Photoelectrochemical Determination of the Absolute Band Edge

Subscriber access provided by University Libraries, University of Memphis

Article

Photoelectrochemical Determination of the Absolute Band Edge Positions as a Function of Particle Size for ZnO Quantum Dots Jesper Tor Jacobsson, and Tomas Edvinsson J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp302220w • Publication Date (Web): 31 May 2012 Downloaded from http://pubs.acs.org on June 10, 2012

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Photoelectrochemical Determination of the Absolute Band Edge Positions as a Function of Particle Size for ZnO Quantum Dots

T. Jesper Jacobsson* and Tomas Edvinsson Dept. of Chemistry - Ångström Laboratry, Uppsala Univ., Box 538, 75121 Uppsala, Sweden [email protected], +46 (0)70-5745116

~1~ ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract The absolute position of the conduction and the valence band edges of ZnO quantum dots (Qdots) has been determined as a function of particle size with potential dependent absorption spectroscopy. The absolute position of the band edges are vital for which catalytic reactions that can occur at the surface. They are also crucial parameters for charge injection and extraction in nanoparticular solar cells and other optoelectronic devices based on nanoparticles. The position of the conduction band edge was determined by potentiostatic population of the conduction band states and monitoring the resulting increase in the optical band gap. This was performed for ZnO particles in the quantum confined region with diameters ranging between 4 and 9 nm. The particles were deposited into thin films giving an ensemble of particles for which the analysis could be performed. The relevant equations were derived and their validity in terms of applied potential and kinetic considerations was quantified. We find that essentially all of the quantum size effect of increased band gap is occurring by a shift of the conduction band edge. The extent of the validity of the parabolic approximation, which is one of the assumptions in the analysis, is investigated, both experimentally and with density functional theory calculations of bulk ZnO. Here we find that the parabolic approximation only is valid in an energy range of slightly less than 0.1 eV from the conduction band edge, but in that regime constitute an excellent approximation. We also demonstrate that the validity of the parabolic approximation follows a rising Fermi level into the conduction band energy levels.


Page 2 of 38

Page 3 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Keywords ZnO, Quantum dots, Nanoparticles, Photolectrochemistry, Burstein-Moss shift, Band edge position



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1. Introduction ZnO has attracted a great deal if interest due to its possible nanoscale applications in fields like catalysis1, optoelectronics and photovoltaics2,3 to mention a few. For nanosized systems there is a great increase of the relative amount of atoms residing at the surface region and for very small dimensions, quantum size effects become apparent. The quantum confinement effects comprise both electronic properties such as a band gap shift4-7, fluorescence properties6, as well as change in vibrational properties such as phonon confinement8,9. The electronic quantum effects start to emerge when the dimensions of the system are in the order of the effective Bohr radius of the free charge carriers in the system and are important for ZnO particles when the diameter is smaller than approximately 9 nm. One of the more striking quantum size effects is the band gap shift, where the optical band gap is larger for smaller particles. This has been quantified with different approaches5,10,11 and also in our group6. As the band gap is shrinking when the particles get larger, by necessity also the absolute position of the band edges will change. In this paper we have quantified the absolute positions of the band edges with respect to particle size for quantum dots of ZnO. The absolute positions of the band edges are important both from a fundamental as well as a technological perspective as the positions have a direct influence on which redox reactions that can occur at the particle surface, as well as on charge transfer into and out from the particles. These in turn are central concepts for catalysis and molecular solar cells. The problem of determining the band edges for nanocrystalline materials has therefore attracted a lot of consideration in the literature12-23. It have turned out to be problematic, especially for mesoporous nanocrystalline materials like the ones used in for example dye sensitized solar cells, due to limited band bending and more complex potential distribution of individual particles in a porous matrix21,22,24. Different strategies for measuring the band positions have


Page 4 of 38

Page 5 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


been described in the literature, but all have their own limitations, and for some of them, the validity for their use under different conditions is not quantified. The purpose with this paper is to clarify a few aspects connected to some electrochemical and photoelectrochemical techniques in order to make them more accessible and to point out to which extent they could be applied. For this purpose we give a brief overview of different methods for determining the band edge positions and we also derive the relevant equations for a potential dependent shift of the absorption edge. We then utilize this to determine the absolute conduction and valence band edge positions of ZnO quantum dots as a function of particle size. This is compared with previous results in the literature. We also clarify the applicability of the methods which should be directly transferable to other nanoparticle systems. The paper starts with a theory section where the absorption in semiconductors and different techniques for determining the band edge positions are shortly reviewed. In the result section we apply these equations together with potential dependent absorption spectroscopy on increasingly larger ZnO particles.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 38

2. Theory 2.1 Absorption in semiconductors A property of importance for semiconductors is their interaction with light, which unfortunately tend to be rather hard to accurately describe in a simple way. A common starting point for a theoretical treatment of absorption in semiconductors is the Fermi golden rule25, or some modification of it, stated in equation 1

Ti  f 

2 

f H if1 i

2

g if E 

(1)

where Tif is the transition probability per unit time from an initial state, i, to a final state, f, H1if is the Hamiltonian for the first order perturbation, gif(E) is the overlap density of the initial and final states and ħ is the reduced Planck constant. This can be seen as an analogue to the simpler case of a small molecule where the optical absorption readily can be described in the framework of time dependent perturbation theory. The application of this approach is not straight forward, and simplifications are needed in the treatment of both the form of the initial states, the final states and the nature of the perturbation26. This has been done several times in the literature27-30, but the validity of the assumptions has not been discussed so much since then. A full treatment including the considerations above will in the end generate an expression for the absorption coefficient, α(hν), and its energy dependence. One of the most commonly applied approximations is the parabolic approximation. It basically imply that the electrons and holes can be described as free carriers with a parabolic momentum dependence close to the conduction and valence band edges, but with different effective masses as in eq 2-3,


