Electronic States of Semiconductor Nanocrystals. • Optical ... Different solutions made out of the same semiconductor l hibi iki l diff l ! ? ..... Core/shell. CdSe/ZnS ...
Phys y 2235
Physics of NanoMaterials The University of Hong Kong S. J. Xu Department of Physics (Lecture 9)
C t t off Lecture Contents L t Ni Nine • • • • • •
Semiconductor Nanocrystals: Quantum Dots Exciton Effect in Bulk Semiconductors Electronic States of Semiconductor Nanocrystals Optical Transitions of Excitons in QDs Applications of Semiconductor QDs References
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Color Changes as Size of Semiconductor Nanocrystals Changes
Different solutions made out of the same semiconductor nanocrystall exhibit hibi strikingly iki l different diff colors! l ! WHY??
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In the Intermediate Regime g Between Bulk and Molecular Properties
Size-dependent Quantum Effect! 4
Fundamental Scale where Quantum Q Size Effect Occurs • Semiconductor nanocrystals keep the structural features of the bulk solid but pparticularly y different electronic properties as a function of their size. • Quantum size effect occurs when the nanostructures themselves becomes smaller than a fundamental scale. • Such a fundamental scale is determined by the exciton B h radius. Bohr di • Excitons are coupled electron-hole pairs via Coulomb attraction. tt ti
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N Nanocrystal t l Properties P ti • Many physical phenomena in both organic and inorganic materials have natural length scales between 1 and 100 nm (102 to 107 atoms). • Controlling the physical size of materials can be used to tune the material properties. properties In the nanometer size regime, the electronic and optical properties of metals and semiconductors strongly depend on crystallite size in the nanometer size regime. • To understand properties - prepare homologous size series of monodisperse nanometer size crystals, known as nanocrystals (NCs). NC samples must be monodisperse in terms of size, shape, internal structure, and surface chemistry. • A diverse set of structural probes is combined to characterize and develop consistent structural models of NC samples. samples Optical, Optical electrical, electrical and magnetic studies of well welldefined NC samples reveal the unique size-dependent properties of materials in this intermediate, nanometer size regime between molecular species and bulk solid. • When atoms or molecules organize into condensed systems, new collective phenomena develop. Cooperative interactions produce the physical properties we recognize as characteristic of bulk materials. Like atoms or molecules, but in the next level of hierarchy, NCs may also be used as the building blocks of condensed matter. 6
Definition of Exciton/ Bohr Exciton Radius • The optical properties of a material are usually determined by electronic transitions within the material and light scattering effects. Due to Coulomb interaction, the electrons and holes existing in a material are known to form excitons. Therefore, the optical nature of semiconductors can be understood by investigating the properties of the excitons. excitons • An exciton is composed of an electron and a hole. The distance between the electron and the hole within an exciton is called Bohr radius of the exciton. Typical exciton Bohr radius of semiconductors is of a few nanometers. In bulk semiconductors, the exciton can move freely f l in i all ll directions. di i When h the h length l h off a semiconductor i d i reduced is d d to the h same order as the exciton radius, i.e., to a few nanometers, quantum confinement effect occurs and the exciton properties are modified. Depending on the dimension of the confinement, three kinds of confined structures are defined: quantum well (sometimes termed QW), quantum wire (QWR) and quantum dot (QD). In a QW, the material size is reduced only in one direction and the exciton can move freely in other two directions. In a QWR, the material size is reduced in two directions and the exciton can move freely in one direction di i only. l In I a QD, QD the h material i l size i is i reduced d d in i all ll directions di i andd the h exciton i can not move freely in any direction. • In these confined structures, the exciton nature is modified and novel optical properties are expected. As a result, these structures are good candidates for developing high highperformance optoelectronic devices such as semiconductor light-emitting diodes and 7 laser diodes.
Exciton Effect in Bulk Semiconductors Semiconductor materials have their electronic structures organized in bands: a valence band generated by the overlap of the occupied energetic levels of the individual structural units and a conduction band generated by the overlap of the h unoccupied i d levels. l l Generally G ll the h conduction d i andd the h valence l b d in band i bulk b lk semiconductors are continuous if intraband energetic spacing is smaller than kBT (T is the temperature).
