Band Alignment and Graded Heterostructures

Guofu Niu Auburn University Outline

• Concept of electron affinity • Types of heterojunction band alignment • Band alignment in strained SiGe/Si • Cusps and Notches at heterojunction • Graded bandgap • Impact of doping on equilibrium band diagram in graded heterostructures Reference

• My own SiGe book – more on npn SiGe HBT base grading • The proc. Of the IEEE review paper by Nobel physics winner Herb Kromer – part of this lecture material came from that paper • The book chapter of Prof. Schubert of RPI – book can be downloaded online from docstoc.com • http://edu.ioffe.ru/register/?doc=pti80en/alfer_e n.tex - by Alfreov, who shared the noble physics prize with Kroemer for heterostructure laser work

3 Band alignment

• So far I have intentionally avoided the issue of band alignment at heterojunction interface • We have simply focused on – ni^2 change due to bandgap change for abrupt junction – Ec or Ev gradient produced by Ge grading • We have seen in our Sdevice simulation that the final band diagrams actually depend on doping – A Ec gradient favorable for electron transport is obtained for forward Ge grading in p-type (npn HBT) – A Ev gradient favorable for hole transport is obtained for forward Ge grading in n-type (pnp HBT) Electron Affinity – rough picture • Neglect interface between vacuum and semiconductor, vacuum level is drawn to be position independent • The energy needed to move an electron from Ec to vacuum level is called electron affinity.

5 Electron Affinity Model • The electron affinity model is the oldest model invoked to calculate the band offsets in semiconductor heterostructures (Anderson, 1962). • This model has proven to give accurate predictions for the band offsets in several semiconductor heterostructures, whereas the model fails for others. We first outline the basic idea of the electron affinity model and then discuss the limitations of this model.

6 Semiconductor vacuum interface

• The band diagram of a semiconductor-vacuum interface is shown. • Near the surface, the n-type semiconductor is depleted of free electrons due to the pinning of the Fermi level near the middle of the forbidden gap at the semiconductor surface. Such a pinning of the Fermi level at the surface occurs for most semiconductors. • The energy required to move an electron from the semiconductor to the vacuum surrounding the semiconductor depends on the initial energy of the electron in the semiconductor. • Promoting an electron from the bottom of the conduction band to the vacuum beyond the reach of image forces requires work called the Electron Affinity. Lifting an electron from the Fermi level requires work called the Work Function, which is defined the same way in semiconductors as it is in metals. Finally, raising an electron from the top of the valence band requires the ionization energy .

7 Interface of Two Semiconductors • Next consider that two semiconductors are brought into physical contact. • The two semiconductors are assumed to have an electron affinity of chi1 and chi2 and a bandgap energy of Eg1 and Eg2 , respectively, as illustrated below. • Near Surface bending and image force has been neglected – which actually can change the conclusion (this effect is typically neglected though in practice) • Energy balance of moving an electron from vacuum to semiconductor “1”, from “1” to “2”, and from “2” to vacuum must be zero, that is

8 Interface of 2 semiconductors: Delta Ev

• The valence band discontinuity naturally follows:

 By convention delta E_v definition is Ev1 – Ev2.

9 Limitations of electron affinity model

• The delta E_c and delta E_v equations from electron affinity model are valid only if the potential steps caused by atomic dipoles at the semiconductor surfaces and the heterostructure interfaces can be neglected. • In this case, the knowledge of the electron affinities of two semiconductors provides the band offsets between these two semiconductors.

10 Band Alignment Types

• Abrupt “heterojunction” band diagram – the abrupt changes in Ec / Ev are determined by Electron Affinity and bangap changes • Three distinct band alignments are possible – type I, II, and III

Straddling, e.g. Staggering, Broken-gap, SiGe/Si, e.g. InP/InSb GaSb/InAs AlGaAs/GaAs

11 Strained SiGe/Si band alignment

• Type 1 • Mostly valence band offset • To first order – Delta Ev = 0.74 * xMole eV – Delta Ec \approx 0 Connecting Hetero Materials of Opposite Doping • Electrons / holes will move around, like in homojunction • Potential drops will develop, until Fermi level is the same on both sides • Far away from interface, potentials are flat

13 Cusps / Notches at Heterojunction

• Some cusp or notch must form in the conduction or valence band, depending on the details of the system. • Exactly what happens and what the cusps look like depends on many details, you must solve the Poisson equation properly for a specific case.

14 General Band Alignment PN Heterojunction • Assuming material 1 is p-type, material 2 is n- type – for drawing purpose, chi1 > chi2 • We will make these assumptions to allow “quantitative” drawing

P-SiGe/n-Si Heterojunction (EB, CB junction of NPN HBT) • Before connection P-SiGe/n-Si Heterojunction (EB junction of NPN HBT) • After connection P-Si / n-SiGe (EB, CB of PNP SiGe HBT)

forward Ge grading in p-type

Retrograding of Ge in p-type

Forward Ge grading in n-type

Retrograding of Ge in n-type

p-SiGe/p-Si abrupt heterojunction

N-SiGe/n-Si abrupt heterojunction

Connecting Materials • Electrons / holes will move around, like in homojunction • Potential drops will develop, until Fermi level is the same on both sides • Far away from interface, potentials are flat

26 Cusps / Notches at Heterojunction

• Some cusp or notch must form in the conduction or valence band, depending on the details of the system. • Exactly what happens and what the cusps look like depends on many details, you must solve the Poisson equation properly for a specific case.

