Statistics of Particles

In the microscopic treatment of matter, the particles, for example, molecules and elec­trons, are distributed in a system of energy levels. How the particles distribute among the energy levels determines the behavior of the system in a significant way. Regarding to applications with solar energy, an understanding of Maxwell-Boltzmann statistics and Fermi-Dirac statistics is essential. Maxwell-Boltzmann statistics is valid for a system of distinguishable particles in a system of energy levels allowing unlimited occu­pancy; whereas Fermi-Dirac statistics is valid for a system of indistinguishable particles in a system of energy levels allowing limited occupancy, that is, systems satisfying the Pauli exclusion principle.

The starting point of the derivation is the Boltzmann expression of entropy,

S = kB ln W, (D.1)

where kB is Boltzmann’s constant and W is the total number of configurations of the system.

Consider a situation of N particles in a system consisting of a series of energy levels. The occupancy number of the *th level is N^ The energy of the *th level is Ei, and the total energy of the system is E. We have the conditions

E = £* Ni Ei.

The condition of equilibrium is that the entropy, Eq. D.1, reaches maximum under the two constraints in Eq. D.2.

Intuitively, the more uniform the distribution, the greater the randomness, or the greater the entropy. However, the condition of constant total energy adds another condition: It is preferable to have more particles in the levels of lower energy and less particles in the levels of higher energy. The problem can be resolved using Fermat’s theorem together with the Lagrange multiplier method. Introducing two Lagrange multipliers a and в, the condition of equilibrium is

The distribution can be found by combining the solution of Eq. D.3 with the two constraints in Eqs. D.1 and D.2.

During the calculation, one needs an approximate value of the derivative of ln N! for large N. This can be simply taken as

dN in N! « ln N! – ln(N – 1)! « ln N. (D.4)

D. 1 Maxwell—Boltzmann Statistics

In the case of Maxwell-Boltzmann statistics, applicable to classical atomic systems, the particles are distinguishable, and the number of particles per energy level is unlimited. The number of possible configurations W can be determined as follows.

Suppose that N particles are placed into containers with occupation numbers N1, N2, … Ni, … and so on. Consider now the number of ways to place N1 particles into con­tainer 1. First, choose one of the N particles for the first position in container 1, that is, N ways. Next, choose one of the remaining N — 1 particles for the second position in container 1. This generates N — 1 ways. Altogether there are N(N — 1) ways. By continuing the process, the total number of ways is N(N — 1)(N — 2)…(N — N1 — 1). However, the placement of particles in the container is arbitrary. Therefore, there is a N1 !-fold redundancy. The net ways of placement are given as

W N (N — 1)(N — 2)…(N — ,¥, — 1) N! (D5)

Wl =————————- N!———————- = (N — N1)! N,!. (D’5)

Similarly, the number of ways of placing the remaining N — N1 particles into container 2 generates a factor

(N — N1)! (N — N1 — N2)! N2!.

(D.6)

W2

Similarly,

(N — N1 — N2)! (N — N1 — N2 — N3)! N3!.
W3	(D.7)

Continuing the process further, finally we find

N!

W = W1 W2 …Wi… = —

i Ni!

The entropy is

S = kB ln W = kB ln N! —J2 ln Ni! The condition of thermal equilibrium gives

d dN

S + Jn — ^ N + /ЗІЕ — J2 NiEi

(D.10)

The result is

kB ln Ni = —a — ЗЕі, (D.11)

The meaning of the parameter З can be interpreted based on thermodynamics. Be­cause the system is under the condition of constant temperature and constant volume, according to Eq. 6.28, we have

dE = TdS. (D.12)

By treating S and E as variables, comparing Eq. D.3 and Eq. D.12, we find, heuristi – cally,

З = T. (D.13)

The constant a can be determined by the condition that the total number of particles is N. Equation D.11 can be rewritten as

N (-ЕЛ ni = z ex4 k^) where Z is a constant determined by the condition

(D.14) = N, (D.15) (D.16) Ni/N, Eq. D.16 can be written

in other words,

, —Ei

pi = Z ехП kBBTT

Obviously, the sum of all probabilities is 1,

^ Pi =1

D. 2 Fermi—Dirac Statistics

Electrons are fermions obeying the Pauli exclusion principle. Each state can only be occupied by one electron. The electrons satisfies Fermi-Dirac statistics.

