , where subscripts 1 and 2 denote the two materials. These offsets manifest as abrupt steps in the band edges, with the direction and magnitude depending on the relative positions of the bands in the two semiconductors.[11] The construction often begins by aligning the vacuum levels or using electron affinity differences to position the bands before contact.[10] The relationship between the offsets and the bandgaps of the materials is given by
where represents the difference in bandgaps () between the two semiconductors, assuming a type I alignment where both offsets contribute additively to the gap difference. This equation holds under the assumption of conserved charge neutrality far from the interface and is fundamental to predicting carrier confinement in heterostructures.[12]
At equilibrium, the Fermi level aligns continuously across the heterojunction, driving electron and hole diffusion until a built-in potential balances the offsets. This charge redistribution creates space charge regions (depletion or accumulation layers) on either side of the interface, inducing electrostatic band bending where the bands slope linearly or parabolically over distances of tens to hundreds of nanometers, depending on doping levels. The bending reflects the internal electric field $ \mathcal{E} = -\nabla \phi $, with potential governed by Poisson's equation in the space charge region.[13] For instance, in an n-type GaAs/AlGaAs heterojunction, electrons transfer from the narrower-gap GaAs to the wider-gap AlGaAs, forming a depletion region in GaAs and upward band bending there.[3]
Ideal band diagrams employ the flat-band approximation, depicting uniform bands far from the interface with sharp offsets and no bending, which simplifies analysis but neglects charge transfer. In reality, this approximation is limited by interface effects such as surface states—localized energy levels within the bandgap arising from dangling bonds or defects—that can pin the Fermi level and introduce interface dipoles, altering the effective offsets by 0.1–0.5 eV. These states lead to additional band bending even in undoped structures and are particularly pronounced in lattice-mismatched interfaces, as seen in Ge/GaAs heterojunctions where experimental diagrams show smoothed transitions over atomic layers rather than ideal steps.[10]
Types of Band Alignments
Type I (Straddling Gap)
In Type I band alignment, also referred to as straddling gap alignment, the conduction and valence bands of the narrower-bandgap semiconductor are fully nested within the bandgap of the wider-bandgap semiconductor at the heterojunction interface. This arrangement ensures that both the valence band offset and the conduction band offset have the same sign, typically positive when defining offsets from the narrower-gap material to the wider-gap barrier, leading to effective potential wells for both charge carriers.[14][9] The nested band structure confines both electrons and holes to the narrower-bandgap region, promoting strong spatial overlap of their wavefunctions and minimizing leakage into the barrier material. This carrier confinement is particularly advantageous for quantum well designs, where it enhances radiative recombination efficiency and supports the development of low-threshold optoelectronic devices.[14][15] The close proximity of electron and hole wavefunctions in Type I alignments facilitates exciton formation, with the binding energy approximated by the hydrogenic model as
where is the exciton reduced mass, is the free electron mass, and is the relative dielectric constant of the confining material; in quantum-confined structures, this energy can be enhanced due to reduced dimensionality.[16][17]
A classic example is the GaAs/AlGaAs heterostructure, where the narrower-bandgap GaAs layer (bandgap ~1.42 eV) is surrounded by wider-bandgap AlGaAs barriers (bandgap tunable up to ~2.16 eV for AlAs), enabling robust confinement for applications in high-performance lasers and quantum well lasers.[14][9]
Type II (Staggered Gap)
In type II band alignment, also known as staggered gap alignment, the conduction band minimum of one semiconductor material lies above the conduction band minimum of the adjacent material, while the valence band maximum of the first lies below that of the second, resulting in partial overlap of the bandgaps at their heterojunction interface.[18] This staggered arrangement does not fully confine both charge carriers within a single material but instead promotes their distribution across the interface, with the conduction band offset (ΔE_c) and valence band offset (ΔE_v) having opposite signs.[19] The spatial separation of electrons and holes in type II alignment drives electrons toward one material and holes toward the other, facilitating efficient charge separation.[20] This separation reduces the probability of radiative recombination between electrons and holes, as they are localized in different spatial regions, thereby suppressing non-productive carrier losses.[21] However, the indirect nature of the electron-hole overlap across the interface leads to weaker optical transitions, characterized by reduced oscillator strength compared to direct-gap configurations.[22] The effective bandgap in a type II heterostructure, denoted as $ E_g^{II} $, is given by the expression
where $ E_{g1} $ and $ E_{g2} $ are the bandgaps of the two materials, and $ \Delta E_v $ and $ \Delta E_c $ are the respective band offsets.[23] This effective bandgap determines the minimum energy required for interband excitations across the staggered interface and is typically smaller than the individual material bandgaps due to the offset contributions.[24]
A representative example of type II alignment is found in InAs/GaSb heterostructures, where the staggered configuration enables absorption in the mid-infrared regime.[7] These structures are particularly advantageous for long-wavelength photodetectors, as the carrier separation enhances responsivity while the reduced oscillator strength minimizes dark current contributions from unwanted radiative processes.[25]
Type III (Broken Gap)
