How Albert Einstein contributed to the development of the most accessible episodic source

Have you ever wondered why Albert Einstein published the first of four papers in 1905 explaining the photoelectric effect and why, paradoxically, he presented the foundations of quantum physics before Einstein physics?

Disclaimer: The purpose of this article was to clarify the effectiveness of one occasional source and its theoretical limits. The English mutation is partialy the result of chance and partialy the request of my friends who want to enjoy my ridiculous English potented by translator [26]. The author bears no responsibility for the fact that such a portion of physics and mathematics has been squeezed into the text, but with a little effort it could be even worse. Nor does he bear any responsibility for any errors and mistakes, whether his own or not. All the facts stated are fictional and are based only on the current state of knowledge. If you do not want to read the whole detective story, read only the beginning and the end. Small plastic parts of toys do not belong in children’s mouths!

Sometime before March 18, 1905, Albert Einstein sent his first article to the Annalen Der Physik, „Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristicischen Gesichtspunkt“ – „On a heuristic view of the creation and transformation of light“. The word heuristic can be understood as something that has arisen through observation and does not yet have a sufficient theoretical basis. The idea of ​​converting a photon into another type of energy was still very revolutionary at the time. [13]

In the past 120 years or so, quantum physics has enriched us with further knowledge, so we know that a photon, when colliding with a molecule or atom in a semiconductor, can not only excite an electron when absorbed, which, when recombining to a stable energy level, can emit a photon again, but can also emit several photons, the total energy of which is equal to (or smaller than) the energy of the original photon, or this energetic, „hot“ electron can gradually transfer its energy to surrounding atoms in the form of quanta of mechanical energy (phonons) and thus increase the temperature before it is drawn into the cathode on the semiconductor or recombined back into the atom. As can be seen, the photovoltaic effect (analogous to the photoelectric effect), or in very simple terms - the conversion of photon energy into a free electron, is only one of several states that occur with a certain probability depending on the type of material and external conditions.

That same year, around June 30, 1905, Albert Einstein sent the article „Zur Elektrodynamik bewegter Körper“ – „On the Electrodynamics of Moving Bodies“, i.e. the Special Theory of Relativity (STR); please note the special formulation – electrodynamics of moving bodies. Einstein makes it clear that the goal is to connect Newtonian physics with the physics of the electromagnetic field and create a complete physical theory. The third contribution was accepted by the publishing house on September 27, the article „Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?“ – „Does the inertia of a body depend on its energy content?“. Thus, the well-known law of conservation of mass and energy E = γm₀c², where γ - Lorentz coefficient, m₀ - invariant mass of the body, c - speed of light in vacuum (I am giving a notation that more closely corresponds to Einstein’s concept). [14] [15]

The last, fourth article was accepted by the publishing house in a revised form on December 19th, „Zur Theorie der Brownschen Bewegung“ – „On the Theory of Brownian Motion“. This fourth article not only provides indirect evidence of the existence of atoms through the kinetic energy of molecules, but also describes the relationship between temperature and energy. For our considerations on the photovoltaic effect, it is illogically later, because Brownian motion is a fundamental phenomenon necessary for understanding many physical phenomena of the microworld. [16]

Small nipple. Why illogically and non-chronologically? Maybe because Einstein first imposes on us an occasional source of energy and then even allows himself to explain how global warming occurs on CO₂ molecules? Shouldn’t it be on the contrary because the photoelectric effect also has consequences in the conversion of light into heat? Or, God forbid, what if it all applies together and something more?

Although all of Einstein’s articles have an irreplaceable impact on quantum physics and solid-state physics, we will focus on the first and fourth.

From Brownian motion to the entropy of the system

Let’s start with Brownian motion, where Einstein also addressed the relationship between temperature and energy. The square of the mean distance ⟨x₂⟩ of the movement of macroscopic particles in a solution is equal to the following relationship

⟨x²⟩ = 2k_BTt / 6πηr (1) * **

⟨x2⟩ - mean square distance (mean square displacement) of a particle at time t (m²)
k_B - Boltzmann’s constant, a physical constant relating temperature to energy (1.380649×10⁻²³ JK^-1)
T – absolute temperature of the liquid (K)
t – time (s)
η - dynamic viscosity of the liquid in pascal seconds ( Pa⋅s, Nsm^-2)
r - radius of a (spherical) particle (m)