For each energy value, there can be multiple states. For example, each electron can have two spin states that have the same energy level. Let the degeneracy, that is, the number of states at energy Ei, be gi. The number of electrons staying at that energy level, Ni, should not exceed gi. The number of different ways of occupation in the gi states is

gi ■

(gi – Ni)! Ni! ’

which is a special case of Eq. D.5. Following Eq. D.3, we obtain

кв [ln(gi – Ni) – lnNi] = a + f3Ei,

or

gi a + /3Ei кв

(D.21)

Ni

exp

+ 1

Using Eq. D.13 and introducing the probability pi = Ni/gi,

1

(D.22)

pi

Ei — Ep квТ

exp

+ 1

Equation D.22 is called the Fermi function. At the Fermi level Ep, the probability is 1/2. Apparently,

pi ^ t Ei ^ ep 5

pi ^ ° Ei ^ ep ■

At low temperature, where квТ ^ Ep, the Fermi-Dirac statistics becomes

At high temperatures, or at high energy levels (Ei — Ep)/квТ ^ 1, the Fermi-Dirac statistics reduces to Maxwell-Boltzmann statistics,

Ei

квТ

AM1.5 Reference Solar Spectrum

To facilitate testing of solar photovoltaic devices, American Society for Testing and Materials (ASTM) in conjenction with the photovoltaic (PV) industry and government research and development laboratories developed and defined two standard terrestrial solar spectral irradiance distributions: The standard direct normal spectral irradi – ance and the standard global spectral irradiance, incorporated into a single document, ASTM G-173-03; see Section 5.2.1.

The original data was presented in wavelength scale. The following table is pre­sented in photon energy scale. The first column is photon energy in eV. The second column is the extraterrestrial solar radiation spectrum, AM0. The third column is the direct normal solar radiation spectrum. The fourth column is the global solar radiation spectrum, including radiation scattered from the sky. All spectral data are presented in watts per square meter per eV (W/m2-eV).

Table E.1: AM1.5 Reference Solar Spectrum (W/m2 eV) c (eV) AM0 Direct Global c (eV) AM0 Direct Global 0.32 116 94 94 0.47 232 0 0 0.33 128 105 105 0.48 240 0 0 0.34 135 104 104 0.49 251 11 11 0.35 141 110 109 0.50 265 104 105 0.36 146 75 74 0.51 274 163 164 0.37 156 37 37 0.52 283 199 201 0.38 163 27 26 0.53 292 247 250 0.39 168 52 52 0.54 301 268 271 0.40 178 25 25 0.55 313 294 299 0.41 184 32 32 0.56 322 294 299 0.42 192 16 16 0.57 328 310 315 0.43 200 0 0 0.58 338 321 326 0.44 208 0 0 0.59 345 286 290 0.45 216 0 0 0.60 358 252 256 0.46 223 0 0 0.61 370 154 156

e (eV)	AM0	Direct	Global	e (eV)	AM0	Direct	Global
0.62	380	214	218	1.02	577	487	510
0.63	385	46	47	1.03	579	485	508
0.64	399	2	2	1.04	577	461	483
0.65	409	0	0	1.05	582	467	489
0.66	406	0	0	1.06	577	398	418
0.67	427	0	0	1.07	584	245	257
0.68	433	31	31	1.08	579	204	213
0.69	447	214	219	1.09	583	141	148
0.70	456	367	376	1.10	578	129	135
0.71	460	396	406	1.11	581	298	314
0.72	477	448	460	1.12	578	445	468
0.73	482	459	471	1.13	571	506	534
0.74	494	470	483	1.14	576	523	553
0.75	497	472	485	1.15	578	532	563
0.76	507	489	503	1.16	584	539	571
0.77	513	479	493	1.17	586	546	578
0.78	516	474	487	1.18	586	550	582
0.79	523	507	522	1.19	589	553	587
0.80	528	505	520	1.20	591	555	589
0.81	529	472	486	1.21	589	552	586
0.82	530	392	404	1.22	597	558	593
0.83	533	212	218	1.23	592	553	587
0.84	535	147	151	1.24	599	559	595
0.85	536	116	119	1.25	600	546	581
0.86	538	52	53	1.26	599	484	514
0.87	541	20	21	1.27	596	454	482
0.88	544	1	1	1.28	595	360	382
0.89	547	0	0	1.29	593	268	284
0.90	547	0	0	1.30	601	233	247
0.91	549	4	4	1.31	593	229	243
0.92	551	204	211	1.32	607	149	158
0.93	553	327	340	1.33	603	351	374
0.94	558	429	447	1.34	596	471	503
0.95	564	494	516	1.35	602	408	435
0.96	562	528	552	1.36	599	422	450
0.97	575	488	510	1.37	596	430	460
0.98	573	516	540	1.38	598	471	504
0.99	574	547	573	1.39	596	545	585
1.00	576	545	571	1.40	591	544	584
1.01	579	522	547	1.41	590	544	584

