I've always assumed that kanji characters are almost always morphemes, but
I've hardly seen anyone explicitly state that. I think the idea that kanji
usually represent ideas, not pronunciation, played a role in my assumption.

Are most kanji no bigger than, and no smaller than, a single morpheme?

Also, are most words containing multiple kanji characters based on a
combination of the meanings of individual characters, like "volcano" being 火山,
a combination of the kanji for "fire" and for "mountain"?

Related questions looking for exceptions to "one kanji = one morpheme": [Does
Japanese have morphemes that span two
kanji?](https://japanese.stackexchange.com/questions/8239/does-japanese-have-
morphemes-that-span-two-kanji) and [Kanji for native Japanese concepts:
Kun'yomi spanning multiple
morphemes](https://japanese.stackexchange.com/questions/6581/kanji-for-native-
japanese-concepts-kunyomi-spanning-multiple-morphemes)
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kanji etymology
Are most kanji characters morphemes?
No. 「火山」is one morpheme.

If you believe not, please find an article written by a linguist of Japanese.
I've already introduced a few articles that support my idea.

All Kanji, as far as I know, are one syllable in old Chinese.
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Short answer: Yes, most of the time each kanji represents a single morpheme.
There are exceptions.

Long answer:

The basic principle of kanji _in the original Chinese system_ is that 1
character = 1 morpheme. This plays along well the property of Chinese
languages where, in most cases, 1 syllable = 1 morpheme (source: Packard, _The
Morphology of Chinese_ ). So in the typical Chinese case, 1 character = 1
morpheme = 1 syllable. There are exceptions even in Chinese (some of them
disputed, some undeniable); but that’s the basic principle. This is accepted
by all major linguistic works on writing systems including Rogers, Sproat,
DeFrancis etc.; the technical term is **morphography**.

In Japanese, some of the _on'yomi_ became two syllables, so the last part of
the equation (1 kanji = 1 syllable) breaks even for _on'yomi_. And then there
are _kun'yomi_ which are, historically speaking, translations. Of course
_kun'yomi_ throw syllabic regularity out of the window (政 = _matsurigoto_ ).
But more than that; since morpheme count doesn’t always translate 1:1, the 1
character = 1 morpheme rule got a lot more exceptions, too:

  * In some cases, a Chinese single morpheme, like _léi_ "thunder", translates to two or more in Japanese, like _kami-nari_ gods-voice = thunder. As a result, the character for _léi_ , 雷, represents two morphemes in _kun-yomi_. Other examples include 唇 _kuchi-biru_ , 卵 _tama-go_ , 政 _matsuri-goto_ above, etc. This is the morphographical equivalent of the letter 'x' representing two sounds, /ks/, in English. These exceptional cases aren't morphographical but logographical (they represent whole words).
  * In some cases, a polymorphemic Chinese word, like _dà-rén_ big-person = "adult", translates to a single Japanese morpheme, like _otona_ "adult". In these cases you need a string of multiple kanji to represent a single morpheme: 大人= _otona_. This is the morphographic equivalent of a digraph. 
    * Note that the original Chinese morpheme breakdown is usually indicative of the meaning (if the compound is transparent, which is normally the case). This means that, by historical coincidence and _in these cases only_ , kanji can be said to be representing meaning, and not morphemes or sounds. An _otona_ is a "big person", in meaning though not in etymology. Once the practice was established, it was used productively, too; like writing the Portuguese word _tabaco_ (tobacco) as 煙草 smoke-herb. These are called "mature _kun_ readings" 熟字訓 _jukujikun_ in Japanese ("mature" in the sense of "conventional"). They’re a minor part of the system, but they’re part of the system.

Still, the _most common_ use of kanji is _on’yomi_ (e.g. according to my own
corpus research, around 83% of Jōyō readings used in the Japanese wikipedia
are _on_ and not _kun_ or _jukujikun_ ). And even for _kun’yomi_ , 1:1
morpheme equivalence is the typical case ( _ken_ → _inu_ , _fuu_ → _kaze_ ,
etc.) Which means that the _basic principle_ of Japanese kanji is still
morphography. It’s just messy morphography. See Terry Joyce, _The significance
of the morphographic principle for the classification of writing systems_ ,
for more discussion.

There’s another sense in which someone may argue that kanji don’t represent
morphemes but ideas. It’s because a single kanji can _potentially_ represent
different morphemes (typically one _on_ and one _kun_ reading, but up to 12
alternatives in 生); and these multiple possibilities are normally related by
meaning (because the _kun_ are conventionalized translations of _on_ ). So 犬
can be said to represent the "idea" of dog because both the morphemes _inu_
and _ken_ (as in _ai-ken_ , "beloved dog") can be represented by it, and these
two morphemes both mean "dog".

This fact is interesting; it distinguishes Japanese writing from simpler
morphography like Chinese, and it has to be acknowledged. But claiming that
it's not morphography is like saying that English letters don’t represent
sounds because the letter ‹a› has multiple possible pronunciations. There’s
still a clear underlying principle. In any specific _use_ of 犬, it will
represent one morpheme, either _inu_ or _ken_ , specifically in these
sound/meaning pairings (and cause the activation of the specific sound
patterns in the reader’s brain, priming them for further use etc.; this has
been shown in neurolinguistic research). What's more, the idea of "dog" as
represented by _doggu_ or _-ku_ (as in 走狗 _sōku_ "hunting hound") are not
represented by 犬; so even _in potentia_ , 犬 doesn’t actually represent the
idea of "dog" however it may be expressed, it represents the specific
morphemes _inu_ and _ken_.

(Compare 犬 to the algarism ‹1›, which represents the mathematical idea of
'oneness' however it's expressed linguistically: ‹1› "one", ‹10› "ten", ‹11›
"eleven", ‹100› "hundred", ‹1st› "fir-st", ‹ _100 años de soledad_ › _"
**Cién** años de soledad"_… The mechanics of these two kinds of symbol are
similar, but aren’t the same.)
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