Einstein proposed a method for determining Boltzmann’s constant, the value of which was not known. It is therefore not only a theoretical calculation and explanation of the phenomenon, but also a guide to how to measure and verify the validity of the statement. If the measured result is in accordance with the physical theory, the explanation of the phenomenon is most likely correct. Jean Perrin (1909) then determined the constant by measurement. The Boltzmann constant in the concept of Boltzmann is related to the relationship between energy and support based on Maxwell’s distribution (statistical method, equipartition principle). [22]

⟨E_k⟩ =(1/2) m⟨v²⟩ = (3/2) k_BT (2)

⟨E_k⟩ – average energy value
m – particle mass
⟨v²⟩ – root mean square velocity

Ludwig Boltzmann also formulated the relationship between the entropy S and the number of states W in 1877, based on the equipartition principle. This is a very general and still valid concept that connects microstates into a macroscopic whole. Quantum physics did not fall from the sky, it developed inconspicuously for many decades before it was established as a separate discipline.

S = k_B ln(W) (3)

S – entropy (JK^-1)
W – the number of microstates that will create a macroscopic result

Josiah Willard Gibbs generalized this relationship to the form of Gibbs entropy

S = -k_B ∑_i (p_i ln(p_i)) (3')

p_i – probability of occurrence of the i-th microstate

Cit. From the point of view of physics, entropy is a key quantity for the formulation of the second law of thermodynamics. This law sets the principle limits on the possibility of extracting useful work from a thermodynamic system. [1]

After determining Boltzmann’s constant, Perrin also calculated Avogadro’s constant (N_A).

R = N_Ak_B (4)

R – gas constant (8.31446261815324 JK⁻¹mol⁻¹)

The gas constant was derived from Avogadro’s law (more precisely, there were several sources - Boyle, Charles, Gay-Lussac, Avogadro). The measurement and calculation were performed by Henri Victor Regnault (1847). Among other gas measurements, he measured the volume of 2g (the hydrogen molecule is diatomic) H₂ at standard pressure and temperature. Later, this amount of substance was defined as the unit 1 mol. One mole of any gas will always have the same volume at standard pressure and temperature (atmospheric pressure, ≈ 0˚C (≈ 300K)); 22.4 dm3 (22.4 l). From this he got the gas constant R.

R = pV​ / nT (5)

R – gas constant (JK⁻¹mol⁻¹)
p – pressure
V – volume
T – temperature
n – substance quantity (mol)

To better understand the meaning of (5), we need to go back in time. Amedeo Avogadro (1811) found by comparing the volume of gases that the ratio of the volume of an ideal gas to its number of molecules is constant under the same conditions (pressure and temperature). However, he could not determine the number of molecules, he only determined the stoichiometric ratio of the volumes of gases, specifically for H₂, O₂ and H₂O when combining. If we know that we have the same number of molecules in the same volume, then the stoichiometric formula for combining H and O ensures the combination without a remainder:

2H₂ + O₂ → 2H₂O (6)

The ratio is 2 + 1 → 2. In other words, 2 mol H₂ + 1 mol O₂ → 2 mol H₂O.

As can be seen above, the stoichiometric calculation does not correspond to ordinary mathematics. The plus operator here gives different results. If we think about it, it is the temperature that represents the kinetic energy of molecules on a macroscopic scale (the total thermal energy of the system is the sum of kinetic (translational) + rotational + vibrational energy) that will obviously affect the gas pressure, and since the average distance that a molecule travels to the next collision is large under normal conditions (atmospheric pressure, temperature≈ 0˚C (≈ 300K)), the dimensions of the molecules can be neglected (macroscopically the difference is almost immeasurable). If we look at the nuclear number A of the H and O atoms, we get the sum 2×2 + 2×16 = 36. Given that the mass of protons and neutrons differs slightly and atoms also contain electrons, and binding forces also act here (as we know from STR, the mass also depends on the binding energy), the mass ratios are only approximate. Therefore, the unit of atomic mass constant m_u (u) was artificially introduced as 1/12 of the mass of ¹²C₆ and the relative atomic mass of A_r.

A_r(X) = ​m(X) / m_u (7)

A_r(X) – relative atomic mass of element X (dimensionless)
m(X) – atomic mass of an element X
m_u = (1 / 12) m(¹²C₆) – atomic mass constant (kg)

From this it follows that the relative atomic mass A_r(X) ≈ A(X), but only for carbon A_r(C) = A(C) and by definition A_r(C) = 12. Analogously, for example, A_r​(H) ≈ 1.008, where A_r is the dimensionless ratio between the mass of a hydrogen atom and the atomic constant m_u.

Please note the huge shift in thinking that is contained in the previous text. From the volume and Avogadro’s guess of a diatomic molecule of gases and their reactance, we have moved on to the masses of any substances, regardless of their state. At the same time, neither Avogadro nor Regnault could determine the number of molecules in 1 mol of a substance, they knew that it was the same, but they could not determine how many there were.