Page 7 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Ev (k )  Evb 

2k 2 2m *p

(2)

Ec (k )  Ecb 

2k 2 2mn*

(3)

where Evb is the energy for the valence band edge at the Γ-point, Ecb is the corresponding energy for the conduction band and mn* and mp* are the effective masses of the electrons and holes respectively. If the energy increases, the E(k) dependence will eventually deviate from a parabola and the effective masses becomes energy dependent and the above approximation is no longer valid. The extent of this validity is quantified for ZnO later in the article. In the case of a direct semiconductor, like ZnO, assuming that only momentum preserving transitions are allowed and that the parabolic approximation is valid the transition probability in equation 1 can be solved. If properly done this generates an expression for the absorption coefficient as a function of excitation energy. The expressions found in the literature derived by this approach do in general not predict numbers in accord to experimental results, but give some insight into which parameters that may affect the absorption and the functional dependence. In most practical work the functional form of the theoretical expressions are kept, but the material parameters are lumped together into one constant, C1, treated as an empirical parameter. For a direct band gap semiconductor this gives the well known expression in eq 4, valid under the assumption of parabolic bands and photon energies close to the band gap energy.

 h   C1 h  E g

(4)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

This expression is a good approximation in a limited energy range, as shown later in the article. The measured absorption is linear to the absorption coefficient, meaning that the relation can be used for determining the band gap. This is accomplished by plotting the square of the measured absorption as a function of photon energies, and extrapolating the straight line close to the band gap energy down to zero absorption. In indirect semiconductor the crystal momentum is not conserved (k ≠ 0) and there is a requirement of a phonon assisted transition coupled to the electronic transition resulting in a lower total transition probability, and thus slower increase in absorption expressed in

(h)=C1(h-Eg). Some materials also have dipole forbidden transitions further diminishing the increase in absorption with energy after to the band gap. A generalized expression for the absorption can be written as31 (h)=C1(h-Eg)R where R = 1/2, 2/3, 2 and 3 for a direct allowed, direct forbidden, indirect allowed and indirect forbidden transition respectively. The theoretical treatment of these situations are out of the scope for this paper but is commonly used as the experimental way of determining if a semiconductor has a direct or indirect optical band gap, both of which can be allowed or forbidden.

2.2 Burstein-Moss shift In a degenerate semiconductor the doping has reached such a level that the Fermi level no longer is within the band gap. For n-doped semiconductors the Fermi level will then lie within the conduction band. This result in an apparent increase of the optical band gap as there no longer are any empty states close above the conduction band edge to where electrons can be exited. This effect goes under the name Burstein-Moss shift after the works of E. Burstein32 and T. S Moss33. Under the assumption of parabolic bands close to the band edges, simple geometry gives a relation between the intrinsic band gap, Eg, and the optical band gap, Egopt, given in eq 532, ~8~ ACS Paragon Plus Environment

Page 8 of 38

Page 9 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


 m* E gopt  E g  1  *n  m p 

 E f  Ecb  4k BT  while E f  Ecb   

(5)

where Ef is the energy at the Fermi level, T is temperature and kB is Boltzmann’s constant. The derivation can be found in the supporting information. A sketch of the concept is given in Figure 1.a.

Suggested position of figure1

Degeneracy by doping is not a necessary precondition for the effect of increased band gap to be apparent. It can also be achieved by populating empty states in the conduction band with electrons by electrochemically applying a potential to the material with a potentiostat. The system is not strictly under equilibrium when measuring the optical band gap due to the incoming photon flux and the corresponding response in electronic excitation under a certain wavelength. Under these conditions there will be a quasi-Fermi level working under steady state conditions instead of under equilibrium. The applied potential will then to a good approximation equal the quasi-Fermi level in the material. Equation 5 then states that a plot of the optical band gap as a function of the quasi-Fermi level, or applied potential, will give a straight line, which if extrapolated down to the intrinsic band gap gives the absolute position of the conduction band edge. By knowing Eg, the valence band position, Evb, will also be known. In the following we will use the Fermi level notation throughout, but it could be replaced by the quasi-Fermi level without lack of generality for the conditions mentioned above.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 38

According to the relation from Burstein, the intercept between the optical band gap as a function of potential and the intrinsic band gap in the undisturbed case, will equal the position of the conduction band edge shifted by 4kBT, or 0.103 V at room temperature. This parameter is a consequence of thermal excitations according to Fermi Dirac statistics in the semiconductor. When the Fermi level is inside the conduction band there will still be empty states below the Fermi level to which electrons from the valence band can be exited. The measured absorption onset, or band gap, will thus be due to excitations to states below the Fermi level. The value 4kBT from Burstein’s article is somewhat arbitrary, but roughly corresponds to the energy at which the Fermi Dirac distribution noticeable begins to take off, as seen in figure 1.b. The approach to determine the conduction band edge with potential dependent absorption has previously been utilized by Fitzmarice and co workers14,15,21,22. They make the assumption that when a sample is under potentiostatic control all the states which lie below the Fermi level will be filled and all those that lie above will be vacant21. Consequently they neglect the 4kBT term in their analysis. Whatever or not this is justifiable will be a question of kinetics, and the relative rate of the thermal equilibration, kthermal, the charge injection and extraction from the potentiostat, kpot,, and losses to the solution, kloss. These parameters are indicated in figure 1.a. The potentiostat used in our experiments update the potential on a 100 ns timescale, but the thermal equilibration of charge carriers is in the picosecond regime34 and thus faster. We do not see any sharp absorption change which would be a consequence of zero thermal relaxation. In our analysis we therefore retain the 4kBT term and will return to this issue later in the article. The value of this term should be seen as a part of the systematical error of the method.