e (eV)	AM0	Direct	Global	e (eV)	AM0	Direct	Global
1.42	591	544	586	1.82	556	471	520
1.43	579	532	572	1.83	554	467	517
1.44	596	544	586	1.84	553	464	513
1.45	554	504	543	1.85	555	464	512
1.46	589	534	577	1.86	556	456	504
1.47	593	530	573	1.87	552	443	489
1.48	589	507	548	1.88	519	422	467
1.49	590	471	508	1.89	519	420	465
1.50	586	453	488	1.90	538	432	477
1.51	587	439	474	1.91	536	425	471
1.52	594	481	521	1.92	542	439	486
1.53	587	521	566	1.93	533	430	474
1.54	585	514	558	1.94	539	430	475
1.55	584	514	557	1.95	535	422	467
1.56	581	509	555	1.96	526	403	446
1.57	592	525	572	1.97	537	403	446
1.58	586	526	572	1.98	519	401	442
1.59	587	526	572	1.99	528	406	450
1.60	583	519	567	2.00	521	404	448
1.61	576	452	493	2.01	511	396	439
1.62	587	212	229	2.02	519	400	443
1.63	584	414	452	2.03	518	397	441
1.64	583	517	564	2.04	522	396	440
1.65	578	511	560	2.05	513	386	430
1.66	577	508	557	2.06	510	383	427
1.67	560	488	535	2.07	513	379	422
1.68	575	485	532	2.08	511	373	413
1.69	570	447	490	2.09	507	369	411
1.70	565	414	454	2.10	481	347	386
1.71	570	427	469	2.11	504	376	420
1.72	559	384	421	2.12	509	380	424
1.73	562	477	526	2.13	499	369	411
1.74	566	485	535	2.14	492	356	398
1.75	567	479	527	2.15	487	351	393
1.76	561	462	509	2.16	496	358	400
1.77	561	467	514	2.17	473	342	383
1.78	564	444	491	2.18	484	354	396
1.79	565	420	462	2.19	466	342	383
1.80	555	396	434	2.20	468	343	385
1.81	556	471	519	2.21	458	336	377

e (eV)	AM0	Direct	Global	e (eV)	AM0	Direct	Global
2.22	458	337	379	2.62	365	241	284
2.23	466	342	385	2.63	349	229	271
2.24	460	337	381	2.64	356	233	277
2.25	455	333	376	2.65	339	221	263
2.26	448	326	368	2.66	349	228	271
2.27	451	328	371	2.67	355	230	272
2.28	443	322	365	2.68	355	229	273
2.29	415	302	342	2.69	341	218	261
2.30	434	316	358	2.70	342	218	261
2.31	447	323	366	2.71	347	220	265
2.32	415	299	339	2.72	333	212	253
2.33	443	318	361	2.73	324	204	245
2.34	433	310	354	2.74	344	215	260
2.35	392	282	320	2.75	338	210	254
2.36	429	309	351	2.76	325	201	244
2.37	409	295	335	2.77	320	198	240
2.38	394	283	323	2.78	298	182	221
2.39	356	254	291	2.79	304	184	226
2.40	403	288	330	2.80	302	182	225
2.41	392	279	320	2.81	284	171	209
2.42	414	294	337	2.82	248	148	183
2.43	406	286	328	2.83	274	163	201
2.44	396	277	319	2.84	286	170	210
2.45	410	286	328	2.85	248	146	181
2.46	386	268	309	2.86	255	149	185
2.47	377	263	304	2.87	219	127	158
2.48	385	269	311	2.88	180	104	130
2.49	394	274	316	2.89	234	135	168
2.50	395	274	318	2.90	241	137	172
2.51	383	266	308	2.91	252	143	179
2.52	371	255	297	2.92	247	139	175
2.53	372	256	296	2.93	251	141	178
2.54	357	244	283	2.94	258	144	182
2.55	340	231	269	2.95	237	131	166
2.56	378	256	298	2.96	238	130	166
2.57	385	258	302	2.97	248	135	172
2.58	381	254	297	2.98	249	135	173
2.59	378	250	293	2.99	236	127	163
2.60	367	243	285	3.00	244	130	168
2.61	371	246	289	3.01	234	123	159