Avogadro’s constant N_A was determined, as we already know, by Perrin (later also by other methods) as 6.022 140 76×10²³ molecules and this number was identified with the unit 1 mol, which was already used earlier (without knowing the number of molecules in 1 mol).

N_A = 6.02214076×10²³ mol^-1

We can understand why the unit mol and then Avogadro’s constant N_A were determined this way if we look at the (boring) conversion table.

Conversion table between relative molecular mass and molar mass (Tab1)

Substance	Nucleon number (A)	Relative atomic (molecular) mass of Ar​ (Mr)	Mass of 1 mol
Hydrogen (H)	1 (¹H: 1p) +	1.008	1.008 g
Oxygen (O₂)	32 (2×16O: 8p+8n)	32.00	32.00 g
Carbon (C)	12 (12C : 6p + 6n)	12.00 (definice)	12.00 g
Water (H₂O)	18 (2×¹H+16O: 10p +8n)	18.02	18.02 g
...

+ p – number of protons, n – number of neutrons in the nucleus of an atom (in the nuclei of atoms of a molecule)

It is thanks to Avogadro’s intuition and genius. We (humanity) arrived at Avogadro’s constant through a series of iterations. Avogadro discovered the dependence of the volume of a gas on the number of molecules. Regnault determined the gas constant (R). Perrin determined Boltzmann’s and then also Avogadro’s constants.

Michael Faraday, who discovered a similar dependence of solids as Avogadro did for gases, determined in 1834 that for the same charge Q (Q = It), an amount of substance with the same number of molecules (1 mol) is released during electrolysis if the ion carries exactly 1 positive charge (hole). Otherwise, the amount of substance released by electrolysis is directly proportional to the total charge that has flowed during electrolysis. [24]

Analogously, as in the case of calculating the gas constant, Faraday calculated Faraday’s constant without knowing N_A from the ratios (direct proportion). After determining the elementary charge of the electron by Millikan’s experiment (quantum experiment, 1913), the constant N_A could then also be calculated macroscopically over 1 mol of substance.

N_A ​= F​ / e (8)

N_A – number of electrons in one mol (Avogadro’s constant)
e – electron charge (elementary charge, see below)
F – Faraday constant ≈ 96485.332 C⋅ mol-1

There are other methods to determine Avogadro’s constant (for example, X-ray crystallography), but here we have two independent procedures, different substances, different states of matter. Both arrived at the same value (with a small error); Avogadro’s and Boltzmann’s constants are apparently correct. Confirmed.

Why does a photovoltaic panel have similar efficiency to thermodynamic machines

Einstein’s first paper on the photoelectric effect builds on Planck’s relation of energy to wavelength and extends it to light (photons). This very simple relation is the fundamental basis for quantum physics. Extended Planck’s relation for the energy of a photon E_ph

E_ph​ = hν = hc / λ (9)

h - Planck’s constant (h = 6.626×10⁻³⁴ Js)
ν – radiation frequency (Hz)
c – speed of light (c = 299 792 458 ms^-1)
λ - wavelength (m)

Energy of an excited electron from the metal surface, i.e. the photoelectric equation according to Einstein

E_ph = hν = E_th + E_max^⁡ (10)

E_th – threshold energy, minimum energy required to excite an electron (for metals it is E_th ≈ 2 until 5 eV)

E_max ⁡= (1/2) m_ev_max² (11)

⁡E_max – maximum kinetic energy of a free electron (the calculation is non-relativistic, v_max « c; relativistic phenomena can be neglected)
m_e – electron mass
v_max – maximum speed

Einstein’s fundamental contribution to the explanation of the photoelectric effect lies in understanding the quantum nature of electron excitation.

• A sufficiently energetic photon (short enough wavelength λ) is needed for excitation to occur see (9). Below this energy threshold (E_th) excitation does not occur.
• The released electrons have the same kinetic energy for a particular wavelength (E_max^⁡) independent of the intensity of the luminous flux.
• As the photon flux intensity increases, the number of excited electrons increases linearly.

R_e ​= ηΦ_in ​= ηP_in​ / hν (12)

R_e​ – number of electrons released per second (electron emission rate (s^-1)
η – quantum efficiency
Φ_in​ – tok fotonů, počet fotonů dopadajících za sekundu (s^-1)
P_in​ – power (intensity) of incident light (W, Js^-1)

Electron excitation occurs in semiconductors (photovoltaic effect) similarly to conductors (photoelectric effect). The threshold energy required to create an electron-hole pair is denoted by E_g – bandgap. The name is derived from the fact that an electron in a semiconductor needs to overcome the forbidden energy band and excite from the valence band, which is the energy level of the electron in the crystal lattice, to the conduction band (energy level above the energy valence band of the crystal lattice), where it can move in the semiconductor lattice.