Page 11 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.3 Alternative method for determination of the band edges When a negative potential, versus the normal hydrogen electrode (NHE), is applied to a semiconductor under illumination, the system transcend from a situation where electrons are exited from a full valence band to an empty conduction band, to a situation where electrons are exited from a full valence band to a partly filled conduction band. This cause a decrease in the absorption which is the potential analogue to the Burstein-Moss shift discussed in section 2.2. As a starting point in the analyses we will use the reduced absorption expression which describes the behavior of the unperturbed system for a direct band gap semiconductor given in the theory section and in equation 4. When a negative potential is applied there will be fewer available empty states in the conduction band, and the probability, f, that a state at energy E, will be populated by an electron can be described by the Fermi-Dirac distribution in equation 6.

1 f E , E f   E  E f  k BT e 1

(6)

The absorption can under an applied potential as a good approximation be modified by multiplying the original expression with the probability that the final state in the conduction band is empty as in equation 7.

 h , E f   C1 h  E g 1  f E, E f 

(7)

Under the assumption that the transition conserves crystal momentum, by far the most probable case in a direct band gap semiconductor, and that the valence and conduction bands



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 38

are parabolic, the energy E in equation 6-7 can by geometry as in figure 1.a be written as eq 8. The geometry is analogous to the Burstein Moss case, and the expression for the absorption at a certain wavelength under applied potential can then be expressed as eq 9.

 m*p  h  E g  E  Ecb   *  m  m*  n   p

(8)



 h , E f   C1 h  E g 1  E 

e

cb  E f







1

h  E g m*p

/



m*p  mn*

/ k T B

   1

(9)

After initial loses and thermal processes during an electrochemical experiment, a steady state condition will be reached and the Fermi level in the material will to a good approximation equal the applied potential. This means that the absolute position of the conduction band can be determined by utilizing the fact that at certain photon energy, hυ, the absorption will be 50 % of the undisturbed case. This will occur for an applied potential that equals the potential of the conduction band plus a fraction of the photon energy exceeding the band gap energy. That fraction depends on the quotients of the effective masses of the electrons in the conduction band and the electrons in the valence band. This constitute an alternative determination of the apparent band gap increase, The photon energy utilized to determine the absorption reduction should not be too close to the band gap as the undisturbed absorption there is relatively low. It should neither be taken too far from the band gap energy as the parabolic approximation then no longer is valid. Suitable vales for the photon energy are discussed further in the result section. We have not found any records in the literature describing this approach, or something similar, as a mean to determine the conduction band edge. We therefore consider it as being a


Page 13 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


new method. From an experimental perspective this is a very simple, fast and convenient approach to get the band edge positions, given that a spectrophotometer and a potentiostat are available. It is simply to measure the absorption at a suitable wavelength as a function of applied potential and see where the absorption reaches half of the undisturbed value. The band edge position is then given as that potential plus a small correction factor.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 38

2.4 Mott-Schottky A concept related to the conduction band edge is the flat band potential, which refers to the potential where no band bending occurs. This corresponds to the potential of the Fermi level at the undisturbed system. For a n-doped semiconductor the flat band potential is rather close to the conduction band edge, and experimentally the two concepts are sometimes treated as interchangeable. A commonly used method to determine the flat band potential is to measure the space charge capacitance with impedance measurements, which under a certain set of assumptions are directly related to the flat band potential. The most important assumptions to be valid are that the majority and minority carriers are negligible compared to the doping level, that the electron and hole concentration follow a Boltzmann distribution, that the double layer capacitance are insignificant compared to the space charge capacitance and that there are no leakage currents or electron transfer over the semiconductor-electrolyte interface. The MottSchottky relation given in equation 10 then holds 35,

1 2 k T    E  E fb  B  2 2 Csc  0 A eN D  e 

(10)

where Csc is the space charge capacitance, ε is the dielectric constant of the semiconductor, ε0 is the permittivity of free space, A is the area of the electrode, e is the elementary charge, ND is the donor concentration, E is the applied potential and Efb is the position of the flat band potential. A derivation of this relation can be found in the literature36. For a quantum confined system one also expects small additional contributions to the capacitance37 which in principle comes from the Pauli principle and the extra energy needed to add an electron to the confined states within a low dimensional system38,39. This is not investigated in detail here where we


Page 15 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


instead analyze the resulting flat band potential from equation 10. To ensure that the charge transfer over the interface is small enough the measurements are performed at kHz frequencies. The flat band potential could then be given if 1/Csc2 is plotted against applied potential as an extrapolated intercept with zero with an offset of 26 mV. The donor density could also be extracted from the slope. The Mott Schottky method works best for smooth, flat, planar films, and it is commonly recognized as difficult to get reliable data for nanoporous films like the ones found in for example dye sensitized solar cells21,22,24. The approach is also considered to be problematic for small nanoparticles which are too small for the bands to bend in the individual particles. It is thus expected that samples of quantum dots may be hard to accurately analyze with the Mott-Schottky approach and that the alternative methods described earlier are more applicable.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 38