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Once the electron is promoted into the conduction band a hole is left behind in the valence alence band; the semiconductor semicond ctor becomes conductive cond cti e and the electron and hole move freely on the expense of their kinetic energy. The transition from the ground state to the excited state occurs as a result l off some externall perturbation, b i e.g. a photon. h The electron in the conduction band and the hole in the valence band can be held together by the electrostatic attraction, to form exciton. The interaction between electron and hole can be described by a hydrogenlike Hamiltonian (cgs units)
(1) where the M is the total mass M= me*+mh* and μ is the reduced mass μ= me*m * h*/ (m ( e*+m * h*); *) me* and d mh* are the h effective ff i masses off the h electron and hole, respectively. Above Hamiltonian can be solved in the same way as in the case of hydrogen atom. The energy level set of excitons can be found as
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E(∞) - R *y n 2 ,
(2)
where E(∞) is the minimum energy of the continuum state, i.e., the band gap Eg. Here n=1, n=1 2 2, 3 … is the principal quantum number number.
⎛ μ ⎞ ⎟× Ry R = ⎜⎜ 2 ⎟ ⎝ m0ε r ⎠ * y
(3)
is the binding energy of exciton or called excitonic Rydberg energy. Ry=13.6 =13 6 eV, eV atomic Rydberg energy. energy εr is the relative dielectric constant of the material.
m0 a ex = a H ⋅ ε r μ
(4)
is the Bohr radius of exciton exciton. Here aH =0.53 =0 53 Å Å is the Bohr radius of hydrogen atom. For typical semiconductors, one finds 10
1meV ≤ R *y ≤ 200meV and
50nm ≥ a ex ≥ 1nm. Finally, y, the dispersion p relationship p of exciton can be written as
R *y
h 2K 2 E x (K ) = E g - 2 + . n 2M
(5)
Therefore, we have a physical picture about excitons in semiconductors Therefore semiconductors, as shown in next page.
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Single Particle Picture and Two Single-Particle Two-Particle Particle (Excitons) Picture of Bulk Semiconductors energy
energy continuum
conduction band
exciton levels R *y
Eg
wave r vector K
Ex
valence band
0
r K
E g = E x + R *y Single g p particle p picture
Two-particle (bound electron-hole pair---exciton) i it ) picture i t 12
Optical p Transitions of Excitons in Semiconductors
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absorpttion
n=1 n=2 n=3
Eg
Photon energy
Theoretical absorption of free exciton in direct gap semiconductors
Experimental absorption of free exciton in GaAs 14
Electronic States of Semiconductor Nanocrystals When quantum dot size is smaller than the exciton Bohr radius, r < αex, the electron hole pair energy levels in quantum dots cannot be treated further based on hydrogen model. model The lowest energy level of the exciton is now delocalized over the entire quantum dot. The exciton levels are given by solving the classical quantum mechanical h i l problem bl off a particle ti l in i a box. b F the For th case where h th the electron and the hole are confined in a small space, the Coulomb attraction is negligibly small compared to a potential U(r) that describes a spherically symmetric potential well of length r. The corresponding Hamiltonian is: (6) 15
The band gap (the minimum energy necessary to excite an electron from the valence level to the conduction level) of semiconductor nanocrystals (quantum dots) is given by
E g, QD
h2π2 . = E g,bulk + 2 2μr
(7)
where the second term describes the energy levels of a particle of mass μ i a spherically in h i ll symmetric t i potential t ti l box. b Above equation explains qualitatively well the quantum size effects in quantum dots: the increase of interband energy separation with the decrease of quantum dot size. In fact, a more accurate equation for the calculation of exciton energy in semiconductor nanocrystals can be derived as E g, QD = E g,bulk
h 2 π 2 1.786e 2 μe 4 + − − 0.248 2 2 2 . 2 ε rε 0 r 2μr 2h ε r ε 0
(8) 16
Correlation diagram of the electron energy states that should exist in bulk semiconductors i d andd quantum dots. d Wh When nanocrystall size i b becomes commensurable with aex (Bohr exciton radius), the distinction between shallow traps and broad electronic bands ceases. The deep traps are correlated with the existence it off the th lattice l tti defects. d f t 17
Table 1. Calculated exciton Bohr radii for various semiconductor quantum dots as a function of effective mass and dielectric constant based on Eq. 4.