The valence band (VB) is not a single energy level, but many very close energy levels—an energy band composed of the possible combinations of valence states of all the atoms in a crystal. The electrons in it are not bound to individual atoms, but to the entire crystal lattice as a whole. (14) [20]

The conduction band (CB) is an energy band in the crystal lattice of a solid that lies above the bandgap and is composed of all possible energy levels in which electrons can move freely through the crystal. (14) [20]

In order for voltage to be generated, it is necessary that, in addition to the photovoltaic effect, there is a separation of electrons and holes. This is ensured by the spatial distribution of the PV cell, which is made of a p-n semiconductor. The interface between p-n is called the depletion zone, which is bounded by an accumulated charge of the opposite polarity than the charge in p (+) and n (-). This natural capacitor without an external source functions as a charge separator, see U_oc (23) (24).

E_ph​ = E_g​ + E_heat + E_loss (13)

E_ph​ – photon energy see (9)
E_g​ – bandgap, the minimum energy required to excite an electron from the valence band to the conduction band across the band gap
E_heat​ – energy lost to heating (relaxation of an electron in the conduction band by releasing phonons)
E_loss​ – other losses (recombination, imperfect separation, optical losses)

E_g ​= E_CB​ - E_VB (14)

E_VB – Valence Band maximum; the highest energy that an electron bound in an atom of a crystal lattice can have
E_CB – Conduction Band minimum; the lowest energy an electron must have to move freely in a crystal lattice

Let’s finally look at the efficiency of photovoltaic cells. To calculate it, we need to know in more detail what happens in the semiconductor when photons hit it. Let’s repeat this once again (not just verbally as in the introductory text) as a probable possibility that some phenomenon will occur.

P_i = R_i / ∑_jRj​ = R_iτ_eff (15)

P_i​ – probability of event i
It indicates how large a fraction of all photons, electrons, or excitations will end up in this particular way (e.g., the photovoltaic effect).

R_i – process speed (rate), frequency of transitions to a state i
Otherwise, how many interactions (e.g. excitations, recombination, emission, thermal relaxations) occur per unit time or per unit volume per unit time (s⁻¹, m⁻³s⁻¹).

∑_jR_j​ – the total rate of all possible processes that can occur after the absorption of a photon
That is, the sum of the number of interactions over the average duration of all possible states into which the energy of a photon in a semiconductor can transition.

τ_eff – the effective mean time is the time that a phenomenon (excitation) exists before it is terminated by any of the other states
Inverse value τ_eff is equal to the sum of the reciprocals of the individual mean times, of all states (τ_i ≈ 1 ps to 100 ns). In other words, the shortest period has the greatest weight – see exponential decay. [18]

1/τ_eff​ ​= R_tot​ = ∑_j​R_j = ∑​_j(1/τ_j) (16)

The written sum of the rate of process changes R_j

∑_j​R_j ​= R_exc​+R_abs+R_rad​+R_nonrad​+R_heat​+R_refl+… (17)

Table of possible phenomena and frequency of interactions after a photon hits a semiconductor layer (Tab2)

Process	Description	Typical mechanism	Speed ​​of interactions
Rexc	Excitation / Excitation with transition	photon excites free electron to CB (exciton singlet, exciton triplet)	Rexc(λ) = Φ(λ)⋅[1−e−α(λ)d] #
Rabs	Photon absorption	electron moves to CB	Rabs ~ αIph
Rrad	Radiative recombination see Prec	photon back radiation	Rrad ​= Bnp
Rnonrad	Non-radiative recombination see Prec	excitation energy into phonons (heat)	Rnonrad​ = An+Cn2p
Rsep	Charge separation (useful current)	electron transfer to the conduction band (CB)	Rsep​ ~ μEdn
Rheat	Hot electron relaxation	transferring excess energy to the semiconductor grid	Rheat​ ~ e−kBTE​
Rrefl	Reflection (reflection of a photon)	some of the photons are reflected from the surface	Rrefl​(λ) = Φ(λ)R(λ)
Rrec	Recombination rate see Prec (21)	mean distance to the electrode, diffusion length, see Rrad , Rnonrad	Rrec = e−d/l

# Description of quantities below

Beer–Lambert relation for absorbed photon flux [25]