3. Methods 3.1 Synthesis The ZnO nanoparticels were synthesized by a wet chemical method that we in detail have describe before6, and that originally is based on earlier work of Meulekamp40 and Spanhel et al41. In the synthesis 2.5 mmol Zn(OAc)2 · 2 H2O are dissolved in 25 ml boiling ethanol under vigorous stirring for about a minute, where after the solution is cooled to room temperature and mixed with 3.5 mmol LiOH · H2O dissolved in 25 ml ethanol. When the two solutions are mixed, ZnO quantum dots begin to grow in the solution which could be monitored by measuring the band gap shift with UV-vis spectroscopy6. At certain times, a small volume of solution is used in order to make films of particles with distinct sizes. 2.5 ml of reaction solution are and then mixed with approximately 5 ml hexane which causes the particles to agglomerate and precipitate. In order to speed up the sedimentation the solution is centrifuged at 5000 rpm for five minutes. The particles where then redispersed in one drop of methanol and ultrasonicated. This concentrated solution was then doctor bladed on substrate of conductive glass (FTO) whereupon smooth and transparent films were formed.

3.2 Measurements and characterizations The UV-vis absorption measurements were performed on an Ocean Optics spectrophotometer HR-2000 γ with deuterium and halogen lamps. In all measurements, a full spectrum from 190 to 1100 nm with 2048 evenly distributed points was sampled, and in order to obtain good statistics an average over 100 consecutive spectra were done. The electrochemical measurements were performed with a CH Instrument model 760. If not other is stated potentials are given with respect to the Ag/AgCl reference electrode, which is shifted 0.197 V with respect to the normal hydrogen electrode (NHE). ~16~ ACS Paragon Plus Environment

Page 17 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The potential dependent absorption has been measured on films of ZnO particles deposited on conductive glass (FTO). The electrolyte was 0.5 M NaSO4 and potential scans were performed from -0.3 V vs Ag/AgCl down to -1.3 V which is just before problems with ZnO reduction begins. The sweep rate has been chosen sufficiently low to ensure that steady state with respect to absorption is achieved at every potential, and 10 mV/s was found to be appropriate for the electrodes in this study. Absorption spectra of the films were measured at every second during the potential scan. XRD measurements where performed with a Siemens D5000 Diffractometer using parallel beam geometry with x-ray mirror and a parallel plate collimator of 0.4°. Cukα with a wavelength of 1.54 Å was used as x-ray source. The angle of incidence was 0.5° and 2Θ scans were performed between 10° and 90° with a step size of 0.1°.

3.3 Computational method Energy dispersive band diagrams where calculated with periodic density functional theory, DFT, calculations in Crystal0942, using the B3LYP functional. The basis set for oxygen is from work of Towler et al43, and for zinc of Jaffe et al44. A mesh of (6x6x6) k-points in the irreducible Brillouin zone in the reciprocal space was generated according to the MonkhorstPack scheme45 and 10-7 Hartree was used as convergence criteria.

4. Results 4.1 X-ray diffraction and optical band gap Measurements were performed on 18 samples of different sizes ranging from 4 to 8.5 nm in diameter. XRD measurements verified that the only crystalline phase present in the samples where wurtzite zinc oxide, and a representative diffractogram for some of the larger particles ~17~ ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 38

are given in figure 2.a together with literature values of the ZnO diffraction peaks46. In a previous study we showed that particles synthesized in this way are fairly isotropic in shape and that the size distribution is reasonably narrow6. UV-vis spectra where measured on all samples and the band gap of the particles where determined as outlined in the theory section. The band gap is correlated to the particle diameter, d, according to eq 11, where the first term is the intrinsic band gap, the second is due to Coulomb effects and the third is due to the increased kinetic energy of the localization of the electron-hole pair6. In figure 2.b the measured band gap and the corresponding particle diameter for the samples synthesized and analyzed in this study are given.

Suggested position of figure 2

E g  3.30 

0.293 3.94  2 d d

(11)

The key measurement in this study is potential dependent optical absorption and a representative measurement is given in figure 3.a, for particles with a diameter of 6.2 nm (sample 8). Figure 3.a displays the 3D surface of absorption against potential and wavelength and captures all the data in the measurement but may be somewhat hard to interpret. The data have therefore been projected down to two dimensions as in figure 3.b. The apparent optical band gap as a function of potential has been extracted as outlined in figure 3.c. The corresponding figures for the entire dataset with 18 different sizes between 4 nm and 8.5 nm can be found in the supporting information.