27 • POINT, APEX: as a : a point of transition (as from one historical period to the next) : TURNING POINT; also : EDGE, VERGE b : either horn of a crescent moon c : a fixed point on a mathematical curve at which a point tracing the curve would exactly reverse its direction of motion d : an ornamental pointed projection formed by or arising from the intersection of two arcs or foils e (1) : a point on the grinding surface of a tooth (2) : a fold or flap of a cardiac valve - cus·pate /'k&s-"pAt, -p&t/ adjective - cusped /'k&spt/ adjective

28 Isotype Heterojunction (n-n or pp)

29 Biased Isotype Heterojunction • Apply bias U -> Fermi level / total potential changes by qU • Majority (electrons here) carriers conduct current • It is easier for electrons to move from left to right than from right to left • We may use an isotype heterojunction to inject majority carriers from the wide band gap material into the small band gap material

30 Strained SiGe on Unstrained Si

bulk SiGe

Si

strained SiGe

31 Graded Bandgap Structures • In regular semiconductor heterostructures, the chemical transition from one semiconductor to another semiconductor structure is abrupt. • In the preceding discussion, we have seen that the periodic potential and the band diagram are nearly as abrupt as the chemical transition. That is, the transition of the periodic potential and of the band diagram occur within a few atomic layers of a chemically abrupt semiconductor heterostructure. • In graded heterostructures, the chemical transition from one semiconductor to another semiconductor is intentionally graded. 32  Assuming two semiconductors “A” and “B” are chemically miscible, the mixed compound, also called semiconductor alloy, is designated by the chemical

formula A1-xBx, where x is the mole fraction of semiconductor A in the mixed compound.  The mole fraction is also designated as the chemical

composition of the compound A1-xBx . Most semiconductors of practical relevance are completely miscible.  Assume further that the gap energy of A and B are different, and that the bandgap energy depends on the composition. The dependence of Eg on the composition is usually expressed in terms of a parabolic (linear plus

quadratic) dependence. The Eg of the alloy A1-xBx is then given by

 where the first two summands describe the linear dependence of the gap and the summand (1 ) b describes the quadratic dependence of the gap. The parameter b is called the Bowing parameter.  For some semiconductor alloys, . (AlAs) (GaAs) , the bowing parameter is vanishingly small. The bandgap of the alloy is then given by

 Equations (17.6) and (17.7) are valid for homogeneous bulk semiconductors. However, the validity of the equations is not limited to bulk semiconductors. They also apply to the local bandgap of graded structures. We have seen in the preceding section that the atomic potentials and the energy bands closely follow the composition in a chemically abrupt heterojunction. Accordingly, the band edges and the gap energy will follow the chemical composition of graded semiconductors.  Example of linearly graded heterostructure is shown below  the figure shows a narrow Eg semiconductor A, a wide Eg semiconductor B, and a linearly graded transition region

“A1-xBx”, with thickness Delta Z.

Quasi-fields and the possibility of moving electrons and holes same direction

37 Graded base bandgap transistor vision • Kroemer also envisioned graded-gap hetero bipolar transistors which enhance the minority carrier transport through the base.

38 Graded SiGe Base NPN HBT – Equilibrium Status • Constant Fermi level -> Ec slope instead of Ev slope as base is p-type, • Electric field will develop such that holes will not move (zero current) • End result is: Quasi-field + electrostatic field = 0, no net hole “drift”, diffusion is zero due to uniform doping • Electrons “drift” due to the e-field, but drift current is cancelled by diffusion, as electron concentration n is higher near collector

39 Graded SiGe Base NPN HBT – with Bias • Electron concentration at end of base is lower, as the EB junction is forward biased and will inject electrons into the base, so electron diffusion and drift will be along the same direction • Hole quasi-fermi level gradient must be small due to high p (uniform p), Jp is always much less than Jn (beta >>1 by design)

40 Lattice constant of semiconductor alloys (unstrained or relaxed) • In graded semiconductor structures, the composition of the semiconductor is varied. This variation in chemical composition is not only accompanied by a change of the bandgap energy, but also (in general, but not necessarily) by a change in the lattice constant. The change in lattice constant is, for all semiconductor alloys, governed by Vegard’s law.

41 lattice-matched graded semiconductors

• For most graded semiconductor structures, it is imperative that the lattice constant does not change as the composition of the alloy is varied. Such structures are called lattice-matched graded semiconductors, e.g. in our SiGe HBT case, strained SiGe on unstrained Si substrate. • If semiconductors are not lattice matched, microscopic defects occur when the composition is varied. These defects degrade the quality, e.g. the radiative efficiency, of the semiconductor. • Lattice matching required for low defect density – This is particularly important for minority carrier devices (HBTs, lasers) – This is not so important for majority carrier devices

42 Relaxation of strain

43 Semiconductor heterostructures

• Ideal: Heterostructures are formed by semiconductors with the same crystal structure and the same lattice constant: An example is Al Ga As on GaAs • Often: Mismatched structures result in misfit dislocations defects which act as recombination centers. An example is GaN on sapphire • Diagrams of energy gap-versus-lattice-constant for of different semiconductors

44 45 Kromer’s Central Design Principle of hetero structure Devices

46Proc IEEE 1982 review paper Double Heterojunction Laser Shockley’s 1948 Patent on Hetero transistor

48 Abstract of 1982 Kromer’s classic HBT review paper

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