R_exc(λ) = Φ(λ)⋅[1−e^−α(λ)d] (26)

λ - wavelength
Φ(λ) – spectral density of photon flux (number of photons of wavelength λ incident on an area and time) (m⁻²·s⁻¹·nm⁻¹)
d – thickness of the absorbing (active) layer of the semiconductor
α – the absorption coefficient determines how strongly a given material absorbs light (m^-1)

I_ph – intensity of incident light (photon flux), the number of photons incident on a surface per unit of time (m⁻²s⁻¹)
B – radiative recombination coefficient (m³s⁻¹)
n – concentration of electrons in the conduction band (type semiconductor n)
p – concentration of holes in the valence band (semiconductor type p)
A – SRH (Shockley–Read–Hall) coefficient characterizes the defect rate
C – Auger coefficient, a measure of multiparticle recombination (energy „heats“ another electron)
μ - charge carrier mobility coefficient (depends on the type of material (m²V⁻¹s⁻¹)
k_B – Boltzmann’s constant gives the relationship between energy and temperature (kinetic energy of molecules in solids, liquids, gases, see subsection above)
T – body (surface) temperature
E_d - electric field in the transition region (internal potential barrier of the depletion region) (Vm⁻¹)
R(λ) – reflection coefficient (reflectance) at a given λ (special case of albedo, perpendicular incidence, one wavelength)
d – active layer thickness or effective distance to the electrode
l – mean distance (diffusion distance that the carrier travels)

For our case of the photovoltaic effect, it is therefore possible to write (this is a simplification, some processes are more complex)

P_ph ​= R_sep / ∑_jR​​​_j (18)

P_ph – probability that a photovoltaic effect will occur in a PV cell
R_sep – rate of separation of electrons into the conduction band

The transition from probability to efficiency is an expression of the ratio between the energy of carriers that have entered the conduction band and the energy of photons. The conceptual transition from probability in the microworld of quantum phenomena to the macroworld of released energy and efficiency is straightforward. We calculate the energy of the incident light and multiply it by the probability of transition into the conductor.

P_el​ = Φ_in​E_ph​P_abs​P_exc​P_sep​(1−P_rec​) (19) compare with (12)

P_el – electrical power (not probability) of a PV cell (W)
Φ_in​ – photon flux (m⁻²·s⁻¹)
E_ph​ – photon energy see (9)

The flux of incident photons Φ_in​ we express as

Φ_in​ = N_ph / At​​ (20)

N_ph – quantity (number) of incident photons
A – area (m²)
t – time

Now let’s look at the probability of recombination P_rec

P_rec ​= P_rad ​+ P_nonrad​ (21)

From here the macroscopic efficiency follows as a proportion of the electrical power P_el and power input in the form of incident light P_in = Φ_inE_ph

η = P_el​ / P_in = P_abs​P_exc​P_sep​(1−P_rec​) (22)

The efficiency is given by the product of the probability of the events that contribute to the production of free electrons multiplied by the probability that the electrons do not recombine ​(1−P_rec​).

As can be seen from Tab2, the calculation of efficiency is not trivial. Each probability is calculated differently. All these other (from our point of view undesirable) phenomena can be included in a coefficient called the fill factor (FF), which can be determined experimentally. Experimental relationship for calculating efficiency

η = ​P_max / P​​_in = ​U_ocJ_sc​FF​ / P_in (23)

P_max – maximum intensity of electrical power – max(P_el) (Wm^-2)
P_in – power light intensity (Wm^-2)
U_oc – open circuit voltage (V)
J_sc – short-circuit current density (Am^-2)
FF – fill factor, see below (dimensionless)

The PV cell has the highest efficiency precisely for the energy E_g (bandgap), i.e. for a specific wavelength. Below this threshold the spectrum is unused and above it undesirable phenomena due to the higher energy of hot excitons (Auger effect, phonon recombination...), see Tab. 2. [3]

These variously probable phenomena in a photovoltaic cell determine the maximum possible theoretical maximum efficiency – the Shockley–Queisser limit – the theoretical (calculated) maximum efficiency of a monofacial and single-junction ideal photovoltaic cell under the following conditions:

• AM1.5G (Air Mass 1.5 Global) - AM1.5G for solar radiation is defined as the radiation from the Sun that strikes a surface at an angle ≈ 48° above the horizon (the ray path passes through such an angle that its path is 1.5× longer than the height of the Earth’s atmosphere), the spectrum is normalized to the radiation intensity 1kW/m²
• the photon flux is diffuse (Lambertian photon flux)
• in the semiconductor there are radiative (ideal) recombination conditions
• equilibrium has been reached between the absorption and emission of photons

The Shockley–Queisser efficiency limit calculation converts the complex calculation of the macroscopic determination of the efficiency of a PV cell into a clear form, where we find the theoretical Eg of the photovoltaic cell at which the efficiency is maximum.