Page 19 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.2 Determination of the band edge positions In order to determine the conduction band position in analogy with the Burstein-Moss shift, Ecb B.M, the apparent band gap has been plotted as a function of applied potential which is display in figure 4.a for a representative sample. Figures and data for all the samples are found in the supporting information. The positions of the band edges given from this approach are given in figure 5. From the figure it is apparent that more or less the entire band gap shift for quantum confined particles occurs in the conduction band. The position of the conduction band is changing from around -0.83 V vs NHE for a 4.4 nm particle to around -0.63 V vs NHE for a particle close to bulk properties. The change in the valence band seems to be rather small and the level is around 2.74 V vs NHE. If the entire shift is occurring in the conduction band it should be possible to describe its size dependence with the same quadratic expression as the one describing the band gap (eq. 11), but shifted in the y-direction. This also seems to be the case as seen in figure 5.a where the non constant part of equation 11 is fitted through the data points. Enright and Fitzmaurice have previously attempted similar measurement on ZnO particles14, which was focused on the pH dependent shift of the band edges. They state that most of the change in the band gap is accounted for by a shift of the valence band, contrary to our results. They do not address possible additional shift in surface charge due to formation of Zn2+ at low pH and Zn(OH)2 at high pH at the surface. The most important difference is however that they only have two measurement points, and consequently have some inherent problems of interpreting the data. From our data, containing 18 samples with increasingly larger particles the gradual shift can be followed and it seems unambiguous that almost the entire shift occurs at the conduction band. In principle the valence band edge could be determined in an analogous way as the conduction band edge by using an anodic sweep. This is however not possible in the case of



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 38

ZnO in water due to the limited potential window of water and the instability of ZnO at the low pH generated by rapid oxidization of water. The valence band edge has therefore been determined by subtracting the band gap energy from the conduction band edge.

Suggested position of figure 4 As described in the theory section, for a certain wavelength the absorption will be half of the undisturbed value at an applied potential according to eq 9, which means that the conduction band edges could be given as in eq 12.

 h , E f   0.5 0  Ecb  E f 

h  E m g

m *p  mn*

* p

0

(12)

There are different values for the effective masses of the electron and holes reported in the literature, but representative values used in this articles are mn* = 0.3me and mp* = 0.45me. A slight shift of the values of the effective masses has only a small impact of the expected shift. The absorption at several different wavelengths were analyzed, and it turns out that an excess energy of 0.06 eV above the band gap energy is a reasonable compromise between the constrains mentioned in the theory section. The situation where the absorption is 50 % of the undisturbed case should then be shifted by 0.036 eV to obtain the conduction band edge. The zero level of the absorption is set to the absorption at -1.3 V vs Ag/AgCl. The measured absorption at this potential is mainly due to reflection and scattering, and is rather close to zero for all particle sizes and implies only small correction. The result of such an analysis on a representative sample is given in figure 4.b. The entire set of figures can be found in the supporting information.


Page 21 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The position of the conduction band edge given by this method, Ecb J, is plotted as a function of particle size in figure 5.a. These data scatter a bit more than for the other method but the trend is the same. When the two methods discussed above, are compared with respect to the position of the conduction band edge as in figure 5.a the methods give slightly different results. When the position of the valence band is compared as in figure 5.b, the second method indicates that some of the change in band gap actually is occurring in the valence band, even though most of the change, about 70 % is occurring in the conduction band. By comparing the two methods, Eb B.M and, Eb J, for determining the conduction band edge as in figure 5.c, it is seen that the difference between them increases with particle size. This is very interesting and can be ascribed to changes in thermal relaxation, charge transport properties and kinetics of the losses at the interfaces. In the Eb B.M-method it is assumed that the rate of thermal relaxation is the same for all the particles. The approach developed here, Eb J, does not make that assumption as the thermal population enters that method by the Fermi-Dirac distribution. It is reasonable to assume that the charge transport are better in the films with larger particles as they contain fewer grain boundaries and also a smaller fraction of possibly defect intense surfaces. This has also been shown for the ZnO system47. It is also reasonable to assume that the thermal equilibration should be slower for the smallest particles as the density of states has more cluster-like character with larger separation between the allowed energy levels in the conduction band. If the charge transport becomes faster than the thermal equilibration in the semiconductor, the assumption made in the article of Redmond et al21 in setting the term due to thermal relaxation in Burstein’s expression to zero becomes more valid. This could be the case for the successively smaller particles, which then would explain the difference between the two methods. This explanation is supported by the fact that the region of absorption change in the figures for the second method in figure 4.b is narrower



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 38

for the smallest particles. This is readily apparent when comparing the figures for all the different samples as can be seen in the supporting information.


Based on the analysis in the theory section the quotient of the effective masses of the electrons and the holes could be determined from the slope in figure 4.a, where the apparent band gap is plotted against applied potential. This have been performed for all samples, and a tendency to a decreased quotient between the effective mass of the electron, mn*, and the hole mp* with particle size is seen. The data does however show a relatively large scattering, wherefore a reliable analysis not could be performed. No quantitative data of the quotient of the effective masses have therefore been possible to obtain. This may be ascribed to the fact that the absorption coefficient in itself, change with applied potential, which obscures the analysis. The associated figures can however be found in the supporting information.

4.3 Mott-Schottky measurements Mott-Schottky measurements were performed on thin films made from the ZnO Qdots. The films are relatively compact which means that the validity of the Mott-Schottky approach may be better than what could be expected for more porous films used in dye sensitized solar cells. The Mott-Schottky measurement was carried out at different frequencies where 1100 Hz and 3200 Hz gave the most reliable data for the film thicknesses used. The choice frequency may shift the data on the order of 10 mV. In figure 6.a, Mott-Schottky plots performed at 3200 Hz are shown. The extracted flat band potential as a function of particle size, and a comparison with the band edge positions are given in figure 6.b.