In type III band alignment, also known as the broken gap configuration, the conduction band edge of one semiconductor lies energetically below the valence band edge of the adjacent semiconductor at the heterojunction interface, resulting in no overlap between the band gaps of the two materials. This misalignment creates a scenario where the valence band maximum of one material exceeds the conduction band minimum of the other, enabling direct spatial overlap of electron and hole states across the interface and facilitating interband transport processes. Such alignments are distinct from overlapping gap types due to the potential barrier that electrons must tunnel through to transition between bands.[26][27] The carrier dynamics in type III heterostructures are dominated by interband tunneling mechanisms, particularly Zener tunneling, where electrons can quantum mechanically tunnel from the valence band of one material directly into the conduction band of the other under an applied electric field. This process leads to unique electrical characteristics, including negative differential resistance (NDR), where the current decreases with increasing voltage beyond a peak value due to the depletion of available tunneling states. The efficiency of this tunneling is highly sensitive to the interface quality and applied bias, making these structures promising for high-speed switching devices.[28][29] A representative example of type III alignment is found in HgTe/CdTe superlattices, where HgTe acts as a zero-gap semimetal paired with the wide-bandgap semiconductor CdTe, resulting in a broken gap that allows tunable band crossing. These superlattices, pioneered through molecular beam epitaxy growth in the 1980s, have been extensively studied for their semimetallic properties and applied in infrared detection owing to the ability to engineer narrow effective gaps via layer thickness control. The historical development highlighted the potential of such systems for optoelectronic applications, building on early theoretical predictions of band discontinuities in semimetal-semiconductor interfaces.[7][30][31] The probability of interband tunneling in these broken gap systems can be estimated using the Wentzel-Kramers-Brillouin (WKB) approximation:
where , is the effective carrier mass, is the local potential profile across the junction, is the carrier energy, and the integral spans the classically forbidden region from turning points to . This semiclassical expression underscores the exponential sensitivity of tunneling to barrier thickness and height, guiding the design of low-power tunneling devices.[32]
Theoretical Determination
Electron Affinity Rule
The electron affinity rule, proposed by R. L. Anderson in 1960, serves as a foundational theoretical model for estimating band offsets at abrupt semiconductor heterojunctions. This rule posits that the vacuum levels of the two isolated semiconductors align upon contact, allowing the conduction band offset to be directly determined by the difference in their electron affinities : . The valence band offset is then derived from the band gap difference : . The model relies on key assumptions, including the absence of charge transfer across the interface and negligible effects from interface dipoles or atomic rearrangements, which simplifies the band alignment to a direct superposition of bulk energy levels referenced to the vacuum. It is particularly suited to ideal, abrupt junctions without significant chemical bonding or strain at the boundary. Despite its simplicity, the electron affinity rule exhibits limitations in real heterojunctions, often overestimating offsets by 0.2-0.5 eV due to unaccounted interface dipoles and surface reconstruction effects. For instance, in the GaAs/Ge system, the rule predicts a valence band offset of 0.49 eV, while experimental measurements yield approximately 0.15 eV. Its predictive accuracy is around 70% for lattice-matched III-V semiconductors, where errors are typically smaller, but it performs less reliably for systems involving different chemical bonding or lattice mismatch. The rule has been refined in later models to incorporate these effects, improving agreement with experimental data.Interface Dipole Models
Interface dipole models address limitations in simpler theories like the electron affinity rule by incorporating charge redistribution and asymmetry at the semiconductor heterojunction interface, which generates an electric dipole layer and a corresponding potential step ΔV. This dipole arises primarily from bond charge asymmetry or electron tunneling between the valence and conduction bands across the interface, leading to a net charge separation. The potential associated with this dipole is given by
where is the elementary charge, is the permittivity, is the interface area, is the charge density profile along the interface normal direction , and the integral captures the dipole moment from the spatial distribution of charge.[33] This correction is essential for realistic predictions, as it accounts for microscopic effects not captured in macroscopic alignments.[34]
Key models within this framework include the common anion rule, which predicts valence band offsets for heterojunctions sharing the same anion, attributing small discontinuities to the dominance of anion p-states in the valence band maximum. Proposed by Wei and Zunger, the rule uses core-level binding energy shifts to estimate offsets, with the valence band discontinuity ΔE_v approximated as the difference between apparent and true chemical shifts in core levels, reflecting cation electronegativity differences. However, the rule's accuracy is limited in systems where cation d-orbitals hybridize strongly with anion states, as seen in examples like CdTe/ZnTe, where calculated ΔE_v ≈ 0.13 eV deviates from the ideal zero offset due to such contributions.[35] Another approach involves self-consistent solutions to the Poisson-Schrödinger equations, which iteratively compute the electrostatic potential and wavefunctions to determine the charge-induced dipole. These solutions model the interface as a quantum system, solving ∇·(ε ∇V) = -ρ/ε_0 coupled with the Schrödinger equation for electron densities, yielding the equilibrium dipole that stabilizes the band alignment.[36]
The overall valence band offset is refined by adding the interface dipole potential to the Anderson rule prediction:
where ΔE_v^{Anderson} derives from differences in electron affinities and band gaps, and ΔV_{interface} provides the microscopic correction. This formulation, as in Tersoff's effective dipole model, stems from charge transfer across the interface, treating the dipole as an effective shift from valence band penetration into the adjacent barrier.[34] Such models enhance accuracy in lattice-mismatched systems like Si/Ge, where strain and intermixing induce dipoles that adjust the nominal 0.74 eV valence band offset by up to 0.2 eV, depending on interfacial composition. Ab initio density functional theory (DFT) calculations further refine these predictions by directly computing ρ(z) and ΔV for specific interfaces, achieving errors below 0.1 eV for Si/Ge through hybrid functionals and slab models that capture charge asymmetry.[34][33][1]
Recent advances include machine learning frameworks, such as graph neural networks integrated with DFT data, enabling rapid and accurate band offset predictions for novel heterostructures as of 2024.[37]