η_SQ ​= ​J_sc​U_oc​FF / P_in (24) compare with (23)

η_SQ – efficiency of an ideal PV cell according to the Shockley–Queisser limit
J_sc – short-circuit current density (Am^-2)
FF – fill factor (0.89 to 0.91 for an ideal cell) is essentially a measure of total losses, i.e. a coefficient that affects overall efficiency.

Ideal fill factor

FF_ideal ​≈ u_oc​−ln(u_oc​ + k_fitt)​ / (u_oc ​+ 1) (25)

Normalized open circuit voltage u_oc

u_oc​ = qU_oc​​ / n_idealk_B​T (24)

U_oc​​ – open circuit voltage (voltage in the depletion zone)
k_fitt – experimental approximation (fitting) constant (0.72) (dimensionless)
q – elementary charge of a proton (same as the charge of an electron, but positive) (C)
n_ideal – ideality factor (≈1 for an ideal radiative diode)
k_B – Boltzmann constant
T – temperature

The elementary charge has the value q = 1.602176634×10⁻¹⁹ C, it is also used to express energy at the microscopic level. One electronvolt ( eV) is the energy gained by a particle with an elementary charge q (i.e. an electron or proton) when accelerated by a voltage of 1 volt 1eV = 1q⋅1V. The definition is inconsistent in the designation of units, because the unit is CV – coulomb volt (the coulomb unit is implicitly hidden in q), i.e. joule, but this is a stable form of notation. The basic unit joule is too large for quantum phenomena.

Theoretically, the maximum efficiency η_SQ ≈ 33% at E_g ≈ 1.34 eV (depending on the spectrum it varies slightly from≈ 32 to 34%). The reason why one discrete value of η_SQ was obtained for discrete E_g lies in the opposite course of the dependence of J_sc and U_oc on the change in E_g. For silicon (Si single) cells according to the Shockley-Queisser limit, the theoretical efficiency η ≈ 32 %, because Eg​ ≈ 1,12 eV. [2] [3]

We could stop here and announce that this theoretical value of the efficiency of a PV cell cannot be exceeded. But this is not true, and this is where a photovoltaic cell differs from a steam engine. Solid state physics hides other theoretical and practical possibilities.

Singlet fission (SF, sometimes also Singlet exciton fission) is a quantum phenomenon that can increase the efficiency of photovoltaic cells beyond the Shockley–Queisser limit by causing a single high-energy photon to create two quantum entangled excitons, and thus two electrons with opposite spins, instead of one. The high-energy singlet exciton decays into 2 triplet quantum entangled excitons. Further fission (Cascade / Higher-order, Quintet mixing) may occur, but by default 2 excitons are produced.

An exciton is a state of an electron in the conduction band and a hole in the valence band bound by the Coulomb force [23]. It is a state that precedes dissociation into free carriers, an electron and a hole. An exciton is an intermediate state – if it separates (drift/diffusion), it contributes to the current; otherwise it recombines (heat/light).

• Singlet exciton S₁ has total spin S_singlet = 0 (magnetic quantum number m_s = 0) (electron and hole have opposite spin – only one state and are quantum entangled), lifetime is ≈ ns
• Triplet exciton T₁ has total spin S_triplet = 1 (m_s = -1, 0 ,+1) (three possible spin states, hence triplet)
• The total spin of both triplet excitons formed is S_double = 0, they are again quantum entangled and have a longer lifetime (≈ μs to ms) than the singlet exciton

This seemingly simple list of spin values ​​contains the knowledge and theory of quantum physics that is needed to understand the symbolism of this notation. The key is the Dirac equation, which introduces spin as a state of fermions; see references [4] [5] [6]. The advantage of SF is that it is enough to deposit a thin layer of≈ 20 až 100 nm on the semiconductor.