Page 23 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



The Mott-Schottky measurements give somewhat lower and slightly noisier values than the method utilizing the analogue with the Burstein-Moss shift. The lower values are expected as these measurements strictly do not measure the conduction band edge, but the flat band potential, which is supposed to be located within the band gap, slightly below the conduction band edge. The trend is however the same and more or less the entire shift occur close to the conduction band. The difference from the Eb B.M-method and the Mott-Schottky method is in the order of 200 mV. From the slope in the Mott-Schottky measurements it is possible to obtain the donor density. The donor density is in the order of 1·1020 cm-1 which is in the higher range of reported values for ZnO. The data scatter a bit but show a weak tendency for decreased donor density with larger particles. Since the films with smaller particles have larger total amount of surface area this indicates a higher density of donors at the surface compared to in the bulk. If we instead for evaluation purposes assume isotropic donor density in the ZnO quantum dots, the potential drop, φd, within a ZnO Qdot can be estimated with equation 13

d 

eN d 2 r 6 0

(13)

where ε is the relative dielectric constant of ZnO, ε0 is the permittivity of vacuum, Nd is the donor density and r is the radius of the particle. The high doping would then lead to a maximum potential drop of 0.15 V and 0.7 V for the 4 nm and 8.5 nm ZnO Qdot respectively. The doping density extracted here from the Mott-Schottky is rather uncertain due to both the questionable validity of only Mott-Schottky behavior of the capacitance with lack of quantum ~23~ ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 38

capacitance and also from the uncertain estimation of the true electrode area due to surface roughness. With a surface roughness of 3 for example, implying a donor density of 1·1019 cm3

, we would have 0.015 V and 0.07 V possible potential difference for the surface and the

center of the 4 nm and 8.5 nm quantum dot respectively. In this situation we would have weak band bending in the larger part of the quantum dots studies whereas very weak band bending for the smallest Qdots. Although a rough estimate it indicates that band bending can occur for these small dimensions due to the high doping level and modest dielectric constant. In the calculations of the doping level, the electrode area has been 0.8 cm2 and a dielectric constant of 8 has been used. The figures are given in the supporting information.

4.4 The validity of the parabolic approximation The Burstein approach to determine the conduction band edge is based on the assumption that the bands and the density of states are parabolic in shape. This is generally assumed as being a good approximation without further motivation. Here we reassess the generality of this assumption by examining when the parabolic approximation is valid and, in particular, when conduction band states are populated potentiostatically. In figure 7 the absorption as well as the square of the absorption is given, together with a parabolic fit to the data. The parabolic approximation there gives an astonishingly good description in an interval of approximately 0.06 eV starting from approximately 0.01 eV above the band gap. For energies closer to the band gap the absorption is better described with an exponential function. This small absorption just before the band gap is often termed the Urbach tail from work by Urbach48 and which mechanistic origin still is debated in the literature. Recent work for Si, however, suggest that the Urbach tail originates in atomistic disorder at the surface49,50.


Page 25 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



The energy interval where the parabolic approximation can be used thus seem to be smaller (a)

(b)

(c)

than the full window of applied potentials used in the experiments. However, by plotting data for different potentials together with a parabolic fit as in figure 7.c it is apparent that the absorption can be approximated with a parabolic expression in an energy interval of at least 0.06 eV regardless of the applied potential. The parabolic expression thus seems to be a good approximation from the first empty conduction band state and 60 meV above. For the more negative potentials the Urbach tail become somewhat more pronounced but at the same time the parabolic region widens somewhat. To analyze the applicability of the parabolic approximation from a more theoretical perspective, the band diagram of bulk wurtzite ZnO has been calculated with DFT using the B3LYP hybrid functional. The band diagram in figure 8.b and the close up in figure 8.c reveals the expected direct band gap of 3.2 eV for Wurtzite ZnO. Slightly below the valence band another band is appearing with a more sharp curvature. Holes in these two bands are referred to as light and heavy holes respectively. In order to see in which energy interval the theory predicts the band energy to be parabolic, the square root of the energy have been plotted against crystal momentum in the vicinity of the Γ-point. This should give straight lines as long as the parabolic approximation is valid according to equation 2 and 3. This could be seen in figure 8.d where the conduction band has been shifted down by the band gap energy to simplify the reading of the figure. It is apparent that the parabolic approximation breaks down for absorption energies approximately 0.25 eV in exess of the band gap energy. This is more than we find experimentally, but not very much more. The theoretical value can be expected to be larger than the experimental value as it does not capture the Urbach tail, and is calculated for the ground state energies. When the conduction band is populated by electrons ~25~ ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 38

in an optical transition or by a potentiostat, the additional shielding within the 4s, and 4p orbitals in the conduction band will inevitably change the local interaction and thus change the conduction band energy levels. The shielding will lead to less interaction and thus to less dispersion and more shallow parabolic band at the Γ-point. With this in mind the theoretical values can be seen as an upper asymptotic limit in an ideal situation for bulk with no electrons populating the conduction band.