e (eV)	AM0	Direct	Global	e (eV)	AM0	Direct	Global
3.02	208	109	142	3.42	106	39	59
3.03	236	123	160	3.43	101	36	55
3.04	217	112	146	3.44	110	39	60
3.05	221	114	148	3.45	82	29	45
3.06	229	117	153	3.46	84	29	46
3.07	227	115	151	3.47	92	32	50
3.08	235	119	157	3.48	108	37	58
3.09	226	113	150	3.49	115	39	62
3.10	213	106	140	3.50	111	37	59
3.11	161	79	106	3.51	95	32	50
3.12	94	46	61	3.52	99	32	52
3.13	156	75	101	3.53	105	34	55
3.14	114	54	73	3.54	95	30	49
3.15	98	46	63	3.55	90	28	46
3.16	160	75	102	3.56	90	28	46
3.17	157	73	100	3.57	93	29	47
3.18	139	64	88	3.58	92	28	46
3.19	121	56	77	3.59	82	25	41
3.20	124	56	78	3.60	87	26	43
3.21	121	54	75	3.61	98	29	48
3.22	114	51	71	3.62	95	28	47
3.23	85	38	53	3.63	90	26	44
3.24	104	46	64	3.64	98	27	46
3.25	140	61	85	3.65	90	25	43
3.26	130	56	79	3.66	86	23	41
3.27	150	64	91	3.67	77	20	35
3.28	149	63	90	3.68	75	20	34
3.29	126	53	76	3.69	84	22	39
3.30	117	49	70	3.70	90	23	41
3.31	105	43	62	3.71	85	21	36
3.32	119	48	70	3.72	86	21	38
3.33	131	53	77	3.73	88	21	38
3.34	130	52	76	3.74	87	20	36
3.35	141	56	82	3.75	94	22	39
3.36	127	50	73	3.76	91	21	38
3.37	135	52	77	3.77	81	18	32
3.38	139	53	79	3.78	84	17	31
3.39	123	47	70	3.79	86	18	34
3.40	113	43	64	3.80	81	16	30
3.41	112	42	63	3.81	70	13	23

e (eV)	AM0	Direct	Global	e (eV)	AM0	Direct	Global
3.82	64	12	23	4.12	31	0	0
3.83	56	10	19	4.13	33	0	0
3.84	59	10	18	4.14	35	0	0
3.85	60	10	19	4.15	33	0	0
3.86	65	11	20	4.16	37	0	0
3.87	64	9	16	4.17	34	0	0
3.88	59	8	16	4.18	38	0	0
3.89	57	7	14	4.19	39	0	0
3.90	65	8	14	4.20	36	0	0
3.91	56	7	13	4.21	36	0	0
3.92	47	5	9	4.22	38	0	0
3.93	52	5	10	4.23	36	0	0
3.94	55	5	9	4.24	38	0	0
3.95	57	4	8	4.25	40	0	0
3.96	54	4	8	4.26	41	0	0
3.97	55	3	7	4.27	38	0	0
3.98	58	3	6	4.28	32	0	0
3.99	50	2	5	4.29	24	0	0
4.00	39	1	3	4.30	20	0	0
4.01	46	1	3	4.31	22	0	0
4.02	49	1	2	4.32	23	0	0
4.03	47	1	2	4.33	15	0	0
4.04	43	0	1	4.34	10	0	0
4.05	44	0	1	4.35	13	0	0
4.06	48	0	1	4.36	19	0	0
4.07	46	0	0	4.37	20	0	0
4.08	47	0	0	4.38	20	0	0
4.09	41	0	0	4.39	17	0	0
4.10	33	0	0	4.40	13	0	0
4.11	34	0	0	4.41	9	0	0

xThe mathematics of Hubbert’s theory is similar to the equations created by Pierre Francois Ver – hurst in 1838 to quantify Malthus’s theory on population growth [85].

[2]By definition, sechx = 1/cosh x = 2/(ex + e x).

[3]In the literature, the notation FF is often used to represent the fill factor, a factor of efficiency in the treatment of solar cells by Shockley and Queisser. To maintain consistency of notations, we use nf instead.

[4] , „ x. xc + ln C

nd — — In (C eXc) ———————– — 1,

xc xc

because the expression C varies much less than the exponential. Therefore, at very low cell temperatures, if there is no other recombination mechanism other than radiative recombination, the open-circuit voltage approaches the band gap energy in volts and the nominal efficiency approaches the ultimate efficiency.

The reverse saturation current in Eq. 9.38 is the absolute limit of the observed reverse saturation current (Eq. 9.29). In addition to radiative recombination, there are other types of recombination processes that increase the reverse saturation current and then reduce the efficiency of solar cells; see Section 9.3.