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EPM E EPM: Empirical i i l Pseudopotential P d t ti l Method M th d
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Optical p Transitions of Excitons in QDs Optical transitions in QDs are associated with their electronic states. Quantum size effects that are very well predicted by the particle-in-box model can be also observed in absorption spectroscopy. The absorption bands shift to higher energies with decreasing quantum dot radius, “blue shift”. It can be seen that the optical band gap is blue shifting dramatically from th bulk the b lk size i value l to t the th quantum sizes, e.g. in right figure, for the CdSe with bulk value of the optical band gap at the 716 nm ((~1.68 eV). ) The absorption p spectrum for a quantum dot
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of 115 Å in size is broad and featureless, characteristic to the bulk absorption spectrum. spectrum Below exciton Bohr radius the absorption spectra exhibit a fine structure. Appearance of a fine structure in absorption spectrum is due to the presence of the discrete energy levels. Another example: colloidal QDs.
InP
Absorption (solid line) and photoluminescence ((dotted line)) p spectra at 298 K for colloidal ensembles of InP QDs with different mean diameters are shown in right figure. All QD colloidal samples were photoexcited at 2.48 2 48 eV. eV The emission peaks also blueshift with decreasing the QD size. 21
PL Spectra p of Excitons in Single g QDs Like ik isolated i l d atoms, QDs possess a number of discrete energy levels, which can be verified by the measurement of PL spectrum from individual QDs. Q The right figure shows the PL spectra of individual GaAs QDs All of the lines are QDs. extremely narrow. These lines are well fitted by Lorentzians, and the linewidths vary from 32 to 51 µeV (FWHM). The sharp lines reflect the deltaf function i like lik density d i off states in QD structures.
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A li ti Applications off Semiconductor S i d t QDs QD • Bioimaging. Fluorescence labeling of specific compartments in cells is a
widely used method in biology to visualize structural units. Core/shell CdSe/ZnS semiconductor nanocrystals are more robust fluorescent probes with ith size i t bl emission tunable i i properties. ti A shell h ll off higher hi h b d gap band semiconductor material can increase dramatically the brightness of quantum dots and protect them from photo bleaching. The surface of the quantum dots can also be chemically modified in such a way that the quantum dot can be chemically h i ll linked li k d with i h a biomolecule bi l l that h binds bi d specifically ifi ll to the h target.
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LEDs, tunable lasers. Quantum dot materials can provide superior
performance f i lasing in l i applications li i comparedd to bulk b lk semiconductors. i d I the In h zero dimensional (0D) quantum dots the separation between energy states is greater than the thermal energy of the charge carriers; this inhibits the thermal depopulation of the lowest, “emitting” transition, phenomenon that confers high-temperature stability and a narrow spectral emission width. In addition, the quantum size effect can be employed to turn the wavelengths of emitting photons. 23
• Quantum Computing. A proposed arrangement of four quantum dots that represent a logic state “1” in a) and “0” in b); c) arrangement of GaAs quantum dots on a solid substrate achieved by e-beam lithography and conductive AFM. The electron can travel f from one corner to the h next corner off the h square by b tunneling. li
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R f References • L.E. Brus, J. Phys. Chem., 1986, 90, 2555. • Y. Wang, and N. J. Herron, Phys. Chem., 1991, 95, 525. • M.G. MG B Bawendi, di et al., l Phys. Ph Rev. R Lett., L 1990 65 (30) 1990, (30), 1623. • D. Gammon,, E. S. Snow,, B. V. Shanabrook,, D. S. Katzer,, and D. Park, Phys. Rev. Lett., 1996, 76 (16), 3005.
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