5. Conclusions We have presented a brief survey of electrochemical and photoelectrochemical methods for determining the band edge positions in semiconductor nanoparticles, which all rely on a parabolic approximation of the orbitals close to the band edges. We both revise existing approaches as well as develop a new method. These have then been used to determine the absolute positions of the conduction and valence band edges of quantum confined ZnO Qdots between 4-9 nm in diameter. The measurements revealed that almost all the band gap shift seen for the quantum confined ZnO can be ascribed to a shift of the conduction band energy level. The position of the conduction band changed from to -0.83 V to -0.63 V vs NHE when the particle size increases from 4.4 to 8.6 nm. The valence band was rather constant around 2.81 V vs NHE regardless of particle size. Mott-Schottky measurements gave values that were in line with the other two methods, but they were shifted approximately 200 mV. This is expected as the Mott-Schottky measurements give the flat band potential rather than the conduction band edge. The measurements showed that Mott-Schottky could be applicable in relatively dense nanoparticle


Page 27 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


assemblies with reasonably good results, if the doping density is high enough and the dielectric constant of the material is not too high. The methods described should be applicably for determining the absolute positions of the band edges on different semiconductor nanoparticles as well, and should be rather easy to implement. The parabolic approximation of the conduction band orbitals was also investigated both with DFT calculations of bulk ZnO and experimentally for the ZnO Qdots. Experimentally we find that the parabolic approximation only is valid in an energy range of slightly less than 0.10 eV from the conduction band edge. This is in line with what we find from the theory. The calculation does however predict a slightly larger domain of applicability which is expected as they deal with the undisturbed ground state case and not the more complicated situation under illumination and a temperature above zero. We also demonstrate that the validity of the parabolic approximation follows the rising Fermi level into the conduction band energy levels, which was utilized in the analysis.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 38

6. Supporting information A derivation of equation 5 based on the geometry in figure 1.a. A table summarizing the numerical key data. The entire set of figures corresponding to absorption data, figure 3.b, band gap determinations, figure 3.c, determination of the conduction band edge, figure 4.a and figure 4.b. Conduction band edge from Mott-Schottky measurements. Donor density as a function of particle size from Mott-Schottky measurements. Additional figures concerning the parabolic approximation. Results and discussion concerning the effective masses given from the slope in figure 4.a. Cyclic voltametry data. These data are available free of charge via the internet at http://pups.acs.org

7. Acknowledgment We are grateful for financial support from Carl Tryggers foundation, the Royal Swedish Academy of Science and Ångpanneföreningens Forskningsstiftelse.


Page 29 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8. References

(1)

Woll, C. Prog. Surf. Sci. 2007, 82, 55.

(2)

Baxter, J. B.; Aydil, E. S. Sol. Energy Mater. Sol. Cells 2006, 90, 607.

(3)

Quintana, M.; Edvinsson, T.; Hagfeldt, A.; Boschloo, G. J. Phys. Chem. C 2007, 111, 1035.

(4)

Brus, L. J. Phys. Chem. 1986, 90, 2555.

(5)

Brus, L. E. J. Chem. Phys. 1984, 80, 4403.

(6)

Jacobsson, T. J.; Edvinsson, T. Inorg. Chem. 2011, 50, 9578.

(7)

Koch, U.; Fojtik, A.; Weller, H.; Henglein, A. Chem. Phys. Lett. 1985, 122, 507.

(8)

Fonoberov, V. A.; Balandin, A. A. J. Nanoelectron. Optoe 2006, 1, 19.

(9)

Raymand, D.; Jacobsson, T. J.; Hermansson, K.; Edvinsson, T. J. Phys. Chem. C 2012, 116, 6893.

(10)

Spanhel, L. J. Sol-Gel Sci. Technol. 2006, 39, 7.

(11)

Viswanatha, R.; Sapra, S.; Satpati, B.; Satyam, P. V.; Dev, B. N.; Sarma, D. D. J. Mater. Chem. 2004, 14, 661.

(12)

Beranek, R. Adv. Phys. Chem 2011.

(13)

Dang, X. J.; Massari, A. M.; Hupp, J. T. Electrochem. Solid. St 2000, 3, 555.

(14)

Enright, B.; Fitzmaurice, D. J. Phys. Chem. 1996, 100, 1027.

(15)

Fitzmaurice, D. Sol. Energy Mater. Sol. Cells 1994, 32, 289.

(16)

Flood, R.; Enright, B.; Allen, M.; Barry, S.; Dalton, A.; Doyle, H.; Tynan, D.; Fitzmaurice, D. Sol. Energy Mater. Sol. Cells 1995, 39, 83.

(17)

Jasieniak, J.; Califano, M.; Watkins, S. E. ACS Nano 2011, 5, 5888.

(18)

Oregan, B.; Gratzel, M.; Fitzmaurice, D. Chem. Phys. Lett. 1991, 183, 89.

(19)

Oregan, B.; Gratzel, M.; Fitzmaurice, D. J. Phys. Chem. 1991, 95, 10525. ~29~ ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(20)

Page 30 of 38

Radecka, M.; Rekas, M.; Trenczek-Zajac, A.; Zakrzewska, K. J. Power Sources 2008, 181, 46.

(21)

Redmond, G.; Okeeffe, A.; Burgess, C.; Machale, C.; Fitzmaurice, D. J. Phys. Chem. 1993, 97, 11081.

(22)

Rothenberger, G.; Fitzmaurice, D.; Gratzel, M. J. Phys. Chem. 1992, 96, 5983.

(23)

Xu, Y.; Schoonen, M. A. A. Am. Mineral. 2000, 85, 543.

(24)

Cao, F.; Oskam, G.; Searson, P. C.; Stipkala, J. M.; Heimer, T. A.; Farzad, F.; Meyer, G. J. J. Phys. Chem. 1995, 99, 11974.

(25)

Dirac, P. A. M. Proc. R. Soc. A 1927, 114, 243.

(26)

Lax, M. J. Chem. Phys. 1952, 20, 1752.

(27)

Basu, P. K. Theory of Optical Processes in Semiconductors; Oxford University Press: Oxford, 2003.

(28)

Hall, L. H.; Bardeen, J.; Blatt, F. J. Phys. Rev 1954, 95, 559.