Before we continue with the further enumeration of efficiency-enhancing mechanisms, let us stop at one type of nanomaterial that was predicted by Richard Feynmann in his lecture „There’s Plenty of Room at the Bottom“. Quantum Dot (QD) are nanoparticles whose size is comparable to the de Broglie wavelength of an electron (or exciton) [19]; ​​these dimensions (≈ 1 to 10 nm) cause quantum phenomena to manifest that are otherwise suppressed in the macroscopic distribution of matter. Depending on the size of the particles, it is possible to change E_g and create hot excitons that interact in the quantum dot by the inverse Auger mechanism and the Coulombic interaction [23] of the hot exciton with the surrounding molecules to form another exciton pair. Each exciton (more excitons can be created, but by default +1, i.e. 2) then has approximately half the energy of the original exciton ( E_ph ≈ nE_g, where n = 2, 3...). For quantum dots, the concept of EQE (External Quantum Efficiency) has been introduced, the internal efficiency of the QD; for example, if the EQE is 200% (η_QD = 2), this means that on average 2× as many excitons were created as there were originally hot exciton singlets. The actual efficiency of an EQD (not a PV cell) can be up to 190%, see Tab3. [17]

Multiple Exciton Generation (MEG) is another quantum phenomenon that allows the creation of multiple electron-hole pairs (excitons) from a single photon in semiconductor nanomaterials. The inverse Auger process in MEG transfers the excess energy of a hot exciton to a neighboring valence electron, thereby creating another exciton instead of converting it into heat. This process can be reduced in quantum dots. In the context of semiconductor PV cells based on QD (e.g. PbS, CdSe or InP), MEG is another mechanism for exceeding the theoretical Shockley-Queisser limit for single-layer cells. The theoretical limit for MEG isη ≈ 66% [7] [8]. Physically, a thin layer (≈ 5 to 50 nm) is deposited on an n-type semiconductor.

Spectral compatibility allows combining SF with MEG on a single cell. SF efficiently processes photons with energy E_ph ≈ 1 to 1.5 E_g (visible/IR), while MEG works for E_ph >= 2 E_g (UV).

Up conversion (UC) is a method of combining the energy of infrared photons below the E_g (bandgap) to create a photon with an energy higher than the E_g threshold. Up conversion can take place in the layer on the back of the PV cell either by sequentially exciting an electron with several IR photons through metastable levels (a metastable level is an excited energy state of an electron that has a long lifetime, typically ≈ 10⁻⁴ to 10⁻² s) and releasing an energetic photon during deexcitation or by combining two exciton triplets into a singlet (Triplet-Triplet annihilation, TTA). The recombination then results in the emission of a photon. Macroscopically, luminescence occurs. The advantage is that UC takes place in a separate layer (≈ 10 to 100 nm), and QDs are also used, which have high conversion efficiency. [9] [10]

Down conversion (DC) occurs in the opposite way to UC, i.e. one high-energy photon (typically UV with E_ph >= 2E_g) is converted into one or two photons with lower energy through excitation and deexcitation of an electron in a molecule (conversion to multiple photons is unlikely, but possible). An electron excited by the absorption of a photon during deexcitation transfers energy to another electron by energy transfer between ions in the lattice (cross-relaxation). By choosing suitable materials, the resulting red/IR photons have E_ph ≈ E_g. The luminescence of these less energetic photons falls on the photosensitive layer of the semiconductor with approximately half the photon energy, but with twice the intensity. The advantage is that DC again occurs in a separate layer (≈ 10 to 100 nm), and QDs are also used, which have high conversion efficiency. [11]

Another method to increase the efficiency of cells above the Shockley–Queisser limit is the Hot-Carrier Solar Cells (HCSC) technology, which is still only in theoretical and experimental concept. If it is possible to transfer a high-energy (hot) electron to the conduction band before it transfers its energy in the form of phonons to the lattice in the valence band, this will prevent losses in the form of heat and the electron energy will be used as electrical energy. To do this, it is necessary to extend the time for which the hot electron can retain its energy (gain time for the electron to drift/diffuse) – phonon barrier and at the same time find a material (selective interface directly in the crystalline lattice of the semiconductor structure) that only allows hot electrons to pass through, otherwise the hot electrons, after mixing with the cold ones in the conduction band, will be freed from the potential difference again by unwanted conversion to heat. The difficulty of the solution is that everything takes place in one structure, which can only be doped in some suitable way; however, hybrid structures with QDs are also being tested. The theoretical efficiency of such a PV cell was again calculated to be η ≈ 66 %. [12]

All of the above methods can be suitably layered (according to E_g). Of course, the energy that was used in one layer cannot be used in the next. The theoretical efficiency of the next layer must then be calculated from the total energy, from which the energy used in the previous layers is subtracted. The result is a theoretically (and practically) highly efficient PV cell Tab3.