(29)

Hall, L. H.; Bardeen, J.; Blatt, F. J. Proc. of Atlantic City Photoconductivity ConferenceNew York. 1956, 1954.

(30)

Pankove, J. I. Optical processes in semiconductors; Dover Publications: Minerola N. Y. USA, 1975, 1971.

(31)

Elliott, R. J. Phys. Rev 1957, 108, 1384.

(32)

Burstein, E. Phys. Rev 1954, 93, 632.

(33)

Moss, T. S. Proc. Phys. Soc. London, Sec. B 1954, 67, 775.

(34)

Djurisic, A. B.; Leung, Y. H. Small 2006, 2, 944.

(35)

Bard, A. J.; Faulkner, L. R. Electrochemical Methodes; Wiley: USA, 2001; Vol. Second edition.

(36)

Gelderman, K.; Lee, L.; Donne, S. W. J. Chem. Educ. 2007, 84, 685.

(37)

Schubert, E. F.; Kuo, J. M.; Kopf, R. F. J. Electron. Mater. 1990, 19, 521.


Page 31 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(38)

Ando, T.; Fowler, A. B.; Stern, F. Reviews of Modern Physics 1982, 54, 437.

(39)

Luryi, S. Appl. Phys. Lett. 1988, 52, 501.

(40)

Meulenkamp, E. A. J. Phys. Chem. B 1998, 102, 5566.

(41)

Spanhel, L.; Anderson, M. A. J. Am. Chem. Soc. 1991, 113, 2826.

(42)

Dovesi, R.; Orlando, R.; Civalleri, B.; Roetti, C.; Saunders, V. R.; ZicovichWilson, C. M. Z. Krist 2005, 220, 571.

(43)

Towler, M. D.; Allan, N. L.; Harrison, N. M.; Saunders, V. R.; Mackrodt, W. C.; Apra, E. Phys. Rev. B 1994, 50, 5041.

(44)

Jaffe, J. E.; Hess, A. C. Phys. Rev. B 1993, 48, 7903.

(45)

Monkhorst, H. J.; Pack, J. D. Phys. Rev. B 1976, 13, 5188.

(46)

McMurdie, H. F.; Morris, M. C.; Evans, E. H.; Paretzkin, B.; Wong-Ng, W. Powder Diffr 1986, 1, 40.

(47)

Roth, A. P.; Williams, D. F. J. Appl. Phys. 1981, 52, 6685.

(48)

Urbach, F. Phys. Rev 1953, 92, 1324.

(49)

Dong, J. J.; Drabold, D. A. Phys. Rev. Lett. 1998, 80, 1928.

(50)

Drabold, D. A.; Li, Y.; Cai, B.; Zhang, M. Phys. Rev. B 2011, 83.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 38

Figure captions Figure 1. (a) Sketch of the parabolic band approximation with relevant energy levels marked. (b) The Fermi Dirac distribution at room temperature with the position of 4kBT indicated.

Figure 2. (a) XRD data for a representative sample of some of the largest particles. Diffraction peak positions from literature values for wurtzite ZnO are given as vertical lines. (b) Measured band gap and related particle diameter for the analyzed samples. (c) Color coding of the samples.

Figure 3. (a) Absorption as a function of applied potential and wavelength for sample 8, with particles of 6.2 nm in diameter. (b) The corresponding two dimensional figure. (c) Determination of the optical band gap as a function of applied potential for the same data by plotting the square of the absorption.

Figure 4. (a) Determination of the position of the conduction band edge by plotting the optical band gap as a function of applied potential. (b) Alternative method for determining the position of the conduction band edge. Absorption against applied potential for certain wavelengths at defined energies in excess of the band gap energy. The figures are for sample 8.

Figure 5. (a) Absolute position of the conduction band edge vs NHE as a function of particle size according the two different methods used. (b) The corresponding valence band edges. To facilitate the comparison between the (a) and (b) figures the y-scale is the same. (c) Difference in conduction band edge from the two methods as a function of particle size.


Page 33 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 6. (a) Mott-Schottky measurements performed at 3200 Hz. The color coding is the same as in previous figures. (b) The flat band potential as a function of particle size from the Mott-Schottky measurements compared to the conduction band edge.

Figure 7. (a) Absorption data and (b) Square of absorption, illustrating that the parabolic approximation gives an excellent description in an energy interval of approximately 0.06 eV. The data is representative for all the analyzed samples. (c) An illustration that the parabolic approximation holds even under an applied potential.

Figure 8. (a) Symmetry points in the first Brillouin zone. (b) Calculated band diagram for bulk ZnO. (c) A magnified subset of figure8.(b). (d) Square root of the energy to illustrate where the parabolic approximation is valid. The energy scale is relative, and the position of the valence band edge is at zero and the conduction band is shifted down by the band gap energy to ease the reading of the figure. The portion of the band diagram is centered around the Γpoint between L and K in figure 8.b.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 38

Figures Figure 1

Figure 2 (a)

(b)

Figure 3 (b)

(a)


(c)

Page 35 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4 (a)

(b)

Figure 5 (a)

(b)

Figure 6

(a)

(b)


(c)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 38

Figure 7

(a)

(b)

(c)

Figure 8 (a)

(b)

(c)

(d)


Page 37 of 38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table of content image



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

88x35mm (300 x 300 DPI)

ACS Paragon Plus Environment

Page 38 of 38

Photoelectrochemical Determination of the Absolute Band Edge

Recommend Documents