If we neglect all other quantum phenomena in the semiconductor except for blackbody radiation, then according to the Carnot efficiency of the heat engine η_Carnot = 1− T_cold / T_hot the theoretical maximum efficiency of the PV cell after substituting T_hot ≈ 6000 to 6500 K for the Sun, T_cold ≈ 300 K for the earth’s surfaceη_Carnot ≈ 99%. This calculation shows the upper limit of how far we could go. The photon temperature is a greeting from our hot solar parent, which was not lost along the way.[21]

Table of methods to increase the efficiency of solar cells (Tab3)

Method	Description	Theoretical profit	Current record (2025)	Examples
Tandem (multi-junction) cells	Layers with different Eg: top absorbs blue/UV, bottom red/IR	>45% (2 layers), >60% (3+ layers)	39.5% (perovskite/Si tandem, LONGi)	Perovskite/Si, GaAs/Ge (for space)
Concentration PV (CPV)	Concentration of sunlight (lenses / mirrors) on the cell	+5 to +15% above SQ (1+ layers)	36.1% (Fraunhofer-ISE) 47.6% (concentration multilayer, Varian)	Fresnel lenses + GaAs cells
Singlet fission (SF)	Organic nano layer: 1 photon (visible) → 2 triplet excitons	>42%	5,1 % (Cavendish Cambridge)	Pentacene, Tetracene, Hexacene layers on Si
Multiple exciton generation (MEG) in QD	Inorganic quantum dots: 1 photon (blue, UV) → 2+ pairs of excitons	>60 %	18.1% (UNIST South Korea), (EQE QD up to 120%)	PbS QD or CsPbI₃ QD layers
Up/down conversion	UC 2 IR photons → 1 visible / DC 1 UV photon → 2 visible	> 60%	+3,8% Up (Rubrene TTA, Oxford) / +10% Down (National School Morocco)	Rubrene TTA Tb³⁺, Yb³⁺ Up / Pr³⁺, Yb³⁺ Down
Hot carrier cells (HCSC)	Preservation of excess energy (hot carriers) from thermalization	>60 %	phonon barrier UNSW (Australia); selective anode - UNSW (Australia)	CsPbI₃, MoS₂, WSe₂, InN, GaN barrier; HfN, GaAs, AlAs separation
Radiative optimization limit	Reduction of non-radiative recombination	Approaching SQ	29,4 % (Si, Kaneka)	Surface passivation Al₂O₃

As can be seen from Tab3, there is still a lot of room for improvement and further development. „There’s Plenty of Room at the Bottom“.

On March 18, 1905, Albert Einstein significantly contributed with his article to the theoretical basis not only of quantum physics, but also the basis for research and development of the currently most available energy source, thereby clearly upsetting a significant portion of the current population, which mockingly calls renewable but unstable sources sources episodic.

* I usually list quantities and their dimensions only once, and I do not list them where the dimension is obvious (d – length, etc.).

** I usually do not explain the derivation of relationships in detail, for a deeper understanding, please research.

Original article in Czech language is there Jak Albert Einstein přispěl k vývoji nejdostupnějšího občasného zdroje [26]

Thanks Google Translator, AI Grok AI Chat GPT AI Canva for help in data search and image generation. Special thanks to AI Grok and ChatGPT for their intellectual abilities in solving questions and for their mistakes. Without intensive research with AI, this article would not have been created; and at the same time, one important reminder. Always measure twice. Verify everything with the source, do not blindly rely on AI. It is creative like a human. If it does not know, it will invent something.

[1] Entropie

[2] Tabulated values ​​and theoretical calculations for the Shockley–Queisser limit

[3] Wikipedia Shockley-Quessier limit

[4] Wikipedia Dirac equation

[5] Wikipedia spin

[6] Wikipedia singlet

[7] Multiple exciton generation in nano-crystals

[8] Theory of highly efficient multiexciton generation in type-II nanorods

[9] Upconversion as a spear carrier for tuning photovoltaic efficiency

[10] Enhancing Solar Cell Efficiency Using Photon Upconversion Materials

[11] Comprehensive Review on Downconversion/Downshifting Silicate-Based Phosphors for Solar Cell Applications

[12] Pathways to hot carrier solar cells

[13] Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt

[14] Zur Elektrodynamik bewegter Körper

[15] Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?

[16] Zur Theorie der Brownschen Bewegung

[17] Electron transport in quantum dots

[18] Wikipedia exponenciální rozpad

[19] De Broglieova vlna

[20] Electronic band structure

[21] Absolutně černé těleso

[22] Equipartition principle

[23] Coulomb Interaction

[24] Faradayovy zákony elektrolýzy

[25] Beer-Lambert law

[26] Jak Albert Einstein přispěl k vývoji nejdostupnějšího občasného zdroje